id	sid	tid	token	lemma	pos
ejpam-5875	1	1	european	european	PROPN
ejpam-5875	1	2	journal	journal	PROPN
ejpam-5875	1	3	of	of	ADP
ejpam-5875	1	4	pure	pure	ADJ
ejpam-5875	1	5	and	and	CCONJ
ejpam-5875	1	6	applied	applied	ADJ
ejpam-5875	1	7	mathematics	mathematic	NOUN
ejpam-5875	1	8	2025	2025	NUM
ejpam-5875	1	9	,	,	PUNCT
ejpam-5875	1	10	vol	vol	NOUN
ejpam-5875	1	11	.	.	PROPN
ejpam-5875	1	12	18	18	NUM
ejpam-5875	1	13	,	,	PUNCT
ejpam-5875	1	14	issue	issue	NOUN
ejpam-5875	1	15	2	2	NUM
ejpam-5875	1	16	,	,	PUNCT
ejpam-5875	1	17	article	article	NOUN
ejpam-5875	1	18	number	number	NOUN
ejpam-5875	1	19	5875	5875	NUM
ejpam-5875	1	20	issn	issn	VERB
ejpam-5875	1	21	1307	1307	NUM
ejpam-5875	1	22	-	-	SYM
ejpam-5875	1	23	5543	5543	NUM
ejpam-5875	1	24	–	–	PUNCT
ejpam-5875	1	25	ejpam.com	ejpam.com	X
ejpam-5875	1	26	published	publish	VERB
ejpam-5875	1	27	by	by	ADP
ejpam-5875	1	28	new	new	PROPN
ejpam-5875	1	29	york	york	PROPN
ejpam-5875	1	30	business	business	NOUN
ejpam-5875	1	31	global	global	VERB
ejpam-5875	1	32	some	some	DET
ejpam-5875	1	33	characterizations	characterization	NOUN
ejpam-5875	1	34	of	of	ADP
ejpam-5875	1	35	quasi	quasi	NOUN
ejpam-5875	1	36	-	-	NOUN
ejpam-5875	1	37	curves	curve	NOUN
ejpam-5875	1	38	in	in	ADP
ejpam-5875	1	39	galilean	galilean	PROPN
ejpam-5875	1	40	3	3	NUM
ejpam-5875	1	41	-	-	PUNCT
ejpam-5875	1	42	space	space	NOUN
ejpam-5875	1	43	ayman	ayman	NOUN
ejpam-5875	1	44	elsharkawy1,∗	elsharkawy1,∗	PROPN
ejpam-5875	1	45	,	,	PUNCT
ejpam-5875	1	46	noha	noha	PROPN
ejpam-5875	1	47	elsharkawy1	elsharkawy1	PROPN
ejpam-5875	1	48	1	1	NUM
ejpam-5875	1	49	department	department	NOUN
ejpam-5875	1	50	of	of	ADP
ejpam-5875	1	51	mathematics	mathematic	NOUN
ejpam-5875	1	52	,	,	PUNCT
ejpam-5875	1	53	faculty	faculty	NOUN
ejpam-5875	1	54	of	of	ADP
ejpam-5875	1	55	science	science	NOUN
ejpam-5875	1	56	,	,	PUNCT
ejpam-5875	1	57	university	university	PROPN
ejpam-5875	1	58	of	of	ADP
ejpam-5875	1	59	tanta	tanta	PROPN
ejpam-5875	1	60	,	,	PUNCT
ejpam-5875	1	61	tanta	tanta	PROPN
ejpam-5875	1	62	,	,	PUNCT
ejpam-5875	1	63	egypt	egypt	PROPN
ejpam-5875	1	64	abstract	abstract	PROPN
ejpam-5875	1	65	.	.	PUNCT
ejpam-5875	2	1	this	this	DET
ejpam-5875	2	2	study	study	NOUN
ejpam-5875	2	3	investigates	investigate	VERB
ejpam-5875	2	4	the	the	DET
ejpam-5875	2	5	theoretical	theoretical	ADJ
ejpam-5875	2	6	basis	basis	NOUN
ejpam-5875	2	7	of	of	ADP
ejpam-5875	2	8	the	the	DET
ejpam-5875	2	9	quasi	quasi	NOUN
ejpam-5875	2	10	-	-	NOUN
ejpam-5875	2	11	frame	frame	NOUN
ejpam-5875	2	12	in	in	ADP
ejpam-5875	2	13	three	three	NUM
ejpam-5875	2	14	-	-	PUNCT
ejpam-5875	2	15	dimensional	dimensional	ADJ
ejpam-5875	2	16	galilean	galilean	PROPN
ejpam-5875	2	17	geometry	geometry	NOUN
ejpam-5875	2	18	.	.	PUNCT
ejpam-5875	3	1	we	we	PRON
ejpam-5875	3	2	derive	derive	VERB
ejpam-5875	3	3	mathematical	mathematical	ADJ
ejpam-5875	3	4	expressions	expression	NOUN
ejpam-5875	3	5	for	for	ADP
ejpam-5875	3	6	the	the	DET
ejpam-5875	3	7	position	position	NOUN
ejpam-5875	3	8	vectors	vector	NOUN
ejpam-5875	3	9	of	of	ADP
ejpam-5875	3	10	curves	curve	NOUN
ejpam-5875	3	11	defined	define	VERB
ejpam-5875	3	12	in	in	ADP
ejpam-5875	3	13	relation	relation	NOUN
ejpam-5875	3	14	to	to	ADP
ejpam-5875	3	15	this	this	DET
ejpam-5875	3	16	quasi	quasi	NOUN
ejpam-5875	3	17	-	-	NOUN
ejpam-5875	3	18	frame	frame	NOUN
ejpam-5875	3	19	and	and	CCONJ
ejpam-5875	3	20	establish	establish	VERB
ejpam-5875	3	21	the	the	DET
ejpam-5875	3	22	quasi	quasi	ADJ
ejpam-5875	3	23	equations	equation	NOUN
ejpam-5875	3	24	that	that	PRON
ejpam-5875	3	25	govern	govern	VERB
ejpam-5875	3	26	their	their	PRON
ejpam-5875	3	27	behavior	behavior	NOUN
ejpam-5875	3	28	.	.	PUNCT
ejpam-5875	4	1	our	our	PRON
ejpam-5875	4	2	findings	finding	NOUN
ejpam-5875	4	3	demonstrate	demonstrate	VERB
ejpam-5875	4	4	the	the	DET
ejpam-5875	4	5	absence	absence	NOUN
ejpam-5875	4	6	of	of	ADP
ejpam-5875	4	7	normal	normal	ADJ
ejpam-5875	4	8	curves	curve	NOUN
ejpam-5875	4	9	in	in	ADP
ejpam-5875	4	10	galilean	galilean	PROPN
ejpam-5875	4	11	3	3	NUM
ejpam-5875	4	12	-	-	PUNCT
ejpam-5875	4	13	space	space	NOUN
ejpam-5875	4	14	,	,	PUNCT
ejpam-5875	4	15	challenging	challenge	VERB
ejpam-5875	4	16	existing	exist	VERB
ejpam-5875	4	17	theories	theory	NOUN
ejpam-5875	4	18	in	in	ADP
ejpam-5875	4	19	the	the	DET
ejpam-5875	4	20	field	field	NOUN
ejpam-5875	4	21	and	and	CCONJ
ejpam-5875	4	22	providing	provide	VERB
ejpam-5875	4	23	new	new	ADJ
ejpam-5875	4	24	insights	insight	NOUN
ejpam-5875	4	25	into	into	ADP
ejpam-5875	4	26	the	the	DET
ejpam-5875	4	27	geometric	geometric	ADJ
ejpam-5875	4	28	structure	structure	NOUN
ejpam-5875	4	29	of	of	ADP
ejpam-5875	4	30	the	the	DET
ejpam-5875	4	31	galilean	galilean	PROPN
ejpam-5875	4	32	3	3	NUM
ejpam-5875	4	33	-	-	PUNCT
ejpam-5875	4	34	space	space	NOUN
ejpam-5875	4	35	.	.	PUNCT
ejpam-5875	5	1	we	we	PRON
ejpam-5875	5	2	explore	explore	VERB
ejpam-5875	5	3	the	the	DET
ejpam-5875	5	4	geometric	geometric	ADJ
ejpam-5875	5	5	properties	property	NOUN
ejpam-5875	5	6	of	of	ADP
ejpam-5875	5	7	quasi	quasi	ADJ
ejpam-5875	5	8	-	-	ADJ
ejpam-5875	5	9	rectifying	rectifying	ADJ
ejpam-5875	5	10	and	and	CCONJ
ejpam-5875	5	11	quasi	quasi	ADJ
ejpam-5875	5	12	-	-	ADJ
ejpam-5875	5	13	osculating	osculating	ADJ
ejpam-5875	5	14	curves	curve	NOUN
ejpam-5875	5	15	,	,	PUNCT
ejpam-5875	5	16	establishing	establish	VERB
ejpam-5875	5	17	the	the	DET
ejpam-5875	5	18	necessary	necessary	ADJ
ejpam-5875	5	19	and	and	CCONJ
ejpam-5875	5	20	sufficient	sufficient	ADJ
ejpam-5875	5	21	conditions	condition	NOUN
ejpam-5875	5	22	for	for	ADP
ejpam-5875	5	23	their	their	PRON
ejpam-5875	5	24	classification	classification	NOUN
ejpam-5875	5	25	.	.	PUNCT
ejpam-5875	6	1	a	a	DET
ejpam-5875	6	2	curve	curve	NOUN
ejpam-5875	6	3	is	be	AUX
ejpam-5875	6	4	identified	identify	VERB
ejpam-5875	6	5	as	as	ADP
ejpam-5875	6	6	quasi	quasi	NOUN
ejpam-5875	6	7	-	-	NOUN
ejpam-5875	6	8	rectifying	rectifying	ADJ
ejpam-5875	6	9	if	if	SCONJ
ejpam-5875	6	10	its	its	PRON
ejpam-5875	6	11	position	position	NOUN
ejpam-5875	6	12	vector	vector	NOUN
ejpam-5875	6	13	can	can	AUX
ejpam-5875	6	14	be	be	AUX
ejpam-5875	6	15	represented	represent	VERB
ejpam-5875	6	16	as	as	ADP
ejpam-5875	6	17	a	a	DET
ejpam-5875	6	18	linear	linear	ADJ
ejpam-5875	6	19	combination	combination	NOUN
ejpam-5875	6	20	of	of	ADP
ejpam-5875	6	21	its	its	PRON
ejpam-5875	6	22	tangent	tangent	NOUN
ejpam-5875	6	23	and	and	CCONJ
ejpam-5875	6	24	quasi	quasi	ADJ
ejpam-5875	6	25	-	-	ADJ
ejpam-5875	6	26	binormal	binormal	ADJ
ejpam-5875	6	27	vectors	vector	NOUN
ejpam-5875	6	28	.	.	PUNCT
ejpam-5875	7	1	in	in	ADP
ejpam-5875	7	2	contrast	contrast	NOUN
ejpam-5875	7	3	,	,	PUNCT
ejpam-5875	7	4	a	a	DET
ejpam-5875	7	5	curve	curve	NOUN
ejpam-5875	7	6	is	be	AUX
ejpam-5875	7	7	classified	classify	VERB
ejpam-5875	7	8	as	as	ADP
ejpam-5875	7	9	quasi	quasi	NOUN
ejpam-5875	7	10	-	-	NOUN
ejpam-5875	7	11	osculating	osculate	VERB
ejpam-5875	7	12	if	if	SCONJ
ejpam-5875	7	13	it	it	PRON
ejpam-5875	7	14	remains	remain	VERB
ejpam-5875	7	15	entirely	entirely	ADV
ejpam-5875	7	16	within	within	ADP
ejpam-5875	7	17	its	its	PRON
ejpam-5875	7	18	quasi	quasi	ADJ
ejpam-5875	7	19	-	-	ADJ
ejpam-5875	7	20	osculating	osculating	ADJ
ejpam-5875	7	21	plane	plane	NOUN
ejpam-5875	7	22	,	,	PUNCT
ejpam-5875	7	23	determined	determine	VERB
ejpam-5875	7	24	by	by	ADP
ejpam-5875	7	25	its	its	PRON
ejpam-5875	7	26	tangent	tangent	NOUN
ejpam-5875	7	27	and	and	CCONJ
ejpam-5875	7	28	quasi	quasi	ADJ
ejpam-5875	7	29	-	-	ADJ
ejpam-5875	7	30	normal	normal	ADJ
ejpam-5875	7	31	vectors	vector	NOUN
ejpam-5875	7	32	.	.	PUNCT
ejpam-5875	8	1	the	the	DET
ejpam-5875	8	2	quasi	quasi	ADJ
ejpam-5875	8	3	-	-	NOUN
ejpam-5875	8	4	frame	frame	NOUN
ejpam-5875	8	5	serves	serve	VERB
ejpam-5875	8	6	as	as	ADP
ejpam-5875	8	7	a	a	DET
ejpam-5875	8	8	generalization	generalization	NOUN
ejpam-5875	8	9	of	of	ADP
ejpam-5875	8	10	the	the	DET
ejpam-5875	8	11	classical	classical	ADJ
ejpam-5875	8	12	frenet	frenet	NOUN
ejpam-5875	8	13	frame	frame	NOUN
ejpam-5875	8	14	,	,	PUNCT
ejpam-5875	8	15	particularly	particularly	ADV
ejpam-5875	8	16	useful	useful	ADJ
ejpam-5875	8	17	in	in	ADP
ejpam-5875	8	18	scenarios	scenario	NOUN
ejpam-5875	8	19	where	where	SCONJ
ejpam-5875	8	20	the	the	DET
ejpam-5875	8	21	curvature	curvature	NOUN
ejpam-5875	8	22	vanishes	vanish	VERB
ejpam-5875	8	23	and	and	CCONJ
ejpam-5875	8	24	the	the	DET
ejpam-5875	8	25	frenet	frenet	ADJ
ejpam-5875	8	26	frame	frame	NOUN
ejpam-5875	8	27	becomes	become	VERB
ejpam-5875	8	28	undefined	undefined	ADJ
ejpam-5875	8	29	.	.	PUNCT
ejpam-5875	9	1	by	by	ADP
ejpam-5875	9	2	introducing	introduce	VERB
ejpam-5875	9	3	the	the	DET
ejpam-5875	9	4	quasi	quasi	ADJ
ejpam-5875	9	5	curvatures	curvature	NOUN
ejpam-5875	9	6	,	,	PUNCT
ejpam-5875	9	7	we	we	PRON
ejpam-5875	9	8	provide	provide	VERB
ejpam-5875	9	9	a	a	DET
ejpam-5875	9	10	robust	robust	ADJ
ejpam-5875	9	11	framework	framework	NOUN
ejpam-5875	9	12	for	for	ADP
ejpam-5875	9	13	analyzing	analyze	VERB
ejpam-5875	9	14	curves	curve	NOUN
ejpam-5875	9	15	in	in	ADP
ejpam-5875	9	16	galilean	galilean	PROPN
ejpam-5875	9	17	3	3	NUM
ejpam-5875	9	18	-	-	PUNCT
ejpam-5875	9	19	space	space	NOUN
ejpam-5875	9	20	.	.	PUNCT
ejpam-5875	10	1	we	we	PRON
ejpam-5875	10	2	derive	derive	VERB
ejpam-5875	10	3	explicit	explicit	ADJ
ejpam-5875	10	4	expressions	expression	NOUN
ejpam-5875	10	5	for	for	ADP
ejpam-5875	10	6	the	the	DET
ejpam-5875	10	7	position	position	NOUN
ejpam-5875	10	8	vectors	vector	NOUN
ejpam-5875	10	9	of	of	ADP
ejpam-5875	10	10	curves	curve	NOUN
ejpam-5875	10	11	with	with	ADP
ejpam-5875	10	12	respect	respect	NOUN
ejpam-5875	10	13	to	to	ADP
ejpam-5875	10	14	the	the	DET
ejpam-5875	10	15	quasi	quasi	ADJ
ejpam-5875	10	16	frame	frame	NOUN
ejpam-5875	10	17	and	and	CCONJ
ejpam-5875	10	18	solve	solve	VERB
ejpam-5875	10	19	for	for	ADP
ejpam-5875	10	20	their	their	PRON
ejpam-5875	10	21	components	component	NOUN
ejpam-5875	10	22	under	under	ADP
ejpam-5875	10	23	specific	specific	ADJ
ejpam-5875	10	24	conditions	condition	NOUN
ejpam-5875	10	25	.	.	PUNCT
ejpam-5875	11	1	furthermore	furthermore	ADV
ejpam-5875	11	2	,	,	PUNCT
ejpam-5875	11	3	we	we	PRON
ejpam-5875	11	4	prove	prove	VERB
ejpam-5875	11	5	that	that	SCONJ
ejpam-5875	11	6	normal	normal	ADJ
ejpam-5875	11	7	curves	curve	NOUN
ejpam-5875	11	8	can	can	AUX
ejpam-5875	11	9	not	not	PART
ejpam-5875	11	10	exist	exist	VERB
ejpam-5875	11	11	in	in	ADP
ejpam-5875	11	12	galilean	galilean	PROPN
ejpam-5875	11	13	space	space	NOUN
ejpam-5875	11	14	,	,	PUNCT
ejpam-5875	11	15	a	a	DET
ejpam-5875	11	16	result	result	NOUN
ejpam-5875	11	17	that	that	PRON
ejpam-5875	11	18	clarifies	clarify	VERB
ejpam-5875	11	19	the	the	DET
ejpam-5875	11	20	limitations	limitation	NOUN
ejpam-5875	11	21	of	of	ADP
ejpam-5875	11	22	certain	certain	ADJ
ejpam-5875	11	23	geometric	geometric	ADJ
ejpam-5875	11	24	classifications	classification	NOUN
ejpam-5875	11	25	in	in	ADP
ejpam-5875	11	26	this	this	DET
ejpam-5875	11	27	context	context	NOUN
ejpam-5875	11	28	.	.	PUNCT
ejpam-5875	12	1	2020	2020	NUM
ejpam-5875	12	2	mathematics	mathematic	NOUN
ejpam-5875	12	3	subject	subject	NOUN
ejpam-5875	12	4	classifications	classification	NOUN
ejpam-5875	12	5	:	:	PUNCT
ejpam-5875	12	6	51a05	51a05	NUM
ejpam-5875	12	7	,	,	PUNCT
ejpam-5875	12	8	53a35	53a35	NUM
ejpam-5875	12	9	key	key	ADJ
ejpam-5875	12	10	words	word	NOUN
ejpam-5875	12	11	and	and	CCONJ
ejpam-5875	12	12	phrases	phrase	NOUN
ejpam-5875	12	13	:	:	PUNCT
ejpam-5875	12	14	quasi	quasi	ADJ
ejpam-5875	12	15	-	-	NOUN
ejpam-5875	12	16	frame	frame	NOUN
ejpam-5875	12	17	,	,	PUNCT
ejpam-5875	12	18	galilean	galilean	PROPN
ejpam-5875	12	19	3	3	NUM
ejpam-5875	12	20	-	-	PUNCT
ejpam-5875	12	21	space	space	NOUN
ejpam-5875	12	22	,	,	PUNCT
ejpam-5875	12	23	quasi	quasi	ADJ
ejpam-5875	12	24	-	-	ADJ
ejpam-5875	12	25	normal	normal	ADJ
ejpam-5875	12	26	curves	curve	NOUN
ejpam-5875	12	27	,	,	PUNCT
ejpam-5875	12	28	quasi	quasi	ADJ
ejpam-5875	12	29	-	-	ADJ
ejpam-5875	12	30	rectifying	rectifying	ADJ
ejpam-5875	12	31	curves	curve	NOUN
ejpam-5875	12	32	,	,	PUNCT
ejpam-5875	12	33	quasi	quasi	ADJ
ejpam-5875	12	34	-	-	ADJ
ejpam-5875	12	35	osculating	osculating	ADJ
ejpam-5875	12	36	curves	curve	NOUN
ejpam-5875	12	37	.	.	PUNCT
ejpam-5875	13	1	1	1	X
ejpam-5875	13	2	.	.	X
ejpam-5875	13	3	introduction	introduction	NOUN
ejpam-5875	13	4	galilean	galilean	PROPN
ejpam-5875	13	5	geometry	geometry	NOUN
ejpam-5875	13	6	,	,	PUNCT
ejpam-5875	13	7	as	as	SCONJ
ejpam-5875	13	8	articulated	articulate	VERB
ejpam-5875	13	9	by	by	ADP
ejpam-5875	13	10	cayley	cayley	NOUN
ejpam-5875	13	11	and	and	CCONJ
ejpam-5875	13	12	klein	klein	PROPN
ejpam-5875	13	13	,	,	PUNCT
ejpam-5875	13	14	encompasses	encompass	VERB
ejpam-5875	13	15	transformations	transformation	NOUN
ejpam-5875	13	16	that	that	PRON
ejpam-5875	13	17	are	be	AUX
ejpam-5875	13	18	fundamental	fundamental	ADJ
ejpam-5875	13	19	to	to	ADP
ejpam-5875	13	20	both	both	CCONJ
ejpam-5875	13	21	classical	classical	ADJ
ejpam-5875	13	22	and	and	CCONJ
ejpam-5875	13	23	modern	modern	ADJ
ejpam-5875	13	24	physics	physic	NOUN
ejpam-5875	13	25	.	.	PUNCT
ejpam-5875	14	1	the	the	DET
ejpam-5875	14	2	group	group	NOUN
ejpam-5875	14	3	of	of	ADP
ejpam-5875	14	4	galilean	galilean	PROPN
ejpam-5875	14	5	transformations	transformation	NOUN
ejpam-5875	14	6	is	be	AUX
ejpam-5875	14	7	pivotal	pivotal	ADJ
ejpam-5875	14	8	within	within	ADP
ejpam-5875	14	9	these	these	DET
ejpam-5875	14	10	theoretical	theoretical	ADJ
ejpam-5875	14	11	frameworks	framework	NOUN
ejpam-5875	14	12	[	[	X
ejpam-5875	14	13	1	1	NUM
ejpam-5875	14	14	]	]	PUNCT
ejpam-5875	14	15	.	.	PUNCT
ejpam-5875	15	1	notably	notably	ADV
ejpam-5875	15	2	,	,	PUNCT
ejpam-5875	15	3	the	the	DET
ejpam-5875	15	4	conventional	conventional	ADJ
ejpam-5875	15	5	frenet	frenet	ADJ
ejpam-5875	15	6	frame	frame	NOUN
ejpam-5875	15	7	becomes	become	VERB
ejpam-5875	15	8	inapplicable	inapplicable	ADJ
ejpam-5875	15	9	at	at	ADP
ejpam-5875	15	10	points	point	NOUN
ejpam-5875	15	11	where	where	SCONJ
ejpam-5875	15	12	curvature	curvature	NOUN
ejpam-5875	15	13	approaches	approach	NOUN
ejpam-5875	15	14	zero	zero	NUM
ejpam-5875	15	15	,	,	PUNCT
ejpam-5875	15	16	specifically	specifically	ADV
ejpam-5875	15	17	at	at	ADP
ejpam-5875	15	18	locations	location	NOUN
ejpam-5875	15	19	where	where	SCONJ
ejpam-5875	15	20	the	the	DET
ejpam-5875	15	21	normal	normal	ADJ
ejpam-5875	15	22	and	and	CCONJ
ejpam-5875	15	23	binormal	binormal	ADJ
ejpam-5875	15	24	vectors	vector	NOUN
ejpam-5875	15	25	are	be	AUX
ejpam-5875	15	26	undefined	undefined	ADJ
ejpam-5875	15	27	[	[	X
ejpam-5875	15	28	2–4	2–4	NUM
ejpam-5875	15	29	]	]	PUNCT
ejpam-5875	15	30	.	.	PUNCT
ejpam-5875	16	1	in	in	ADP
ejpam-5875	16	2	response	response	NOUN
ejpam-5875	16	3	to	to	ADP
ejpam-5875	16	4	∗corresponding	∗corresponde	VERB
ejpam-5875	16	5	author	author	NOUN
ejpam-5875	16	6	.	.	PUNCT
ejpam-5875	17	1	doi	doi	NOUN
ejpam-5875	17	2	:	:	PUNCT
ejpam-5875	17	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5875	https://doi.org/10.29020/nybg.ejpam.v18i2.5875	NOUN
ejpam-5875	17	4	email	email	NOUN
ejpam-5875	17	5	addresses	address	VERB
ejpam-5875	17	6	:	:	PUNCT
ejpam-5875	18	1	ayman_ramadan@science.tanta.edu.eg	ayman_ramadan@science.tanta.edu.eg	PROPN
ejpam-5875	18	2	(	(	PUNCT
ejpam-5875	18	3	a.	a.	NOUN
ejpam-5875	18	4	elsharkawy	elsharkawy	PROPN
ejpam-5875	18	5	)	)	PUNCT
ejpam-5875	18	6	noha_elsharkawy@science.tanta.edu.eg	noha_elsharkawy@science.tanta.edu.eg	PROPN
ejpam-5875	18	7	(	(	PUNCT
ejpam-5875	18	8	n.	n.	NOUN
ejpam-5875	18	9	elsharkawy	elsharkawy	PROPN
ejpam-5875	18	10	)	)	PUNCT
ejpam-5875	18	11	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5875	18	12	1	1	NUM
ejpam-5875	18	13	copyright	copyright	NOUN
ejpam-5875	18	14	:	:	PUNCT
ejpam-5875	18	15	©	©	PROPN
ejpam-5875	18	16	2025	2025	NUM
ejpam-5875	18	17	the	the	DET
ejpam-5875	18	18	author(s	author(s	NOUN
ejpam-5875	18	19	)	)	PUNCT
ejpam-5875	18	20	.	.	PUNCT
ejpam-5875	19	1	(	(	PUNCT
ejpam-5875	19	2	cc	cc	NOUN
ejpam-5875	19	3	by	by	ADP
ejpam-5875	19	4	-	-	PUNCT
ejpam-5875	19	5	nc	nc	PROPN
ejpam-5875	19	6	4.0	4.0	NUM
ejpam-5875	19	7	)	)	PUNCT
ejpam-5875	19	8	a.	a.	NOUN
ejpam-5875	19	9	elsharkawy	elsharkawy	NOUN
ejpam-5875	19	10	,	,	PUNCT
ejpam-5875	19	11	n.	n.	NOUN
ejpam-5875	19	12	elsharkawy	elsharkawy	PROPN
ejpam-5875	19	13	/	/	SYM
ejpam-5875	19	14	eur	eur	PROPN
ejpam-5875	19	15	.	.	PUNCT
ejpam-5875	20	1	j.	j.	PROPN
ejpam-5875	20	2	pure	pure	PROPN
ejpam-5875	20	3	appl	appl	PROPN
ejpam-5875	20	4	.	.	PROPN
ejpam-5875	20	5	math	math	PROPN
ejpam-5875	20	6	,	,	PUNCT
ejpam-5875	20	7	18	18	NUM
ejpam-5875	20	8	(	(	PUNCT
ejpam-5875	20	9	2	2	NUM
ejpam-5875	20	10	)	)	PUNCT
ejpam-5875	20	11	(	(	PUNCT
ejpam-5875	20	12	2025	2025	NUM
ejpam-5875	20	13	)	)	PUNCT
ejpam-5875	20	14	,	,	PUNCT
ejpam-5875	20	15	5875	5875	NUM
ejpam-5875	20	16	2	2	NUM
ejpam-5875	20	17	of	of	ADP
ejpam-5875	20	18	15	15	NUM
ejpam-5875	20	19	this	this	DET
ejpam-5875	20	20	limitation	limitation	NOUN
ejpam-5875	21	1	,	,	PUNCT
ejpam-5875	21	2	various	various	ADJ
ejpam-5875	21	3	researchers	researcher	NOUN
ejpam-5875	21	4	have	have	AUX
ejpam-5875	21	5	devised	devise	VERB
ejpam-5875	21	6	alternative	alternative	ADJ
ejpam-5875	21	7	frames	frame	NOUN
ejpam-5875	21	8	that	that	PRON
ejpam-5875	21	9	effectively	effectively	ADV
ejpam-5875	21	10	handle	handle	VERB
ejpam-5875	21	11	such	such	ADJ
ejpam-5875	21	12	scenarios	scenario	NOUN
ejpam-5875	21	13	in	in	ADP
ejpam-5875	21	14	euclidean	euclidean	ADJ
ejpam-5875	21	15	space	space	NOUN
ejpam-5875	21	16	[	[	X
ejpam-5875	21	17	5	5	NUM
ejpam-5875	21	18	,	,	PUNCT
ejpam-5875	21	19	6	6	NUM
ejpam-5875	21	20	]	]	PUNCT
ejpam-5875	21	21	,	,	PUNCT
ejpam-5875	21	22	minkowski	minkowski	ADJ
ejpam-5875	21	23	space	space	NOUN
ejpam-5875	22	1	[	[	X
ejpam-5875	22	2	7–9	7–9	NOUN
ejpam-5875	22	3	]	]	X
ejpam-5875	22	4	,	,	PUNCT
ejpam-5875	22	5	and	and	CCONJ
ejpam-5875	22	6	galilean	galilean	PROPN
ejpam-5875	22	7	space	space	NOUN
ejpam-5875	22	8	[	[	X
ejpam-5875	22	9	10–14	10–14	NUM
ejpam-5875	22	10	]	]	PUNCT
ejpam-5875	22	11	.	.	PUNCT
ejpam-5875	23	1	these	these	PRON
ejpam-5875	23	2	include	include	VERB
ejpam-5875	23	3	the	the	DET
ejpam-5875	23	4	equiform	equiform	NOUN
ejpam-5875	23	5	frame	frame	NOUN
ejpam-5875	23	6	[	[	X
ejpam-5875	23	7	15	15	NUM
ejpam-5875	23	8	,	,	PUNCT
ejpam-5875	23	9	16	16	NUM
ejpam-5875	23	10	]	]	PUNCT
ejpam-5875	23	11	,	,	PUNCT
ejpam-5875	23	12	the	the	DET
ejpam-5875	23	13	bishop	bishop	PROPN
ejpam-5875	23	14	frame	frame	NOUN
ejpam-5875	24	1	[	[	X
ejpam-5875	24	2	17	17	NUM
ejpam-5875	24	3	]	]	PUNCT
ejpam-5875	24	4	,	,	PUNCT
ejpam-5875	24	5	the	the	DET
ejpam-5875	24	6	darboux	darboux	ADJ
ejpam-5875	24	7	frame	frame	NOUN
ejpam-5875	24	8	[	[	X
ejpam-5875	24	9	17	17	NUM
ejpam-5875	24	10	,	,	PUNCT
ejpam-5875	24	11	18	18	NUM
ejpam-5875	24	12	]	]	PUNCT
ejpam-5875	24	13	,	,	PUNCT
ejpam-5875	24	14	the	the	DET
ejpam-5875	24	15	modified	modify	VERB
ejpam-5875	24	16	frame	frame	NOUN
ejpam-5875	24	17	[	[	X
ejpam-5875	24	18	2	2	NUM
ejpam-5875	24	19	,	,	PUNCT
ejpam-5875	24	20	3	3	NUM
ejpam-5875	24	21	,	,	PUNCT
ejpam-5875	24	22	12	12	NUM
ejpam-5875	24	23	,	,	PUNCT
ejpam-5875	24	24	13	13	NUM
ejpam-5875	24	25	]	]	PUNCT
ejpam-5875	24	26	,	,	PUNCT
ejpam-5875	24	27	and	and	CCONJ
ejpam-5875	24	28	the	the	DET
ejpam-5875	24	29	quasi	quasi	ADJ
ejpam-5875	24	30	frame	frame	NOUN
ejpam-5875	24	31	[	[	X
ejpam-5875	24	32	5	5	NUM
ejpam-5875	24	33	,	,	PUNCT
ejpam-5875	24	34	6	6	NUM
ejpam-5875	24	35	,	,	PUNCT
ejpam-5875	24	36	10	10	NUM
ejpam-5875	24	37	,	,	PUNCT
ejpam-5875	24	38	11	11	NUM
ejpam-5875	24	39	]	]	PUNCT
ejpam-5875	24	40	.	.	PUNCT
ejpam-5875	25	1	in	in	ADP
ejpam-5875	25	2	the	the	DET
ejpam-5875	25	3	context	context	NOUN
ejpam-5875	25	4	of	of	ADP
ejpam-5875	25	5	plane	plane	NOUN
ejpam-5875	25	6	curves	curve	NOUN
ejpam-5875	25	7	,	,	PUNCT
ejpam-5875	25	8	three	three	NUM
ejpam-5875	25	9	primary	primary	ADJ
ejpam-5875	25	10	classifications	classification	NOUN
ejpam-5875	25	11	can	can	AUX
ejpam-5875	25	12	be	be	AUX
ejpam-5875	25	13	identified	identify	VERB
ejpam-5875	25	14	:	:	PUNCT
ejpam-5875	25	15	osculating	osculating	NOUN
ejpam-5875	25	16	,	,	PUNCT
ejpam-5875	25	17	normal	normal	ADJ
ejpam-5875	25	18	,	,	PUNCT
ejpam-5875	25	19	and	and	CCONJ
ejpam-5875	25	20	rectifying	rectifying	NOUN
ejpam-5875	25	21	curves	curve	NOUN
ejpam-5875	25	22	.	.	PUNCT
ejpam-5875	26	1	an	an	DET
ejpam-5875	26	2	osculating	osculating	NOUN
ejpam-5875	26	3	curve	curve	NOUN
ejpam-5875	26	4	is	be	AUX
ejpam-5875	26	5	characterized	characterize	VERB
ejpam-5875	26	6	by	by	ADP
ejpam-5875	26	7	the	the	DET
ejpam-5875	26	8	tangent	tangent	NOUN
ejpam-5875	26	9	and	and	CCONJ
ejpam-5875	26	10	normal	normal	ADJ
ejpam-5875	26	11	vectors	vector	NOUN
ejpam-5875	26	12	residing	reside	VERB
ejpam-5875	26	13	within	within	ADP
ejpam-5875	26	14	the	the	DET
ejpam-5875	26	15	same	same	ADJ
ejpam-5875	26	16	plane	plane	NOUN
ejpam-5875	26	17	defined	define	VERB
ejpam-5875	26	18	by	by	ADP
ejpam-5875	26	19	its	its	PRON
ejpam-5875	26	20	position	position	NOUN
ejpam-5875	26	21	vector	vector	NOUN
ejpam-5875	26	22	at	at	ADP
ejpam-5875	26	23	all	all	DET
ejpam-5875	26	24	instances	instance	NOUN
ejpam-5875	26	25	.	.	PUNCT
ejpam-5875	27	1	in	in	ADP
ejpam-5875	27	2	contrast	contrast	NOUN
ejpam-5875	27	3	,	,	PUNCT
ejpam-5875	27	4	a	a	DET
ejpam-5875	27	5	normal	normal	ADJ
ejpam-5875	27	6	curve	curve	NOUN
ejpam-5875	27	7	is	be	AUX
ejpam-5875	27	8	defined	define	VERB
ejpam-5875	27	9	by	by	ADP
ejpam-5875	27	10	a	a	DET
ejpam-5875	27	11	position	position	NOUN
ejpam-5875	27	12	vector	vector	NOUN
ejpam-5875	27	13	that	that	PRON
ejpam-5875	27	14	consistently	consistently	ADV
ejpam-5875	27	15	maintains	maintain	VERB
ejpam-5875	27	16	a	a	DET
ejpam-5875	27	17	normal	normal	ADJ
ejpam-5875	27	18	orientation	orientation	NOUN
ejpam-5875	27	19	.	.	PUNCT
ejpam-5875	28	1	the	the	DET
ejpam-5875	28	2	plane	plane	NOUN
ejpam-5875	28	3	in	in	ADP
ejpam-5875	28	4	question	question	NOUN
ejpam-5875	28	5	is	be	AUX
ejpam-5875	28	6	formed	form	VERB
ejpam-5875	28	7	by	by	ADP
ejpam-5875	28	8	the	the	DET
ejpam-5875	28	9	curve	curve	NOUN
ejpam-5875	28	10	’s	’s	PART
ejpam-5875	28	11	normal	normal	ADJ
ejpam-5875	28	12	and	and	CCONJ
ejpam-5875	28	13	binormal	binormal	ADJ
ejpam-5875	28	14	vectors	vector	NOUN
ejpam-5875	28	15	.	.	PUNCT
ejpam-5875	29	1	previous	previous	ADJ
ejpam-5875	29	2	studies	study	NOUN
ejpam-5875	29	3	have	have	AUX
ejpam-5875	29	4	explored	explore	VERB
ejpam-5875	29	5	the	the	DET
ejpam-5875	29	6	characteristics	characteristic	NOUN
ejpam-5875	29	7	of	of	ADP
ejpam-5875	29	8	normal	normal	ADJ
ejpam-5875	29	9	,	,	PUNCT
ejpam-5875	29	10	osculating	osculating	NOUN
ejpam-5875	29	11	,	,	PUNCT
ejpam-5875	29	12	and	and	CCONJ
ejpam-5875	29	13	rectifying	rectify	VERB
ejpam-5875	29	14	curves	curve	NOUN
ejpam-5875	29	15	across	across	ADP
ejpam-5875	29	16	various	various	ADJ
ejpam-5875	29	17	geometric	geometric	ADJ
ejpam-5875	29	18	frameworks	framework	NOUN
ejpam-5875	29	19	[	[	X
ejpam-5875	29	20	19–24	19–24	NUM
ejpam-5875	29	21	]	]	PUNCT
ejpam-5875	29	22	.	.	PUNCT
ejpam-5875	30	1	a	a	DET
ejpam-5875	30	2	recent	recent	ADJ
ejpam-5875	30	3	study	study	NOUN
ejpam-5875	30	4	by	by	ADP
ejpam-5875	30	5	dede	dede	NOUN
ejpam-5875	30	6	et	et	PROPN
ejpam-5875	30	7	al	al	PROPN
ejpam-5875	30	8	.	.	PUNCT
ejpam-5875	31	1	[	[	X
ejpam-5875	31	2	25	25	NUM
ejpam-5875	31	3	]	]	PUNCT
ejpam-5875	31	4	introduced	introduce	VERB
ejpam-5875	31	5	an	an	DET
ejpam-5875	31	6	innovative	innovative	ADJ
ejpam-5875	31	7	approach	approach	NOUN
ejpam-5875	31	8	by	by	ADP
ejpam-5875	31	9	constructing	construct	VERB
ejpam-5875	31	10	an	an	DET
ejpam-5875	31	11	adapted	adapted	ADJ
ejpam-5875	31	12	frame	frame	NOUN
ejpam-5875	31	13	that	that	PRON
ejpam-5875	31	14	precisely	precisely	ADV
ejpam-5875	31	15	follows	follow	VERB
ejpam-5875	31	16	a	a	DET
ejpam-5875	31	17	space	space	NOUN
ejpam-5875	31	18	curve	curve	NOUN
ejpam-5875	31	19	,	,	PUNCT
ejpam-5875	31	20	moving	move	VERB
ejpam-5875	31	21	beyond	beyond	ADP
ejpam-5875	31	22	reliance	reliance	NOUN
ejpam-5875	31	23	on	on	ADP
ejpam-5875	31	24	the	the	DET
ejpam-5875	31	25	traditional	traditional	ADJ
ejpam-5875	31	26	serret	serret	ADJ
ejpam-5875	31	27	frenet	frenet	ADJ
ejpam-5875	31	28	frame	frame	NOUN
ejpam-5875	31	29	.	.	PUNCT
ejpam-5875	32	1	this	this	DET
ejpam-5875	32	2	newly	newly	ADV
ejpam-5875	32	3	developed	develop	VERB
ejpam-5875	32	4	framework	framework	NOUN
ejpam-5875	32	5	termed	term	VERB
ejpam-5875	32	6	the	the	DET
ejpam-5875	32	7	quasi	quasi	NOUN
ejpam-5875	32	8	-	-	NOUN
ejpam-5875	32	9	frame	frame	NOUN
ejpam-5875	32	10	(	(	PUNCT
ejpam-5875	32	11	q	q	NOUN
ejpam-5875	32	12	-	-	PUNCT
ejpam-5875	32	13	frame	frame	NOUN
ejpam-5875	32	14	)	)	PUNCT
ejpam-5875	32	15	,	,	PUNCT
ejpam-5875	32	16	enhances	enhance	VERB
ejpam-5875	32	17	precision	precision	NOUN
ejpam-5875	32	18	and	and	CCONJ
ejpam-5875	32	19	applicability	applicability	NOUN
ejpam-5875	32	20	,	,	PUNCT
ejpam-5875	32	21	thereby	thereby	ADV
ejpam-5875	32	22	serving	serve	VERB
ejpam-5875	32	23	as	as	ADP
ejpam-5875	32	24	an	an	DET
ejpam-5875	32	25	expanded	expand	VERB
ejpam-5875	32	26	interpretation	interpretation	NOUN
ejpam-5875	32	27	of	of	ADP
ejpam-5875	32	28	the	the	DET
ejpam-5875	32	29	frenet	frenet	ADJ
ejpam-5875	32	30	frame	frame	NOUN
ejpam-5875	32	31	.	.	PUNCT
ejpam-5875	33	1	the	the	DET
ejpam-5875	33	2	q	q	NOUN
ejpam-5875	33	3	-	-	PUNCT
ejpam-5875	33	4	frame	frame	NOUN
ejpam-5875	33	5	is	be	AUX
ejpam-5875	33	6	distinguished	distinguish	VERB
ejpam-5875	33	7	by	by	ADP
ejpam-5875	33	8	a	a	DET
ejpam-5875	33	9	fixed	fix	VERB
ejpam-5875	33	10	vector	vector	NOUN
ejpam-5875	33	11	and	and	CCONJ
ejpam-5875	33	12	the	the	DET
ejpam-5875	33	13	angle	angle	NOUN
ejpam-5875	33	14	between	between	ADP
ejpam-5875	33	15	the	the	DET
ejpam-5875	33	16	quasi	quasi	ADJ
ejpam-5875	33	17	-	-	ADJ
ejpam-5875	33	18	normal	normal	ADJ
ejpam-5875	33	19	vector	vector	NOUN
ejpam-5875	33	20	and	and	CCONJ
ejpam-5875	33	21	the	the	DET
ejpam-5875	33	22	principal	principal	NOUN
ejpam-5875	33	23	normal	normal	ADJ
ejpam-5875	33	24	of	of	ADP
ejpam-5875	33	25	the	the	DET
ejpam-5875	33	26	frenet	frenet	ADJ
ejpam-5875	33	27	frame	frame	NOUN
ejpam-5875	33	28	.	.	PUNCT
ejpam-5875	34	1	at	at	ADP
ejpam-5875	34	2	points	point	NOUN
ejpam-5875	34	3	where	where	SCONJ
ejpam-5875	34	4	the	the	DET
ejpam-5875	34	5	curvature	curvature	NOUN
ejpam-5875	34	6	is	be	AUX
ejpam-5875	34	7	zero	zero	NUM
ejpam-5875	34	8	,	,	PUNCT
ejpam-5875	34	9	this	this	DET
ejpam-5875	34	10	frame	frame	NOUN
ejpam-5875	34	11	undergoes	undergo	VERB
ejpam-5875	34	12	rotation	rotation	NOUN
ejpam-5875	34	13	by	by	ADP
ejpam-5875	34	14	the	the	DET
ejpam-5875	34	15	specified	specified	ADJ
ejpam-5875	34	16	angle	angle	NOUN
ejpam-5875	34	17	,	,	PUNCT
ejpam-5875	34	18	establishing	establish	VERB
ejpam-5875	34	19	the	the	DET
ejpam-5875	34	20	q	q	NOUN
ejpam-5875	34	21	-	-	ADJ
ejpam-5875	34	22	normal	normal	ADJ
ejpam-5875	34	23	as	as	ADP
ejpam-5875	34	24	orthogonal	orthogonal	ADJ
ejpam-5875	34	25	to	to	ADP
ejpam-5875	34	26	both	both	CCONJ
ejpam-5875	34	27	the	the	DET
ejpam-5875	34	28	tangent	tangent	NOUN
ejpam-5875	34	29	vector	vector	NOUN
ejpam-5875	34	30	and	and	CCONJ
ejpam-5875	34	31	the	the	DET
ejpam-5875	34	32	fixed	fix	VERB
ejpam-5875	34	33	vector	vector	NOUN
ejpam-5875	34	34	.	.	PUNCT
ejpam-5875	35	1	the	the	DET
ejpam-5875	35	2	q	q	ADJ
ejpam-5875	35	3	-	-	PUNCT
ejpam-5875	35	4	binormal	binormal	ADJ
ejpam-5875	35	5	vector	vector	NOUN
ejpam-5875	35	6	is	be	AUX
ejpam-5875	35	7	defined	define	VERB
ejpam-5875	35	8	as	as	ADP
ejpam-5875	35	9	the	the	DET
ejpam-5875	35	10	unit	unit	NOUN
ejpam-5875	35	11	vector	vector	NOUN
ejpam-5875	35	12	orthogonal	orthogonal	NOUN
ejpam-5875	35	13	to	to	ADP
ejpam-5875	35	14	both	both	CCONJ
ejpam-5875	35	15	the	the	DET
ejpam-5875	35	16	tangent	tangent	NOUN
ejpam-5875	35	17	and	and	CCONJ
ejpam-5875	35	18	q	q	ADJ
ejpam-5875	35	19	-	-	ADJ
ejpam-5875	35	20	normal	normal	ADJ
ejpam-5875	35	21	vectors	vector	NOUN
ejpam-5875	35	22	.	.	PUNCT
ejpam-5875	36	1	numerous	numerous	ADJ
ejpam-5875	36	2	studies	study	NOUN
ejpam-5875	36	3	have	have	AUX
ejpam-5875	36	4	examined	examine	VERB
ejpam-5875	36	5	the	the	DET
ejpam-5875	36	6	q	q	NOUN
ejpam-5875	36	7	-	-	PUNCT
ejpam-5875	36	8	frame	frame	NOUN
ejpam-5875	36	9	within	within	ADP
ejpam-5875	36	10	euclidean	euclidean	ADJ
ejpam-5875	36	11	and	and	CCONJ
ejpam-5875	36	12	minkowski	minkowski	ADJ
ejpam-5875	36	13	spaces	space	NOUN
ejpam-5875	36	14	[	[	X
ejpam-5875	36	15	26–30	26–30	NUM
ejpam-5875	36	16	]	]	PUNCT
ejpam-5875	36	17	,	,	PUNCT
ejpam-5875	36	18	while	while	SCONJ
ejpam-5875	36	19	more	more	ADJ
ejpam-5875	36	20	recent	recent	ADJ
ejpam-5875	36	21	investigations	investigation	NOUN
ejpam-5875	36	22	have	have	AUX
ejpam-5875	36	23	focused	focus	VERB
ejpam-5875	36	24	on	on	ADP
ejpam-5875	36	25	position	position	NOUN
ejpam-5875	36	26	vectors	vector	NOUN
ejpam-5875	36	27	in	in	ADP
ejpam-5875	36	28	galilean	galilean	PROPN
ejpam-5875	36	29	threeand	threeand	PROPN
ejpam-5875	36	30	four	four	NUM
ejpam-5875	36	31	-	-	PUNCT
ejpam-5875	36	32	dimensional	dimensional	ADJ
ejpam-5875	36	33	spaces	space	NOUN
ejpam-5875	36	34	using	use	VERB
ejpam-5875	36	35	the	the	DET
ejpam-5875	36	36	frenet	frenet	ADJ
ejpam-5875	36	37	frame	frame	NOUN
ejpam-5875	37	1	[	[	X
ejpam-5875	37	2	31–34	31–34	NUM
ejpam-5875	37	3	]	]	PUNCT
ejpam-5875	37	4	.	.	PUNCT
ejpam-5875	38	1	the	the	DET
ejpam-5875	38	2	structure	structure	NOUN
ejpam-5875	38	3	of	of	ADP
ejpam-5875	38	4	this	this	DET
ejpam-5875	38	5	paper	paper	NOUN
ejpam-5875	38	6	is	be	AUX
ejpam-5875	38	7	organized	organize	VERB
ejpam-5875	38	8	as	as	SCONJ
ejpam-5875	38	9	follows	follow	VERB
ejpam-5875	38	10	:	:	PUNCT
ejpam-5875	38	11	section	section	NOUN
ejpam-5875	38	12	2	2	NUM
ejpam-5875	38	13	details	detail	NOUN
ejpam-5875	38	14	the	the	DET
ejpam-5875	38	15	q	q	NOUN
ejpam-5875	38	16	-	-	PUNCT
ejpam-5875	38	17	frame	frame	NOUN
ejpam-5875	38	18	and	and	CCONJ
ejpam-5875	38	19	its	its	PRON
ejpam-5875	38	20	relationship	relationship	NOUN
ejpam-5875	38	21	to	to	ADP
ejpam-5875	38	22	the	the	DET
ejpam-5875	38	23	frenet	frenet	ADJ
ejpam-5875	38	24	frame	frame	NOUN
ejpam-5875	38	25	.	.	PUNCT
ejpam-5875	39	1	section	section	NOUN
ejpam-5875	39	2	3	3	NUM
ejpam-5875	39	3	delves	delf	NOUN
ejpam-5875	39	4	into	into	ADP
ejpam-5875	39	5	the	the	DET
ejpam-5875	39	6	analysis	analysis	NOUN
ejpam-5875	39	7	of	of	ADP
ejpam-5875	39	8	quasiformulas	quasiformula	NOUN
ejpam-5875	39	9	within	within	ADP
ejpam-5875	39	10	galilean	galilean	PROPN
ejpam-5875	39	11	3	3	NUM
ejpam-5875	39	12	-	-	PUNCT
ejpam-5875	39	13	space	space	NOUN
ejpam-5875	39	14	.	.	PUNCT
ejpam-5875	40	1	section	section	NOUN
ejpam-5875	40	2	4	4	NUM
ejpam-5875	40	3	investigates	investigate	VERB
ejpam-5875	40	4	position	position	NOUN
ejpam-5875	40	5	vectors	vector	NOUN
ejpam-5875	40	6	in	in	ADP
ejpam-5875	40	7	galilean	galilean	PROPN
ejpam-5875	40	8	3	3	NUM
ejpam-5875	40	9	-	-	PUNCT
ejpam-5875	40	10	space	space	NOUN
ejpam-5875	40	11	and	and	CCONJ
ejpam-5875	40	12	determines	determine	VERB
ejpam-5875	40	13	coefficients	coefficient	NOUN
ejpam-5875	40	14	under	under	ADP
ejpam-5875	40	15	specific	specific	ADJ
ejpam-5875	40	16	conditions	condition	NOUN
ejpam-5875	40	17	,	,	PUNCT
ejpam-5875	40	18	covering	cover	VERB
ejpam-5875	40	19	quasi	quasi	NOUN
ejpam-5875	40	20	-	-	ADJ
ejpam-5875	40	21	rectifying	rectifying	ADJ
ejpam-5875	40	22	and	and	CCONJ
ejpam-5875	40	23	quasi	quasi	ADJ
ejpam-5875	40	24	-	-	ADJ
ejpam-5875	40	25	osculating	osculating	ADJ
ejpam-5875	40	26	curves	curve	NOUN
ejpam-5875	40	27	.	.	PUNCT
ejpam-5875	41	1	furthermore	furthermore	ADV
ejpam-5875	41	2	,	,	PUNCT
ejpam-5875	41	3	we	we	PRON
ejpam-5875	41	4	establish	establish	VERB
ejpam-5875	41	5	the	the	DET
ejpam-5875	41	6	non	non	NOUN
ejpam-5875	41	7	-	-	NOUN
ejpam-5875	41	8	existence	existence	NOUN
ejpam-5875	41	9	of	of	ADP
ejpam-5875	41	10	q	q	ADJ
ejpam-5875	41	11	-	-	ADJ
ejpam-5875	41	12	normal	normal	ADJ
ejpam-5875	41	13	curves	curve	NOUN
ejpam-5875	41	14	in	in	ADP
ejpam-5875	41	15	galilean	galilean	PROPN
ejpam-5875	41	16	3	3	NUM
ejpam-5875	41	17	-	-	PUNCT
ejpam-5875	41	18	space	space	NOUN
ejpam-5875	41	19	,	,	PUNCT
ejpam-5875	41	20	outlining	outline	VERB
ejpam-5875	41	21	the	the	DET
ejpam-5875	41	22	necessary	necessary	ADJ
ejpam-5875	41	23	and	and	CCONJ
ejpam-5875	41	24	sufficient	sufficient	ADJ
ejpam-5875	41	25	criteria	criterion	NOUN
ejpam-5875	41	26	for	for	ADP
ejpam-5875	41	27	classifying	classify	VERB
ejpam-5875	41	28	a	a	DET
ejpam-5875	41	29	curve	curve	NOUN
ejpam-5875	41	30	as	as	ADP
ejpam-5875	41	31	either	either	CCONJ
ejpam-5875	41	32	quasi	quasi	ADJ
ejpam-5875	41	33	-	-	ADJ
ejpam-5875	41	34	rectifying	rectifying	ADJ
ejpam-5875	41	35	or	or	CCONJ
ejpam-5875	41	36	quasi	quasi	NOUN
ejpam-5875	41	37	-	-	NOUN
ejpam-5875	41	38	osculating	osculating	NOUN
ejpam-5875	41	39	.	.	PUNCT
ejpam-5875	42	1	2	2	X
ejpam-5875	42	2	.	.	X
ejpam-5875	42	3	preliminaries	preliminary	NOUN
ejpam-5875	42	4	in	in	ADP
ejpam-5875	42	5	this	this	DET
ejpam-5875	42	6	section	section	NOUN
ejpam-5875	42	7	,	,	PUNCT
ejpam-5875	42	8	we	we	PRON
ejpam-5875	42	9	present	present	VERB
ejpam-5875	42	10	essential	essential	ADJ
ejpam-5875	42	11	concepts	concept	NOUN
ejpam-5875	42	12	and	and	CCONJ
ejpam-5875	42	13	definitions	definition	NOUN
ejpam-5875	42	14	that	that	PRON
ejpam-5875	42	15	will	will	AUX
ejpam-5875	42	16	be	be	AUX
ejpam-5875	42	17	crucial	crucial	ADJ
ejpam-5875	42	18	for	for	ADP
ejpam-5875	42	19	our	our	PRON
ejpam-5875	42	20	subsequent	subsequent	ADJ
ejpam-5875	42	21	analysis	analysis	NOUN
ejpam-5875	42	22	.	.	PUNCT
ejpam-5875	43	1	the	the	DET
ejpam-5875	43	2	three	three	NUM
ejpam-5875	43	3	-	-	PUNCT
ejpam-5875	43	4	dimensional	dimensional	ADJ
ejpam-5875	43	5	galilean	galilean	PROPN
ejpam-5875	43	6	space	space	NOUN
ejpam-5875	43	7	,	,	PUNCT
ejpam-5875	43	8	denoted	denote	VERB
ejpam-5875	43	9	as	as	ADP
ejpam-5875	43	10	g3	g3	NOUN
ejpam-5875	43	11	,	,	PUNCT
ejpam-5875	43	12	is	be	AUX
ejpam-5875	43	13	a	a	DET
ejpam-5875	43	14	real	real	ADJ
ejpam-5875	43	15	vector	vector	NOUN
ejpam-5875	43	16	space	space	NOUN
ejpam-5875	43	17	structured	structure	VERB
ejpam-5875	43	18	according	accord	VERB
ejpam-5875	43	19	to	to	ADP
ejpam-5875	43	20	the	the	DET
ejpam-5875	43	21	cayley	cayley	ADJ
ejpam-5875	43	22	-	-	PUNCT
ejpam-5875	43	23	klein	klein	NOUN
ejpam-5875	43	24	model	model	NOUN
ejpam-5875	43	25	,	,	PUNCT
ejpam-5875	43	26	characterized	characterize	VERB
ejpam-5875	43	27	by	by	ADP
ejpam-5875	43	28	a	a	DET
ejpam-5875	43	29	projective	projective	ADJ
ejpam-5875	43	30	metric	metric	NOUN
ejpam-5875	43	31	with	with	ADP
ejpam-5875	43	32	signature	signature	NOUN
ejpam-5875	43	33	(	(	PUNCT
ejpam-5875	43	34	0	0	NUM
ejpam-5875	43	35	,	,	PUNCT
ejpam-5875	43	36	0,+,+	0,+,+	PROPN
ejpam-5875	43	37	)	)	PUNCT
ejpam-5875	43	38	.	.	PUNCT
ejpam-5875	44	1	the	the	DET
ejpam-5875	44	2	absolute	absolute	ADJ
ejpam-5875	44	3	structure	structure	NOUN
ejpam-5875	44	4	of	of	ADP
ejpam-5875	44	5	this	this	DET
ejpam-5875	44	6	threedimensional	threedimensional	ADJ
ejpam-5875	44	7	galilean	galilean	PROPN
ejpam-5875	44	8	space	space	NOUN
ejpam-5875	44	9	can	can	AUX
ejpam-5875	44	10	be	be	AUX
ejpam-5875	44	11	represented	represent	VERB
ejpam-5875	44	12	by	by	ADP
ejpam-5875	44	13	an	an	DET
ejpam-5875	44	14	ordered	order	VERB
ejpam-5875	44	15	triple	triple	ADJ
ejpam-5875	44	16	{	{	PUNCT
ejpam-5875	44	17	ω	ω	PROPN
ejpam-5875	44	18	,	,	PUNCT
ejpam-5875	44	19	l	l	PROPN
ejpam-5875	44	20	,	,	PUNCT
ejpam-5875	44	21	j	j	NOUN
ejpam-5875	44	22	}	}	PUNCT
ejpam-5875	44	23	,	,	PUNCT
ejpam-5875	44	24	where	where	SCONJ
ejpam-5875	44	25	ω	ω	PROPN
ejpam-5875	44	26	signifies	signify	VERB
ejpam-5875	44	27	the	the	DET
ejpam-5875	44	28	absolute	absolute	ADJ
ejpam-5875	44	29	plane	plane	NOUN
ejpam-5875	44	30	within	within	ADP
ejpam-5875	44	31	g3	g3	PROPN
ejpam-5875	44	32	,	,	PUNCT
ejpam-5875	44	33	l	l	PROPN
ejpam-5875	44	34	denotes	denote	VERB
ejpam-5875	44	35	the	the	DET
ejpam-5875	44	36	absolute	absolute	ADJ
ejpam-5875	44	37	line	line	NOUN
ejpam-5875	44	38	contained	contain	VERB
ejpam-5875	44	39	in	in	ADP
ejpam-5875	44	40	ω	ω	NUM
ejpam-5875	44	41	,	,	PUNCT
ejpam-5875	44	42	and	and	CCONJ
ejpam-5875	44	43	j	j	PROPN
ejpam-5875	44	44	represents	represent	VERB
ejpam-5875	44	45	a	a	DET
ejpam-5875	44	46	fixed	fix	VERB
ejpam-5875	44	47	elliptic	elliptic	ADJ
ejpam-5875	44	48	involution	involution	NOUN
ejpam-5875	44	49	of	of	ADP
ejpam-5875	44	50	the	the	DET
ejpam-5875	44	51	points	point	NOUN
ejpam-5875	44	52	along	along	ADP
ejpam-5875	44	53	l.	l.	PROPN
ejpam-5875	44	54	a	a	DET
ejpam-5875	44	55	vector	vector	NOUN
ejpam-5875	44	56	p	p	NOUN
ejpam-5875	44	57	=	=	SYM
ejpam-5875	44	58	(	(	PUNCT
ejpam-5875	44	59	p1	p1	PROPN
ejpam-5875	44	60	,	,	PUNCT
ejpam-5875	44	61	p2	p2	NOUN
ejpam-5875	44	62	,	,	PUNCT
ejpam-5875	44	63	p3	p3	PROPN
ejpam-5875	44	64	)	)	PUNCT
ejpam-5875	44	65	in	in	ADP
ejpam-5875	44	66	g3	g3	PROPN
ejpam-5875	44	67	is	be	AUX
ejpam-5875	44	68	classified	classify	VERB
ejpam-5875	44	69	as	as	ADP
ejpam-5875	44	70	non	non	ADJ
ejpam-5875	44	71	-	-	ADJ
ejpam-5875	44	72	isotropic	isotropic	ADJ
ejpam-5875	44	73	if	if	SCONJ
ejpam-5875	44	74	p1	p1	PROPN
ejpam-5875	44	75	̸=	̸=	PROPN
ejpam-5875	44	76	0	0	NUM
ejpam-5875	44	77	;	;	PUNCT
ejpam-5875	44	78	otherwise	otherwise	ADV
ejpam-5875	44	79	,	,	PUNCT
ejpam-5875	44	80	it	it	PRON
ejpam-5875	44	81	is	be	AUX
ejpam-5875	44	82	termed	term	VERB
ejpam-5875	44	83	isotropic	isotropic	NOUN
ejpam-5875	44	84	[	[	X
ejpam-5875	44	85	31	31	NUM
ejpam-5875	44	86	,	,	PUNCT
ejpam-5875	44	87	32	32	NUM
ejpam-5875	44	88	]	]	PUNCT
ejpam-5875	44	89	.	.	PUNCT
ejpam-5875	45	1	vectors	vector	NOUN
ejpam-5875	45	2	of	of	ADP
ejpam-5875	45	3	the	the	DET
ejpam-5875	45	4	form	form	NOUN
ejpam-5875	45	5	p	p	X
ejpam-5875	45	6	=	=	X
ejpam-5875	45	7	(	(	PUNCT
ejpam-5875	45	8	1	1	NUM
ejpam-5875	45	9	,	,	PUNCT
ejpam-5875	45	10	p2	p2	NOUN
ejpam-5875	45	11	,	,	PUNCT
ejpam-5875	45	12	p3	p3	PROPN
ejpam-5875	45	13	)	)	PUNCT
ejpam-5875	45	14	are	be	AUX
ejpam-5875	45	15	considered	consider	VERB
ejpam-5875	45	16	unit	unit	ADJ
ejpam-5875	45	17	non	non	ADJ
ejpam-5875	45	18	-	-	ADJ
ejpam-5875	45	19	isotropic	isotropic	ADJ
ejpam-5875	45	20	vectors	vector	NOUN
ejpam-5875	45	21	.	.	PUNCT
ejpam-5875	46	1	the	the	DET
ejpam-5875	46	2	galilean	galilean	PROPN
ejpam-5875	46	3	metric	metric	PROPN
ejpam-5875	46	4	g	g	PROPN
ejpam-5875	46	5	for	for	ADP
ejpam-5875	46	6	vectors	vector	NOUN
ejpam-5875	46	7	p	p	NOUN
ejpam-5875	46	8	and	and	CCONJ
ejpam-5875	46	9	q	q	NOUN
ejpam-5875	46	10	in	in	ADP
ejpam-5875	46	11	g3	g3	PROPN
ejpam-5875	46	12	is	be	AUX
ejpam-5875	46	13	defined	define	VERB
ejpam-5875	46	14	as	as	ADP
ejpam-5875	46	15	a.	a.	NOUN
ejpam-5875	46	16	elsharkawy	elsharkawy	PROPN
ejpam-5875	46	17	,	,	PUNCT
ejpam-5875	46	18	n.	n.	NOUN
ejpam-5875	46	19	elsharkawy	elsharkawy	PROPN
ejpam-5875	46	20	/	/	SYM
ejpam-5875	46	21	eur	eur	PROPN
ejpam-5875	46	22	.	.	PUNCT
ejpam-5875	47	1	j.	j.	PROPN
ejpam-5875	47	2	pure	pure	PROPN
ejpam-5875	47	3	appl	appl	PROPN
ejpam-5875	47	4	.	.	PROPN
ejpam-5875	47	5	math	math	PROPN
ejpam-5875	47	6	,	,	PUNCT
ejpam-5875	47	7	18	18	NUM
ejpam-5875	47	8	(	(	PUNCT
ejpam-5875	47	9	2	2	NUM
ejpam-5875	47	10	)	)	PUNCT
ejpam-5875	47	11	(	(	PUNCT
ejpam-5875	47	12	2025	2025	NUM
ejpam-5875	47	13	)	)	PUNCT
ejpam-5875	47	14	,	,	PUNCT
ejpam-5875	47	15	5875	5875	NUM
ejpam-5875	47	16	3	3	NUM
ejpam-5875	47	17	of	of	ADP
ejpam-5875	47	18	15	15	NUM
ejpam-5875	47	19	follows	follow	VERB
ejpam-5875	47	20	:	:	PUNCT
ejpam-5875	47	21	g(p	g(p	PROPN
ejpam-5875	47	22	,	,	PUNCT
ejpam-5875	47	23	q	q	NOUN
ejpam-5875	47	24	)	)	PUNCT
ejpam-5875	47	25	=	=	PRON
ejpam-5875	47	26	{	{	PUNCT
ejpam-5875	47	27	p1q1	p1q1	NOUN
ejpam-5875	47	28	,	,	PUNCT
ejpam-5875	47	29	if	if	SCONJ
ejpam-5875	47	30	p1	p1	PROPN
ejpam-5875	47	31	̸=	̸=	PROPN
ejpam-5875	47	32	0	0	NUM
ejpam-5875	47	33	or	or	CCONJ
ejpam-5875	47	34	q1	q1	PROPN
ejpam-5875	47	35	̸=	̸=	PROPN
ejpam-5875	47	36	0	0	NUM
ejpam-5875	47	37	,	,	PUNCT
ejpam-5875	47	38	p2q2	p2q2	X
ejpam-5875	47	39	+	+	X
ejpam-5875	47	40	p3q3	p3q3	NOUN
ejpam-5875	47	41	,	,	PUNCT
ejpam-5875	47	42	if	if	SCONJ
ejpam-5875	47	43	p1	p1	PROPN
ejpam-5875	47	44	=	=	SYM
ejpam-5875	47	45	0	0	NUM
ejpam-5875	47	46	and	and	CCONJ
ejpam-5875	47	47	q1	q1	PROPN
ejpam-5875	47	48	=	=	SYM
ejpam-5875	47	49	0	0	X
ejpam-5875	47	50	.	.	PUNCT
ejpam-5875	48	1	consequently	consequently	ADV
ejpam-5875	48	2	,	,	PUNCT
ejpam-5875	48	3	the	the	DET
ejpam-5875	48	4	galilean	galilean	PROPN
ejpam-5875	48	5	norm	norm	NOUN
ejpam-5875	48	6	of	of	ADP
ejpam-5875	48	7	the	the	DET
ejpam-5875	48	8	vector	vector	NOUN
ejpam-5875	48	9	q	q	NOUN
ejpam-5875	48	10	is	be	AUX
ejpam-5875	48	11	given	give	VERB
ejpam-5875	48	12	by	by	ADP
ejpam-5875	48	13	:	:	PUNCT
ejpam-5875	48	14	∥q∥	∥q∥	PROPN
ejpam-5875	48	15	=	=	PRON
ejpam-5875	48	16	{	{	PUNCT
ejpam-5875	48	17	|q1|	|q1|	ADV
ejpam-5875	48	18	,	,	PUNCT
ejpam-5875	48	19	if	if	SCONJ
ejpam-5875	48	20	q1	q1	VERB
ejpam-5875	48	21	̸=	̸=	PROPN
ejpam-5875	48	22	0,√	0,√	NUM
ejpam-5875	48	23	q22	q22	NOUN
ejpam-5875	48	24	+	+	CCONJ
ejpam-5875	48	25	q23	q23	PROPN
ejpam-5875	48	26	,	,	PUNCT
ejpam-5875	48	27	if	if	SCONJ
ejpam-5875	48	28	q1	q1	PROPN
ejpam-5875	48	29	=	=	SYM
ejpam-5875	48	30	0	0	NUM
ejpam-5875	48	31	.	.	PUNCT
ejpam-5875	49	1	the	the	DET
ejpam-5875	49	2	galilean	galilean	PROPN
ejpam-5875	49	3	vector	vector	NOUN
ejpam-5875	49	4	product	product	NOUN
ejpam-5875	49	5	of	of	ADP
ejpam-5875	49	6	vectors	vector	NOUN
ejpam-5875	49	7	p	p	NOUN
ejpam-5875	49	8	and	and	CCONJ
ejpam-5875	49	9	q	q	NOUN
ejpam-5875	49	10	is	be	AUX
ejpam-5875	49	11	defined	define	VERB
ejpam-5875	49	12	as	as	ADP
ejpam-5875	49	13	:	:	PUNCT
ejpam-5875	49	14	p×	p×	NOUN
ejpam-5875	49	15	q	q	NOUN
ejpam-5875	49	16	=	=	PUNCT
ejpam-5875	49	17			X
ejpam-5875	49	18	∣∣∣∣∣∣∣	∣∣∣∣∣∣∣	X
ejpam-5875	49	19	0	0	NUM
ejpam-5875	49	20	e2	e2	PROPN
ejpam-5875	49	21	e3	e3	NOUN
ejpam-5875	49	22	p1	p1	NOUN
ejpam-5875	49	23	p2	p2	PROPN
ejpam-5875	49	24	p3	p3	PROPN
ejpam-5875	49	25	q1	q1	PROPN
ejpam-5875	49	26	q2	q2	PROPN
ejpam-5875	49	27	q3	q3	PROPN
ejpam-5875	49	28	∣∣∣∣∣∣∣	∣∣∣∣∣∣∣	PROPN
ejpam-5875	49	29	,	,	PUNCT
ejpam-5875	49	30	if	if	SCONJ
ejpam-5875	49	31	p1	p1	PROPN
ejpam-5875	49	32	̸=	̸=	PROPN
ejpam-5875	49	33	0	0	NUM
ejpam-5875	49	34	or	or	CCONJ
ejpam-5875	49	35	q1	q1	PROPN
ejpam-5875	49	36	̸=	̸=	PROPN
ejpam-5875	49	37	0	0	NUM
ejpam-5875	49	38	,	,	PUNCT
ejpam-5875	49	39	∣∣∣∣∣∣∣	∣∣∣∣∣∣∣	PROPN
ejpam-5875	49	40	e1	e1	PROPN
ejpam-5875	49	41	e2	e2	PROPN
ejpam-5875	49	42	e3	e3	NOUN
ejpam-5875	49	43	p1	p1	NOUN
ejpam-5875	49	44	p2	p2	PROPN
ejpam-5875	49	45	p3	p3	PROPN
ejpam-5875	49	46	q1	q1	PROPN
ejpam-5875	49	47	q2	q2	PROPN
ejpam-5875	49	48	q3	q3	PROPN
ejpam-5875	49	49	∣∣∣∣∣∣∣	∣∣∣∣∣∣∣	PROPN
ejpam-5875	49	50	,	,	PUNCT
ejpam-5875	49	51	if	if	SCONJ
ejpam-5875	49	52	p1	p1	PROPN
ejpam-5875	49	53	=	=	SYM
ejpam-5875	49	54	0	0	NUM
ejpam-5875	49	55	and	and	CCONJ
ejpam-5875	49	56	q1	q1	PROPN
ejpam-5875	49	57	=	=	SYM
ejpam-5875	49	58	0	0	NUM
ejpam-5875	49	59	,	,	PUNCT
ejpam-5875	49	60	where	where	SCONJ
ejpam-5875	49	61	(	(	PUNCT
ejpam-5875	49	62	e1	e1	PROPN
ejpam-5875	49	63	,	,	PUNCT
ejpam-5875	49	64	e2	e2	PROPN
ejpam-5875	49	65	,	,	PUNCT
ejpam-5875	49	66	e3	e3	NOUN
ejpam-5875	49	67	)	)	PUNCT
ejpam-5875	49	68	denotes	denote	VERB
ejpam-5875	49	69	the	the	DET
ejpam-5875	49	70	standard	standard	ADJ
ejpam-5875	49	71	basis	basis	NOUN
ejpam-5875	49	72	of	of	ADP
ejpam-5875	49	73	r3	r3	PROPN
ejpam-5875	50	1	[	[	X
ejpam-5875	50	2	11	11	NUM
ejpam-5875	50	3	,	,	PUNCT
ejpam-5875	50	4	35	35	NUM
ejpam-5875	50	5	]	]	PUNCT
ejpam-5875	50	6	.	.	PUNCT
ejpam-5875	51	1	in	in	ADP
ejpam-5875	51	2	g3	g3	PROPN
ejpam-5875	51	3	,	,	PUNCT
ejpam-5875	51	4	a	a	DET
ejpam-5875	51	5	curve	curve	NOUN
ejpam-5875	51	6	is	be	AUX
ejpam-5875	51	7	defined	define	VERB
ejpam-5875	51	8	as	as	ADP
ejpam-5875	51	9	a	a	DET
ejpam-5875	51	10	mapping	mapping	NOUN
ejpam-5875	51	11	from	from	ADP
ejpam-5875	51	12	an	an	DET
ejpam-5875	51	13	open	open	ADJ
ejpam-5875	51	14	interval	interval	NOUN
ejpam-5875	51	15	j	j	PROPN
ejpam-5875	51	16	in	in	ADP
ejpam-5875	51	17	r	r	NOUN
ejpam-5875	51	18	to	to	ADP
ejpam-5875	51	19	g3	g3	NOUN
ejpam-5875	51	20	,	,	PUNCT
ejpam-5875	51	21	represented	represent	VERB
ejpam-5875	51	22	as	as	ADP
ejpam-5875	51	23	:	:	PUNCT
ejpam-5875	51	24	γ	γ	X
ejpam-5875	51	25	:	:	PUNCT
ejpam-5875	51	26	j	j	PROPN
ejpam-5875	51	27	→	→	SYM
ejpam-5875	51	28	g3	g3	PROPN
ejpam-5875	51	29	,	,	PUNCT
ejpam-5875	51	30	t	t	NOUN
ejpam-5875	51	31	7→	7→	NUM
ejpam-5875	51	32	γ(t	γ(t	NOUN
ejpam-5875	51	33	)	)	PUNCT
ejpam-5875	52	1	=	=	SYM
ejpam-5875	52	2	(	(	PUNCT
ejpam-5875	52	3	x(t	x(t	PROPN
ejpam-5875	52	4	)	)	PUNCT
ejpam-5875	52	5	,	,	PUNCT
ejpam-5875	52	6	y(t	y(t	PROPN
ejpam-5875	52	7	)	)	PUNCT
ejpam-5875	52	8	,	,	PUNCT
ejpam-5875	52	9	z(t	z(t	PROPN
ejpam-5875	52	10	)	)	PUNCT
ejpam-5875	52	11	)	)	PUNCT
ejpam-5875	52	12	.	.	PUNCT
ejpam-5875	53	1	a	a	DET
ejpam-5875	53	2	curve	curve	NOUN
ejpam-5875	53	3	is	be	AUX
ejpam-5875	53	4	considered	consider	VERB
ejpam-5875	53	5	admissible	admissible	ADJ
ejpam-5875	53	6	if	if	SCONJ
ejpam-5875	53	7	it	it	PRON
ejpam-5875	53	8	has	have	VERB
ejpam-5875	53	9	no	no	DET
ejpam-5875	53	10	inflection	inflection	NOUN
ejpam-5875	53	11	points	point	NOUN
ejpam-5875	53	12	(	(	PUNCT
ejpam-5875	53	13	i.e.	i.e.	X
ejpam-5875	53	14	,	,	PUNCT
ejpam-5875	53	15	γ̇(t	γ̇(t	ADJ
ejpam-5875	53	16	)	)	PUNCT
ejpam-5875	53	17	×	×	PROPN
ejpam-5875	53	18	γ̈(t	γ̈(t	PROPN
ejpam-5875	53	19	)	)	PUNCT
ejpam-5875	53	20	̸=	̸=	PROPN
ejpam-5875	53	21	0	0	NUM
ejpam-5875	53	22	)	)	PUNCT
ejpam-5875	53	23	and	and	CCONJ
ejpam-5875	53	24	does	do	AUX
ejpam-5875	53	25	not	not	PART
ejpam-5875	53	26	possess	possess	VERB
ejpam-5875	53	27	isotropic	isotropic	ADJ
ejpam-5875	53	28	tangents	tangent	NOUN
ejpam-5875	53	29	(	(	PUNCT
ejpam-5875	53	30	i.e.	i.e.	X
ejpam-5875	53	31	,	,	PUNCT
ejpam-5875	53	32	ẋ(t	ẋ(t	NOUN
ejpam-5875	53	33	)	)	PUNCT
ejpam-5875	53	34	̸=	̸=	PROPN
ejpam-5875	53	35	0	0	NUM
ejpam-5875	53	36	for	for	ADP
ejpam-5875	53	37	all	all	DET
ejpam-5875	53	38	t	t	NOUN
ejpam-5875	53	39	∈	∈	PROPN
ejpam-5875	53	40	j	j	PROPN
ejpam-5875	53	41	)	)	PUNCT
ejpam-5875	54	1	[	[	X
ejpam-5875	54	2	11	11	NUM
ejpam-5875	54	3	,	,	PUNCT
ejpam-5875	54	4	31	31	NUM
ejpam-5875	54	5	]	]	PUNCT
ejpam-5875	54	6	.	.	PUNCT
ejpam-5875	55	1	for	for	ADP
ejpam-5875	55	2	an	an	DET
ejpam-5875	55	3	admissible	admissible	ADJ
ejpam-5875	55	4	differentiable	differentiable	ADJ
ejpam-5875	55	5	curve	curve	NOUN
ejpam-5875	55	6	γ	γ	X
ejpam-5875	55	7	:	:	PUNCT
ejpam-5875	55	8	j	j	PROPN
ejpam-5875	55	9	⊂	⊂	PROPN
ejpam-5875	55	10	r	r	PROPN
ejpam-5875	55	11	→	→	SYM
ejpam-5875	55	12	g3	g3	PROPN
ejpam-5875	55	13	,	,	PUNCT
ejpam-5875	55	14	parameterized	parameterize	VERB
ejpam-5875	55	15	by	by	ADP
ejpam-5875	55	16	the	the	DET
ejpam-5875	55	17	galilean	galilean	PROPN
ejpam-5875	55	18	invariant	invariant	PROPN
ejpam-5875	55	19	arc	arc	NOUN
ejpam-5875	55	20	length	length	NOUN
ejpam-5875	55	21	s	s	PROPN
ejpam-5875	55	22	,	,	PUNCT
ejpam-5875	55	23	the	the	DET
ejpam-5875	55	24	curve	curve	NOUN
ejpam-5875	55	25	can	can	AUX
ejpam-5875	55	26	be	be	AUX
ejpam-5875	55	27	expressed	express	VERB
ejpam-5875	55	28	as	as	ADP
ejpam-5875	55	29	:	:	PUNCT
ejpam-5875	55	30	γ(s	γ(	NOUN
ejpam-5875	55	31	)	)	PUNCT
ejpam-5875	56	1	=	=	PUNCT
ejpam-5875	56	2	(	(	PUNCT
ejpam-5875	56	3	s	s	PROPN
ejpam-5875	56	4	,	,	PUNCT
ejpam-5875	56	5	y(s	y(s	PROPN
ejpam-5875	56	6	)	)	PUNCT
ejpam-5875	56	7	,	,	PUNCT
ejpam-5875	56	8	z(s	z(s	PROPN
ejpam-5875	56	9	)	)	PUNCT
ejpam-5875	56	10	)	)	PUNCT
ejpam-5875	56	11	.	.	PUNCT
ejpam-5875	57	1	the	the	DET
ejpam-5875	57	2	curvature	curvature	NOUN
ejpam-5875	57	3	κ(s	κ(s	PROPN
ejpam-5875	57	4	)	)	PUNCT
ejpam-5875	57	5	and	and	CCONJ
ejpam-5875	57	6	torsion	torsion	NOUN
ejpam-5875	57	7	τ(s	τ(s	NOUN
ejpam-5875	57	8	)	)	PUNCT
ejpam-5875	57	9	of	of	ADP
ejpam-5875	57	10	the	the	DET
ejpam-5875	57	11	curve	curve	NOUN
ejpam-5875	57	12	γ(s	γ(	NOUN
ejpam-5875	57	13	)	)	PUNCT
ejpam-5875	57	14	are	be	AUX
ejpam-5875	57	15	given	give	VERB
ejpam-5875	57	16	by	by	ADP
ejpam-5875	57	17	the	the	DET
ejpam-5875	57	18	formulas	formula	NOUN
ejpam-5875	57	19	:	:	PUNCT
ejpam-5875	57	20	κ(s	κ(s	NOUN
ejpam-5875	57	21	)	)	PUNCT
ejpam-5875	57	22	=	=	SYM
ejpam-5875	57	23	∥γ′′(s)∥	∥γ′′(s)∥	PROPN
ejpam-5875	57	24	=	=	SYM
ejpam-5875	57	25	√	√	NUM
ejpam-5875	57	26	y′′2(s	y′′2(s	NOUN
ejpam-5875	57	27	)	)	PUNCT
ejpam-5875	57	28	+	+	CCONJ
ejpam-5875	57	29	z′′2(s	z′′2(s	PROPN
ejpam-5875	57	30	)	)	PUNCT
ejpam-5875	57	31	,	,	PUNCT
ejpam-5875	57	32	τ(s	τ(s	NOUN
ejpam-5875	57	33	)	)	PUNCT
ejpam-5875	57	34	=	=	SYM
ejpam-5875	57	35	det(γ′(s	det(γ′(s	PROPN
ejpam-5875	57	36	)	)	PUNCT
ejpam-5875	57	37	,	,	PUNCT
ejpam-5875	57	38	γ′′(s	γ′′(s	NOUN
ejpam-5875	57	39	)	)	PUNCT
ejpam-5875	57	40	,	,	PUNCT
ejpam-5875	57	41	γ′′′(s	γ′′′(s	PROPN
ejpam-5875	57	42	)	)	PUNCT
ejpam-5875	57	43	)	)	PUNCT
ejpam-5875	57	44	κ2(s	κ2(s	PROPN
ejpam-5875	57	45	)	)	PUNCT
ejpam-5875	57	46	.	.	PUNCT
ejpam-5875	58	1	the	the	DET
ejpam-5875	58	2	moving	move	VERB
ejpam-5875	58	3	frenet	frenet	ADJ
ejpam-5875	58	4	frame	frame	NOUN
ejpam-5875	58	5	{	{	PUNCT
ejpam-5875	58	6	t	t	PROPN
ejpam-5875	58	7	(	(	PUNCT
ejpam-5875	58	8	s	s	NOUN
ejpam-5875	58	9	)	)	PUNCT
ejpam-5875	58	10	,	,	PUNCT
ejpam-5875	58	11	n(s	n(s	PROPN
ejpam-5875	58	12	)	)	PUNCT
ejpam-5875	58	13	,	,	PUNCT
ejpam-5875	58	14	b(s	b(	NOUN
ejpam-5875	58	15	)	)	PUNCT
ejpam-5875	58	16	}	}	PUNCT
ejpam-5875	58	17	for	for	ADP
ejpam-5875	58	18	the	the	DET
ejpam-5875	58	19	curve	curve	NOUN
ejpam-5875	58	20	γ(s	γ(	NOUN
ejpam-5875	58	21	)	)	PUNCT
ejpam-5875	58	22	is	be	AUX
ejpam-5875	58	23	defined	define	VERB
ejpam-5875	58	24	as	as	ADP
ejpam-5875	58	25	:	:	PUNCT
ejpam-5875	58	26	t	t	PROPN
ejpam-5875	58	27	(	(	PUNCT
ejpam-5875	58	28	s	s	X
ejpam-5875	58	29	)	)	PUNCT
ejpam-5875	58	30	=	=	SYM
ejpam-5875	58	31	γ′(s	γ′(s	PROPN
ejpam-5875	58	32	)	)	PUNCT
ejpam-5875	58	33	=	=	PUNCT
ejpam-5875	59	1	(	(	PUNCT
ejpam-5875	59	2	1	1	NUM
ejpam-5875	59	3	,	,	PUNCT
ejpam-5875	59	4	y′(s	y′(s	INTJ
ejpam-5875	59	5	)	)	PUNCT
ejpam-5875	59	6	,	,	PUNCT
ejpam-5875	59	7	z′(s	z′(s	PROPN
ejpam-5875	59	8	)	)	PUNCT
ejpam-5875	59	9	)	)	PUNCT
ejpam-5875	59	10	,	,	PUNCT
ejpam-5875	59	11	n(s	n(s	PROPN
ejpam-5875	59	12	)	)	PUNCT
ejpam-5875	59	13	=	=	SYM
ejpam-5875	59	14	1	1	NUM
ejpam-5875	59	15	κ(s	κ(s	NOUN
ejpam-5875	59	16	)	)	PUNCT
ejpam-5875	59	17	γ′′(s	γ′′(s	NOUN
ejpam-5875	59	18	)	)	PUNCT
ejpam-5875	59	19	=	=	SYM
ejpam-5875	59	20	1	1	NUM
ejpam-5875	59	21	κ(s	κ(s	NOUN
ejpam-5875	59	22	)	)	PUNCT
ejpam-5875	59	23	(	(	PUNCT
ejpam-5875	59	24	0	0	NUM
ejpam-5875	59	25	,	,	PUNCT
ejpam-5875	59	26	y′′(s	y′′(s	PRON
ejpam-5875	59	27	)	)	PUNCT
ejpam-5875	59	28	,	,	PUNCT
ejpam-5875	59	29	z′′(s	z′′(s	NOUN
ejpam-5875	59	30	)	)	PUNCT
ejpam-5875	59	31	)	)	PUNCT
ejpam-5875	59	32	,	,	PUNCT
ejpam-5875	59	33	a.	a.	NOUN
ejpam-5875	59	34	elsharkawy	elsharkawy	PROPN
ejpam-5875	59	35	,	,	PUNCT
ejpam-5875	59	36	n.	n.	NOUN
ejpam-5875	59	37	elsharkawy	elsharkawy	PROPN
ejpam-5875	59	38	/	/	SYM
ejpam-5875	59	39	eur	eur	PROPN
ejpam-5875	59	40	.	.	PUNCT
ejpam-5875	60	1	j.	j.	PROPN
ejpam-5875	60	2	pure	pure	PROPN
ejpam-5875	60	3	appl	appl	PROPN
ejpam-5875	60	4	.	.	PROPN
ejpam-5875	60	5	math	math	PROPN
ejpam-5875	60	6	,	,	PUNCT
ejpam-5875	60	7	18	18	NUM
ejpam-5875	60	8	(	(	PUNCT
ejpam-5875	60	9	2	2	NUM
ejpam-5875	60	10	)	)	PUNCT
ejpam-5875	60	11	(	(	PUNCT
ejpam-5875	60	12	2025	2025	NUM
ejpam-5875	60	13	)	)	PUNCT
ejpam-5875	60	14	,	,	PUNCT
ejpam-5875	60	15	5875	5875	NUM
ejpam-5875	60	16	4	4	NUM
ejpam-5875	60	17	of	of	ADP
ejpam-5875	60	18	15	15	NUM
ejpam-5875	60	19	b(s	b(	NOUN
ejpam-5875	60	20	)	)	PUNCT
ejpam-5875	61	1	=	=	SYM
ejpam-5875	61	2	t	t	PROPN
ejpam-5875	61	3	(	(	PUNCT
ejpam-5875	61	4	s)×n(s	s)×n(s	NOUN
ejpam-5875	61	5	)	)	PUNCT
ejpam-5875	61	6	=	=	SYM
ejpam-5875	61	7	1	1	NUM
ejpam-5875	61	8	κ(s	κ(s	NOUN
ejpam-5875	61	9	)	)	PUNCT
ejpam-5875	61	10	(	(	PUNCT
ejpam-5875	61	11	0,−z′′(s	0,−z′′(s	PROPN
ejpam-5875	61	12	)	)	PUNCT
ejpam-5875	61	13	,	,	PUNCT
ejpam-5875	61	14	y′′(s	y′′(s	NOUN
ejpam-5875	61	15	)	)	PUNCT
ejpam-5875	61	16	)	)	PUNCT
ejpam-5875	61	17	.	.	PUNCT
ejpam-5875	62	1	finally	finally	ADV
ejpam-5875	62	2	,	,	PUNCT
ejpam-5875	62	3	the	the	DET
ejpam-5875	62	4	frenet	frenet	ADJ
ejpam-5875	62	5	derivative	derivative	ADJ
ejpam-5875	62	6	formulas	formula	NOUN
ejpam-5875	62	7	can	can	AUX
ejpam-5875	62	8	be	be	AUX
ejpam-5875	62	9	represented	represent	VERB
ejpam-5875	62	10	in	in	ADP
ejpam-5875	62	11	matrix	matrix	NOUN
ejpam-5875	62	12	form	form	NOUN
ejpam-5875	62	13	as	as	SCONJ
ejpam-5875	62	14	follows	follow	VERB
ejpam-5875	62	15	:	:	PUNCT
ejpam-5875	62	16	t	t	NOUN
ejpam-5875	62	17	′	′	NUM
ejpam-5875	63	1	n	n	NOUN
ejpam-5875	63	2	′	′	NUM
ejpam-5875	63	3	b′	b′	NUM
ejpam-5875	63	4			PROPN
ejpam-5875	63	5	=	=	SYM
ejpam-5875	63	6	0	0	ADP
ejpam-5875	63	7	κ(s	κ(s	PROPN
ejpam-5875	63	8	)	)	PUNCT
ejpam-5875	63	9	0	0	NUM
ejpam-5875	63	10	0	0	NUM
ejpam-5875	63	11	0	0	NUM
ejpam-5875	63	12	τ(s	τ(s	NOUN
ejpam-5875	63	13	)	)	PUNCT
ejpam-5875	63	14	0	0	PUNCT
ejpam-5875	64	1	−τ(s	−τ(s	NOUN
ejpam-5875	64	2	)	)	PUNCT
ejpam-5875	64	3	0	0	NUM
ejpam-5875	64	4	t	t	VERB
ejpam-5875	64	5	n	n	PRON
ejpam-5875	64	6	b	b	PROPN
ejpam-5875	64	7			PROPN
ejpam-5875	64	8	.	.	PUNCT
ejpam-5875	65	1	(	(	PUNCT
ejpam-5875	65	2	2.1	2.1	NUM
ejpam-5875	65	3	)	)	PUNCT
ejpam-5875	65	4	3	3	NUM
ejpam-5875	65	5	.	.	X
ejpam-5875	66	1	quasi	quasi	ADJ
ejpam-5875	66	2	-	-	NOUN
ejpam-5875	66	3	frame	frame	ADJ
ejpam-5875	66	4	and	and	CCONJ
ejpam-5875	66	5	quasi	quasi	ADJ
ejpam-5875	66	6	equations	equation	NOUN
ejpam-5875	66	7	this	this	DET
ejpam-5875	66	8	section	section	NOUN
ejpam-5875	66	9	delves	delve	VERB
ejpam-5875	66	10	into	into	ADP
ejpam-5875	66	11	the	the	DET
ejpam-5875	66	12	concept	concept	NOUN
ejpam-5875	66	13	of	of	ADP
ejpam-5875	66	14	the	the	DET
ejpam-5875	66	15	quasi	quasi	NOUN
ejpam-5875	66	16	-	-	NOUN
ejpam-5875	66	17	frame	frame	NOUN
ejpam-5875	66	18	(	(	PUNCT
ejpam-5875	66	19	q	q	NOUN
ejpam-5875	66	20	-	-	PUNCT
ejpam-5875	66	21	frame	frame	NOUN
ejpam-5875	66	22	)	)	PUNCT
ejpam-5875	66	23	and	and	CCONJ
ejpam-5875	66	24	its	its	PRON
ejpam-5875	66	25	relationship	relationship	NOUN
ejpam-5875	66	26	with	with	ADP
ejpam-5875	66	27	the	the	DET
ejpam-5875	66	28	classical	classical	ADJ
ejpam-5875	66	29	frenet	frenet	ADJ
ejpam-5875	66	30	frame	frame	NOUN
ejpam-5875	66	31	,	,	PUNCT
ejpam-5875	66	32	as	as	ADV
ejpam-5875	66	33	well	well	ADV
ejpam-5875	66	34	as	as	ADP
ejpam-5875	66	35	an	an	DET
ejpam-5875	66	36	examination	examination	NOUN
ejpam-5875	66	37	of	of	ADP
ejpam-5875	66	38	the	the	DET
ejpam-5875	66	39	quasi	quasi	ADJ
ejpam-5875	66	40	equations	equation	NOUN
ejpam-5875	66	41	within	within	ADP
ejpam-5875	66	42	the	the	DET
ejpam-5875	66	43	context	context	NOUN
ejpam-5875	66	44	of	of	ADP
ejpam-5875	66	45	galilean	galilean	PROPN
ejpam-5875	66	46	three	three	NUM
ejpam-5875	66	47	-	-	PUNCT
ejpam-5875	66	48	dimensional	dimensional	ADJ
ejpam-5875	66	49	space	space	NOUN
ejpam-5875	66	50	.	.	PUNCT
ejpam-5875	67	1	let	let	VERB
ejpam-5875	67	2	α(s	α(s	PROPN
ejpam-5875	67	3	)	)	PUNCT
ejpam-5875	68	1	represent	represent	VERB
ejpam-5875	68	2	a	a	DET
ejpam-5875	68	3	curve	curve	NOUN
ejpam-5875	68	4	in	in	ADP
ejpam-5875	68	5	g3	g3	PROPN
ejpam-5875	68	6	.	.	PUNCT
ejpam-5875	69	1	the	the	DET
ejpam-5875	69	2	q	q	NOUN
ejpam-5875	69	3	-	-	PUNCT
ejpam-5875	69	4	frame	frame	NOUN
ejpam-5875	69	5	is	be	AUX
ejpam-5875	69	6	established	establish	VERB
ejpam-5875	69	7	using	use	VERB
ejpam-5875	69	8	three	three	NUM
ejpam-5875	69	9	orthonormal	orthonormal	ADJ
ejpam-5875	69	10	vectors	vector	NOUN
ejpam-5875	69	11	:	:	PUNCT
ejpam-5875	69	12	t	t	PROPN
ejpam-5875	69	13	(	(	PUNCT
ejpam-5875	69	14	s	s	PROPN
ejpam-5875	69	15	)	)	PUNCT
ejpam-5875	69	16	,	,	PUNCT
ejpam-5875	69	17	the	the	DET
ejpam-5875	69	18	unit	unit	NOUN
ejpam-5875	69	19	tangent	tangent	PROPN
ejpam-5875	69	20	vector	vector	PROPN
ejpam-5875	69	21	;	;	PUNCT
ejpam-5875	69	22	nq(s	nq(s	NUM
ejpam-5875	69	23	)	)	PUNCT
ejpam-5875	69	24	,	,	PUNCT
ejpam-5875	69	25	the	the	DET
ejpam-5875	69	26	unit	unit	NOUN
ejpam-5875	69	27	q	q	ADJ
ejpam-5875	69	28	-	-	ADJ
ejpam-5875	69	29	normal	normal	ADJ
ejpam-5875	69	30	vector	vector	NOUN
ejpam-5875	69	31	;	;	PUNCT
ejpam-5875	69	32	and	and	CCONJ
ejpam-5875	69	33	bq(s	bq(s	NUM
ejpam-5875	69	34	)	)	PUNCT
ejpam-5875	69	35	,	,	PUNCT
ejpam-5875	69	36	the	the	DET
ejpam-5875	69	37	unit	unit	NOUN
ejpam-5875	69	38	q	q	ADJ
ejpam-5875	69	39	-	-	PUNCT
ejpam-5875	69	40	binormal	binormal	ADJ
ejpam-5875	69	41	vector	vector	NOUN
ejpam-5875	69	42	.	.	PUNCT
ejpam-5875	70	1	the	the	DET
ejpam-5875	70	2	q	q	NOUN
ejpam-5875	70	3	-	-	PUNCT
ejpam-5875	70	4	frame	frame	NOUN
ejpam-5875	70	5	{	{	PUNCT
ejpam-5875	70	6	t	t	PROPN
ejpam-5875	70	7	(	(	PUNCT
ejpam-5875	70	8	s	s	NOUN
ejpam-5875	70	9	)	)	PUNCT
ejpam-5875	70	10	,	,	PUNCT
ejpam-5875	70	11	nq(s	nq(s	NUM
ejpam-5875	70	12	)	)	PUNCT
ejpam-5875	70	13	,	,	PUNCT
ejpam-5875	70	14	bq(s	bq(s	NOUN
ejpam-5875	70	15	)	)	PUNCT
ejpam-5875	70	16	}	}	PUNCT
ejpam-5875	70	17	is	be	AUX
ejpam-5875	70	18	defined	define	VERB
ejpam-5875	70	19	as	as	SCONJ
ejpam-5875	70	20	follows	follow	VERB
ejpam-5875	70	21	:	:	PUNCT
ejpam-5875	70	22	t	t	NOUN
ejpam-5875	70	23	=	=	PUNCT
ejpam-5875	70	24	α′	α′	NUM
ejpam-5875	70	25	∥α′∥	∥α′∥	PUNCT
ejpam-5875	70	26	,	,	PUNCT
ejpam-5875	70	27	nq	nq	PROPN
ejpam-5875	70	28	=	=	SYM
ejpam-5875	70	29	t	t	PROPN
ejpam-5875	70	30	×	×	NOUN
ejpam-5875	70	31	z	z	NOUN
ejpam-5875	70	32	∥t	∥t	PROPN
ejpam-5875	70	33	×	×	NOUN
ejpam-5875	70	34	z∥	z∥	NOUN
ejpam-5875	70	35	,	,	PUNCT
ejpam-5875	70	36	bq	bq	PROPN
ejpam-5875	70	37	=	=	SYM
ejpam-5875	70	38	t	t	PROPN
ejpam-5875	70	39	×nq	×nq	PROPN
ejpam-5875	70	40	,	,	PUNCT
ejpam-5875	70	41	(	(	PUNCT
ejpam-5875	70	42	3.1	3.1	NUM
ejpam-5875	70	43	)	)	PUNCT
ejpam-5875	70	44	where	where	SCONJ
ejpam-5875	70	45	z	z	NOUN
ejpam-5875	70	46	is	be	AUX
ejpam-5875	70	47	a	a	DET
ejpam-5875	70	48	projection	projection	NOUN
ejpam-5875	70	49	vector	vector	NOUN
ejpam-5875	70	50	that	that	PRON
ejpam-5875	70	51	can	can	AUX
ejpam-5875	70	52	be	be	AUX
ejpam-5875	70	53	chosen	choose	VERB
ejpam-5875	70	54	from	from	ADP
ejpam-5875	70	55	(	(	PUNCT
ejpam-5875	70	56	1	1	NUM
ejpam-5875	70	57	,	,	PUNCT
ejpam-5875	70	58	0	0	NUM
ejpam-5875	70	59	,	,	PUNCT
ejpam-5875	70	60	0	0	NUM
ejpam-5875	70	61	)	)	PUNCT
ejpam-5875	70	62	,	,	PUNCT
ejpam-5875	70	63	(	(	PUNCT
ejpam-5875	70	64	0	0	NUM
ejpam-5875	70	65	,	,	PUNCT
ejpam-5875	70	66	1	1	NUM
ejpam-5875	70	67	,	,	PUNCT
ejpam-5875	70	68	0	0	NUM
ejpam-5875	70	69	)	)	PUNCT
ejpam-5875	70	70	,	,	PUNCT
ejpam-5875	70	71	or	or	CCONJ
ejpam-5875	70	72	(	(	PUNCT
ejpam-5875	70	73	0	0	NUM
ejpam-5875	70	74	,	,	PUNCT
ejpam-5875	70	75	0	0	NUM
ejpam-5875	70	76	,	,	PUNCT
ejpam-5875	70	77	1	1	NUM
ejpam-5875	70	78	)	)	PUNCT
ejpam-5875	70	79	.	.	PUNCT
ejpam-5875	71	1	denote	denote	VERB
ejpam-5875	71	2	the	the	DET
ejpam-5875	71	3	standard	standard	ADJ
ejpam-5875	71	4	frenet	frenet	ADJ
ejpam-5875	71	5	frame	frame	NOUN
ejpam-5875	71	6	by	by	ADP
ejpam-5875	71	7	{	{	PUNCT
ejpam-5875	71	8	t	t	PROPN
ejpam-5875	71	9	,	,	PUNCT
ejpam-5875	71	10	n	n	CCONJ
ejpam-5875	71	11	,	,	PUNCT
ejpam-5875	71	12	b	b	NOUN
ejpam-5875	71	13	}	}	PUNCT
ejpam-5875	71	14	,	,	PUNCT
ejpam-5875	71	15	and	and	CCONJ
ejpam-5875	71	16	let	let	VERB
ejpam-5875	71	17	θ(s	θ(s	NOUN
ejpam-5875	71	18	)	)	PUNCT
ejpam-5875	71	19	denote	denote	VERB
ejpam-5875	71	20	the	the	DET
ejpam-5875	71	21	angle	angle	NOUN
ejpam-5875	71	22	between	between	ADP
ejpam-5875	71	23	the	the	DET
ejpam-5875	71	24	vectors	vector	NOUN
ejpam-5875	71	25	n	n	PRON
ejpam-5875	71	26	and	and	CCONJ
ejpam-5875	71	27	nq	nq	PROPN
ejpam-5875	71	28	.	.	PROPN
ejpam-5875	72	1	we	we	PRON
ejpam-5875	72	2	can	can	AUX
ejpam-5875	72	3	express	express	VERB
ejpam-5875	72	4	nq	nq	PROPN
ejpam-5875	72	5	and	and	CCONJ
ejpam-5875	72	6	bq	bq	X
ejpam-5875	72	7	in	in	ADP
ejpam-5875	72	8	terms	term	NOUN
ejpam-5875	72	9	of	of	ADP
ejpam-5875	72	10	n	n	PRON
ejpam-5875	72	11	and	and	CCONJ
ejpam-5875	72	12	b	b	NOUN
ejpam-5875	72	13	as	as	SCONJ
ejpam-5875	72	14	follows	follow	VERB
ejpam-5875	72	15	:	:	PUNCT
ejpam-5875	73	1	nq	nq	PROPN
ejpam-5875	73	2	=	=	PUNCT
ejpam-5875	73	3	cos(θ)n	cos(θ)n	NOUN
ejpam-5875	73	4	+	+	CCONJ
ejpam-5875	73	5	sin(θ)b	sin(θ)b	VERB
ejpam-5875	73	6	,	,	PUNCT
ejpam-5875	73	7	(	(	PUNCT
ejpam-5875	73	8	3.2	3.2	NUM
ejpam-5875	73	9	)	)	PUNCT
ejpam-5875	73	10	bq	bq	NOUN
ejpam-5875	73	11	=	=	SYM
ejpam-5875	73	12	−	−	PROPN
ejpam-5875	73	13	sin(θ)n	sin(θ)n	NOUN
ejpam-5875	73	14	+	+	CCONJ
ejpam-5875	73	15	cos(θ)b	cos(θ)b	NOUN
ejpam-5875	73	16	.	.	PUNCT
ejpam-5875	74	1	(	(	PUNCT
ejpam-5875	74	2	3.3	3.3	NUM
ejpam-5875	74	3	)	)	PUNCT
ejpam-5875	74	4	from	from	ADP
ejpam-5875	74	5	these	these	DET
ejpam-5875	74	6	relationships	relationship	NOUN
ejpam-5875	74	7	,	,	PUNCT
ejpam-5875	74	8	it	it	PRON
ejpam-5875	74	9	is	be	AUX
ejpam-5875	74	10	also	also	ADV
ejpam-5875	74	11	possible	possible	ADJ
ejpam-5875	74	12	to	to	PART
ejpam-5875	74	13	rewrite	rewrite	VERB
ejpam-5875	74	14	n	n	PRON
ejpam-5875	74	15	and	and	CCONJ
ejpam-5875	74	16	b	b	NOUN
ejpam-5875	74	17	in	in	ADP
ejpam-5875	74	18	terms	term	NOUN
ejpam-5875	74	19	of	of	ADP
ejpam-5875	74	20	the	the	DET
ejpam-5875	74	21	qframe	qframe	NOUN
ejpam-5875	74	22	:	:	PUNCT
ejpam-5875	74	23	n	n	PROPN
ejpam-5875	74	24	=	=	SYM
ejpam-5875	74	25	cos(θ)nq	cos(θ)nq	PROPN
ejpam-5875	74	26	−	−	PROPN
ejpam-5875	74	27	sin(θ)bq	sin(θ)bq	NOUN
ejpam-5875	74	28	,	,	PUNCT
ejpam-5875	74	29	(	(	PUNCT
ejpam-5875	74	30	3.4	3.4	NUM
ejpam-5875	74	31	)	)	PUNCT
ejpam-5875	74	32	b	b	NOUN
ejpam-5875	74	33	=	=	SYM
ejpam-5875	74	34	sin(θ)nq	sin(θ)nq	PROPN
ejpam-5875	74	35	+	+	CCONJ
ejpam-5875	74	36	cos(θ)bq	cos(θ)bq	PROPN
ejpam-5875	74	37	.	.	PUNCT
ejpam-5875	75	1	(	(	PUNCT
ejpam-5875	75	2	3.5	3.5	NUM
ejpam-5875	75	3	)	)	PUNCT
ejpam-5875	75	4	utilizing	utilize	VERB
ejpam-5875	75	5	the	the	DET
ejpam-5875	75	6	frenet	frenet	NOUN
ejpam-5875	75	7	formulas	formula	NOUN
ejpam-5875	75	8	,	,	PUNCT
ejpam-5875	75	9	we	we	PRON
ejpam-5875	75	10	have	have	VERB
ejpam-5875	75	11	:	:	PUNCT
ejpam-5875	75	12	t	t	NOUN
ejpam-5875	75	13	′	′	NUM
ejpam-5875	75	14	=	=	PUNCT
ejpam-5875	75	15	κn	κn	NOUN
ejpam-5875	75	16	=	=	PUNCT
ejpam-5875	75	17	κ(cos(θ)nq	κ(cos(θ)nq	PROPN
ejpam-5875	75	18	−	−	PROPN
ejpam-5875	75	19	sin(θ)bq	sin(θ)bq	NOUN
ejpam-5875	75	20	)	)	PUNCT
ejpam-5875	75	21	.	.	PUNCT
ejpam-5875	76	1	by	by	ADP
ejpam-5875	76	2	defining	define	VERB
ejpam-5875	76	3	k1	k1	NOUN
ejpam-5875	76	4	=	=	SYM
ejpam-5875	76	5	κ	κ	NOUN
ejpam-5875	76	6	cos(θ	cos(θ	PROPN
ejpam-5875	76	7	)	)	PUNCT
ejpam-5875	76	8	and	and	CCONJ
ejpam-5875	76	9	k2	k2	PROPN
ejpam-5875	76	10	=	=	PROPN
ejpam-5875	76	11	κ	κ	X
ejpam-5875	76	12	sin(θ	sin(θ	PROPN
ejpam-5875	76	13	)	)	PUNCT
ejpam-5875	76	14	,	,	PUNCT
ejpam-5875	76	15	we	we	PRON
ejpam-5875	76	16	can	can	AUX
ejpam-5875	76	17	express	express	VERB
ejpam-5875	76	18	the	the	DET
ejpam-5875	76	19	derivative	derivative	NOUN
ejpam-5875	76	20	of	of	ADP
ejpam-5875	76	21	t	t	PROPN
ejpam-5875	76	22	as	as	ADP
ejpam-5875	76	23	:	:	PUNCT
ejpam-5875	76	24	t	t	NOUN
ejpam-5875	76	25	′	′	NOUN
ejpam-5875	76	26	=	=	PUNCT
ejpam-5875	76	27	k1nq	k1nq	NOUN
ejpam-5875	76	28	−k2bq	−k2bq	NUM
ejpam-5875	76	29	.	.	PUNCT
ejpam-5875	77	1	(	(	PUNCT
ejpam-5875	77	2	3.6	3.6	NUM
ejpam-5875	77	3	)	)	PUNCT
ejpam-5875	77	4	using	use	VERB
ejpam-5875	77	5	the	the	DET
ejpam-5875	77	6	established	establish	VERB
ejpam-5875	77	7	relationships	relationship	NOUN
ejpam-5875	77	8	,	,	PUNCT
ejpam-5875	77	9	the	the	DET
ejpam-5875	77	10	derivatives	derivative	NOUN
ejpam-5875	77	11	n	n	PRON
ejpam-5875	77	12	′	′	NUM
ejpam-5875	77	13	q	q	NOUN
ejpam-5875	78	1	and	and	CCONJ
ejpam-5875	78	2	b′	b′	NUM
ejpam-5875	78	3	q	q	NOUN
ejpam-5875	78	4	can	can	AUX
ejpam-5875	78	5	be	be	AUX
ejpam-5875	78	6	expressed	express	VERB
ejpam-5875	78	7	as	as	ADP
ejpam-5875	78	8	:	:	PUNCT
ejpam-5875	78	9	a.	a.	NOUN
ejpam-5875	78	10	elsharkawy	elsharkawy	PROPN
ejpam-5875	78	11	,	,	PUNCT
ejpam-5875	78	12	n.	n.	NOUN
ejpam-5875	78	13	elsharkawy	elsharkawy	PROPN
ejpam-5875	78	14	/	/	SYM
ejpam-5875	78	15	eur	eur	PROPN
ejpam-5875	78	16	.	.	PUNCT
ejpam-5875	79	1	j.	j.	PROPN
ejpam-5875	79	2	pure	pure	PROPN
ejpam-5875	79	3	appl	appl	PROPN
ejpam-5875	79	4	.	.	PROPN
ejpam-5875	79	5	math	math	PROPN
ejpam-5875	79	6	,	,	PUNCT
ejpam-5875	79	7	18	18	NUM
ejpam-5875	79	8	(	(	PUNCT
ejpam-5875	79	9	2	2	NUM
ejpam-5875	79	10	)	)	PUNCT
ejpam-5875	79	11	(	(	PUNCT
ejpam-5875	79	12	2025	2025	NUM
ejpam-5875	79	13	)	)	PUNCT
ejpam-5875	79	14	,	,	PUNCT
ejpam-5875	79	15	5875	5875	NUM
ejpam-5875	79	16	5	5	NUM
ejpam-5875	79	17	of	of	ADP
ejpam-5875	79	18	15	15	NUM
ejpam-5875	79	19	n	n	NOUN
ejpam-5875	79	20	′	′	NUM
ejpam-5875	79	21	q	q	NOUN
ejpam-5875	79	22	=	=	SYM
ejpam-5875	79	23	k3bq	k3bq	PROPN
ejpam-5875	79	24	,	,	PUNCT
ejpam-5875	79	25	b′	b′	NOUN
ejpam-5875	79	26	q	q	NOUN
ejpam-5875	79	27	=	=	PUNCT
ejpam-5875	79	28	−k3nq	−k3nq	NOUN
ejpam-5875	79	29	,	,	PUNCT
ejpam-5875	79	30	(	(	PUNCT
ejpam-5875	79	31	3.7	3.7	NUM
ejpam-5875	79	32	)	)	PUNCT
ejpam-5875	79	33	where	where	SCONJ
ejpam-5875	79	34	k3	k3	VERB
ejpam-5875	79	35	=	=	NOUN
ejpam-5875	79	36	θ′	θ′	NOUN
ejpam-5875	79	37	+	+	CCONJ
ejpam-5875	79	38	τ	τ	X
ejpam-5875	79	39	.	.	PUNCT
ejpam-5875	80	1	consequently	consequently	ADV
ejpam-5875	80	2	,	,	PUNCT
ejpam-5875	80	3	the	the	DET
ejpam-5875	80	4	quasi	quasi	ADJ
ejpam-5875	80	5	equations	equation	NOUN
ejpam-5875	80	6	can	can	AUX
ejpam-5875	80	7	be	be	AUX
ejpam-5875	80	8	presented	present	VERB
ejpam-5875	80	9	in	in	ADP
ejpam-5875	80	10	a	a	DET
ejpam-5875	80	11	matrix	matrix	NOUN
ejpam-5875	80	12	form	form	NOUN
ejpam-5875	80	13	:	:	PUNCT
ejpam-5875	80	14	t	t	NOUN
ejpam-5875	80	15	′	′	NUM
ejpam-5875	81	1	n	n	NOUN
ejpam-5875	81	2	′	′	NUM
ejpam-5875	81	3	q	q	PROPN
ejpam-5875	81	4	b′	b′	NOUN
ejpam-5875	81	5	q	q	NOUN
ejpam-5875	81	6			PROPN
ejpam-5875	81	7	=	=	PUNCT
ejpam-5875	81	8	0	0	ADP
ejpam-5875	81	9	k1	k1	PROPN
ejpam-5875	81	10	−k2	−k2	PROPN
ejpam-5875	81	11	0	0	NUM
ejpam-5875	81	12	0	0	NUM
ejpam-5875	82	1	k3	k3	ADJ
ejpam-5875	82	2	0	0	PUNCT
ejpam-5875	83	1	−k3	−k3	NOUN
ejpam-5875	83	2	0	0	NUM
ejpam-5875	83	3			PROPN
ejpam-5875	83	4	t	t	PROPN
ejpam-5875	83	5	nq	nq	INTJ
ejpam-5875	83	6	bq	bq	PROPN
ejpam-5875	83	7			PROPN
ejpam-5875	83	8	.	.	PUNCT
ejpam-5875	84	1	(	(	PUNCT
ejpam-5875	84	2	3.8	3.8	NUM
ejpam-5875	84	3	)	)	PUNCT
ejpam-5875	84	4	the	the	DET
ejpam-5875	84	5	quasi	quasi	ADJ
ejpam-5875	84	6	curvatures	curvature	NOUN
ejpam-5875	84	7	k1	k1	NOUN
ejpam-5875	84	8	,	,	PUNCT
ejpam-5875	84	9	k2	k2	NOUN
ejpam-5875	84	10	,	,	PUNCT
ejpam-5875	84	11	and	and	CCONJ
ejpam-5875	84	12	k3	k3	NOUN
ejpam-5875	84	13	are	be	AUX
ejpam-5875	84	14	associated	associate	VERB
ejpam-5875	84	15	with	with	ADP
ejpam-5875	84	16	the	the	DET
ejpam-5875	84	17	q	q	NOUN
ejpam-5875	84	18	-	-	PUNCT
ejpam-5875	84	19	frame	frame	NOUN
ejpam-5875	84	20	as	as	SCONJ
ejpam-5875	84	21	follows	follow	VERB
ejpam-5875	84	22	:	:	PUNCT
ejpam-5875	84	23	corollary	corollary	ADJ
ejpam-5875	84	24	1	1	X
ejpam-5875	84	25	.	.	PUNCT
ejpam-5875	85	1	if	if	SCONJ
ejpam-5875	85	2	α(s	α(s	PROPN
ejpam-5875	85	3	)	)	PUNCT
ejpam-5875	85	4	is	be	AUX
ejpam-5875	85	5	a	a	DET
ejpam-5875	85	6	curve	curve	NOUN
ejpam-5875	85	7	in	in	ADP
ejpam-5875	85	8	g3	g3	PROPN
ejpam-5875	85	9	,	,	PUNCT
ejpam-5875	85	10	then	then	ADV
ejpam-5875	85	11	the	the	DET
ejpam-5875	85	12	quasi	quasi	ADJ
ejpam-5875	85	13	curvatures	curvature	VERB
ejpam-5875	85	14	k1,k2	k1,k2	PROPN
ejpam-5875	85	15	,	,	PUNCT
ejpam-5875	85	16	and	and	CCONJ
ejpam-5875	85	17	k3	k3	VERB
ejpam-5875	85	18	in	in	ADP
ejpam-5875	85	19	terms	term	NOUN
ejpam-5875	85	20	of	of	ADP
ejpam-5875	85	21	the	the	DET
ejpam-5875	85	22	q	q	NOUN
ejpam-5875	85	23	-	-	PUNCT
ejpam-5875	85	24	frame	frame	NOUN
ejpam-5875	85	25	are	be	AUX
ejpam-5875	85	26	given	give	VERB
ejpam-5875	85	27	by	by	ADP
ejpam-5875	85	28	k1	k1	PROPN
ejpam-5875	85	29	=	=	SYM
ejpam-5875	85	30	g(t	g(t	PROPN
ejpam-5875	85	31	′	′	PROPN
ejpam-5875	85	32	,	,	PUNCT
ejpam-5875	85	33	nq	nq	PROPN
ejpam-5875	85	34	)	)	PUNCT
ejpam-5875	85	35	,	,	PUNCT
ejpam-5875	85	36	k2	k2	NOUN
ejpam-5875	85	37	=	=	SYM
ejpam-5875	85	38	−g(t	−g(t	PROPN
ejpam-5875	85	39	′	′	NOUN
ejpam-5875	85	40	,	,	PUNCT
ejpam-5875	85	41	bq	bq	NOUN
ejpam-5875	85	42	)	)	PUNCT
ejpam-5875	85	43	,	,	PUNCT
ejpam-5875	85	44	k3	k3	PROPN
ejpam-5875	85	45	=	=	PROPN
ejpam-5875	85	46	g(n	g(n	PROPN
ejpam-5875	85	47	′	′	NOUN
ejpam-5875	86	1	q	q	ADJ
ejpam-5875	86	2	,	,	PUNCT
ejpam-5875	86	3	bq	bq	INTJ
ejpam-5875	86	4	)	)	PUNCT
ejpam-5875	86	5	=	=	PUNCT
ejpam-5875	87	1	−g(b′	−g(b′	PROPN
ejpam-5875	87	2	q	q	PROPN
ejpam-5875	87	3	,	,	PUNCT
ejpam-5875	87	4	nq	nq	PROPN
ejpam-5875	87	5	)	)	PUNCT
ejpam-5875	87	6	.	.	PUNCT
ejpam-5875	88	1	corollary	corollary	ADJ
ejpam-5875	88	2	2	2	NUM
ejpam-5875	88	3	.	.	PUNCT
ejpam-5875	89	1	if	if	SCONJ
ejpam-5875	89	2	k2	k2	PROPN
ejpam-5875	89	3	=	=	SYM
ejpam-5875	89	4	0	0	PROPN
ejpam-5875	89	5	,	,	PUNCT
ejpam-5875	89	6	the	the	DET
ejpam-5875	89	7	q	q	ADJ
ejpam-5875	89	8	-	-	PUNCT
ejpam-5875	89	9	frame	frame	NOUN
ejpam-5875	89	10	coincides	coincide	VERB
ejpam-5875	89	11	with	with	ADP
ejpam-5875	89	12	the	the	DET
ejpam-5875	89	13	frenet	frenet	ADJ
ejpam-5875	89	14	frame	frame	NOUN
ejpam-5875	89	15	,	,	PUNCT
ejpam-5875	89	16	demonstrating	demonstrate	VERB
ejpam-5875	89	17	that	that	SCONJ
ejpam-5875	89	18	the	the	DET
ejpam-5875	89	19	q	q	ADJ
ejpam-5875	89	20	-	-	PUNCT
ejpam-5875	89	21	frame	frame	NOUN
ejpam-5875	89	22	generalizes	generalize	VERB
ejpam-5875	89	23	the	the	DET
ejpam-5875	89	24	frenet	frenet	ADJ
ejpam-5875	89	25	frame	frame	NOUN
ejpam-5875	89	26	in	in	ADP
ejpam-5875	89	27	g3	g3	PROPN
ejpam-5875	89	28	.	.	PUNCT
ejpam-5875	90	1	4	4	NUM
ejpam-5875	90	2	.	.	NUM
ejpam-5875	90	3	generalized	generalize	VERB
ejpam-5875	90	4	position	position	NOUN
ejpam-5875	90	5	vector	vector	NOUN
ejpam-5875	90	6	this	this	DET
ejpam-5875	90	7	section	section	NOUN
ejpam-5875	90	8	examines	examine	VERB
ejpam-5875	90	9	the	the	DET
ejpam-5875	90	10	spatial	spatial	ADJ
ejpam-5875	90	11	coordinates	coordinate	NOUN
ejpam-5875	90	12	of	of	ADP
ejpam-5875	90	13	position	position	NOUN
ejpam-5875	90	14	vectors	vector	NOUN
ejpam-5875	90	15	within	within	ADP
ejpam-5875	90	16	the	the	DET
ejpam-5875	90	17	galilean	galilean	PROPN
ejpam-5875	90	18	three	three	NUM
ejpam-5875	90	19	-	-	PUNCT
ejpam-5875	90	20	dimensional	dimensional	ADJ
ejpam-5875	90	21	manifold	manifold	NOUN
ejpam-5875	90	22	.	.	PUNCT
ejpam-5875	91	1	we	we	PRON
ejpam-5875	91	2	derive	derive	VERB
ejpam-5875	91	3	the	the	DET
ejpam-5875	91	4	components	component	NOUN
ejpam-5875	91	5	associated	associate	VERB
ejpam-5875	91	6	with	with	ADP
ejpam-5875	91	7	the	the	DET
ejpam-5875	91	8	tangential	tangential	ADJ
ejpam-5875	91	9	,	,	PUNCT
ejpam-5875	91	10	q	q	ADJ
ejpam-5875	91	11	-	-	ADJ
ejpam-5875	91	12	normal	normal	ADJ
ejpam-5875	91	13	,	,	PUNCT
ejpam-5875	91	14	and	and	CCONJ
ejpam-5875	91	15	q	q	ADJ
ejpam-5875	91	16	-	-	PUNCT
ejpam-5875	91	17	binormal	binormal	ADJ
ejpam-5875	91	18	vectors	vector	NOUN
ejpam-5875	91	19	under	under	ADP
ejpam-5875	91	20	specific	specific	ADJ
ejpam-5875	91	21	conditions	condition	NOUN
ejpam-5875	91	22	.	.	PUNCT
ejpam-5875	92	1	additionally	additionally	ADV
ejpam-5875	92	2	,	,	PUNCT
ejpam-5875	92	3	we	we	PRON
ejpam-5875	92	4	investigate	investigate	VERB
ejpam-5875	92	5	the	the	DET
ejpam-5875	92	6	properties	property	NOUN
ejpam-5875	92	7	of	of	ADP
ejpam-5875	92	8	q	q	NOUN
ejpam-5875	92	9	-	-	PUNCT
ejpam-5875	92	10	rectifying	rectifying	ADJ
ejpam-5875	92	11	and	and	CCONJ
ejpam-5875	92	12	q	q	NOUN
ejpam-5875	92	13	-	-	PUNCT
ejpam-5875	92	14	osculating	osculate	VERB
ejpam-5875	92	15	curves	curve	NOUN
ejpam-5875	92	16	in	in	ADP
ejpam-5875	92	17	g3	g3	NOUN
ejpam-5875	92	18	with	with	ADP
ejpam-5875	92	19	respect	respect	NOUN
ejpam-5875	92	20	to	to	ADP
ejpam-5875	92	21	the	the	DET
ejpam-5875	92	22	q	q	NOUN
ejpam-5875	92	23	-	-	PUNCT
ejpam-5875	92	24	frame	frame	NOUN
ejpam-5875	92	25	of	of	ADP
ejpam-5875	92	26	reference	reference	NOUN
ejpam-5875	92	27	.	.	PUNCT
ejpam-5875	93	1	furthermore	furthermore	ADV
ejpam-5875	93	2	,	,	PUNCT
ejpam-5875	93	3	we	we	PRON
ejpam-5875	93	4	demonstrate	demonstrate	VERB
ejpam-5875	93	5	the	the	DET
ejpam-5875	93	6	absence	absence	NOUN
ejpam-5875	93	7	of	of	ADP
ejpam-5875	93	8	normal	normal	ADJ
ejpam-5875	93	9	curves	curve	NOUN
ejpam-5875	93	10	within	within	ADP
ejpam-5875	93	11	the	the	DET
ejpam-5875	93	12	galilean	galilean	PROPN
ejpam-5875	93	13	spatial	spatial	ADJ
ejpam-5875	93	14	framework	framework	NOUN
ejpam-5875	93	15	.	.	PUNCT
ejpam-5875	94	1	let	let	VERB
ejpam-5875	94	2	α	α	NOUN
ejpam-5875	94	3	=	=	PUNCT
ejpam-5875	94	4	α(s	α(s	PROPN
ejpam-5875	94	5	)	)	PUNCT
ejpam-5875	94	6	be	be	AUX
ejpam-5875	94	7	a	a	DET
ejpam-5875	94	8	unit	unit	NOUN
ejpam-5875	94	9	speed	speed	NOUN
ejpam-5875	94	10	curve	curve	NOUN
ejpam-5875	94	11	in	in	ADP
ejpam-5875	94	12	g3	g3	PROPN
ejpam-5875	94	13	.	.	PUNCT
ejpam-5875	95	1	we	we	PRON
ejpam-5875	95	2	can	can	AUX
ejpam-5875	95	3	write	write	VERB
ejpam-5875	95	4	the	the	DET
ejpam-5875	95	5	position	position	NOUN
ejpam-5875	95	6	vector	vector	NOUN
ejpam-5875	95	7	concerning	concern	VERB
ejpam-5875	95	8	the	the	DET
ejpam-5875	95	9	q	q	NOUN
ejpam-5875	95	10	-	-	PUNCT
ejpam-5875	95	11	frame	frame	NOUN
ejpam-5875	95	12	in	in	ADP
ejpam-5875	95	13	g3	g3	PROPN
ejpam-5875	95	14	as	as	ADP
ejpam-5875	95	15	α	α	NOUN
ejpam-5875	95	16	=	=	SYM
ejpam-5875	95	17	α(s	α(s	PROPN
ejpam-5875	95	18	)	)	PUNCT
ejpam-5875	96	1	=	=	PUNCT
ejpam-5875	96	2	m1(s)t	m1(s)t	PUNCT
ejpam-5875	96	3	+	+	NOUN
ejpam-5875	96	4	m2(s)nq	m2(s)nq	NOUN
ejpam-5875	96	5	+	+	ADV
ejpam-5875	96	6	m3(s)bq	m3(s)bq	NOUN
ejpam-5875	96	7	,	,	PUNCT
ejpam-5875	96	8	(	(	PUNCT
ejpam-5875	96	9	4.1	4.1	NUM
ejpam-5875	96	10	)	)	PUNCT
ejpam-5875	96	11	for	for	ADP
ejpam-5875	96	12	some	some	DET
ejpam-5875	96	13	differentiable	differentiable	ADJ
ejpam-5875	96	14	functions	function	NOUN
ejpam-5875	96	15	m1(s),m2(s	m1(s),m2(s	NOUN
ejpam-5875	96	16	)	)	PUNCT
ejpam-5875	96	17	and	and	CCONJ
ejpam-5875	96	18	m3(s	m3(s	NOUN
ejpam-5875	96	19	)	)	PUNCT
ejpam-5875	96	20	.	.	PUNCT
ejpam-5875	97	1	by	by	ADP
ejpam-5875	97	2	differentiating	differentiate	VERB
ejpam-5875	97	3	equation	equation	NOUN
ejpam-5875	97	4	(	(	PUNCT
ejpam-5875	97	5	4.1	4.1	NUM
ejpam-5875	97	6	)	)	PUNCT
ejpam-5875	97	7	and	and	CCONJ
ejpam-5875	97	8	using	use	VERB
ejpam-5875	97	9	equation	equation	NOUN
ejpam-5875	97	10	(	(	PUNCT
ejpam-5875	97	11	3.8	3.8	NUM
ejpam-5875	97	12	)	)	PUNCT
ejpam-5875	97	13	,	,	PUNCT
ejpam-5875	97	14	we	we	PRON
ejpam-5875	97	15	have	have	VERB
ejpam-5875	97	16	m′	m′	NUM
ejpam-5875	97	17	1	1	NUM
ejpam-5875	97	18	=	=	SYM
ejpam-5875	97	19	1	1	NUM
ejpam-5875	97	20	,	,	PUNCT
ejpam-5875	97	21	(	(	PUNCT
ejpam-5875	97	22	4.2	4.2	NUM
ejpam-5875	97	23	)	)	PUNCT
ejpam-5875	97	24	m1k1	m1k1	PRON
ejpam-5875	98	1	+	+	NOUN
ejpam-5875	98	2	m′	m′	NUM
ejpam-5875	98	3	2	2	NUM
ejpam-5875	98	4	−m3k3	−m3k3	NOUN
ejpam-5875	98	5	=	=	SYM
ejpam-5875	98	6	0	0	PROPN
ejpam-5875	98	7	,	,	PUNCT
ejpam-5875	98	8	(	(	PUNCT
ejpam-5875	98	9	4.3	4.3	NUM
ejpam-5875	98	10	)	)	PUNCT
ejpam-5875	98	11	−m1k2	−m1k2	ADP
ejpam-5875	98	12	+	+	PROPN
ejpam-5875	98	13	m2k3	m2k3	NOUN
ejpam-5875	98	14	+	+	NOUN
ejpam-5875	98	15	m′	m′	NOUN
ejpam-5875	98	16	3	3	NUM
ejpam-5875	98	17	=	=	SYM
ejpam-5875	98	18	0	0	NUM
ejpam-5875	98	19	.	.	PUNCT
ejpam-5875	99	1	(	(	PUNCT
ejpam-5875	99	2	4.4	4.4	NUM
ejpam-5875	99	3	)	)	PUNCT
ejpam-5875	99	4	from	from	ADP
ejpam-5875	99	5	equation	equation	NOUN
ejpam-5875	99	6	(	(	PUNCT
ejpam-5875	99	7	4.2	4.2	NUM
ejpam-5875	99	8	)	)	PUNCT
ejpam-5875	99	9	,	,	PUNCT
ejpam-5875	99	10	we	we	PRON
ejpam-5875	99	11	get	get	VERB
ejpam-5875	99	12	:	:	PUNCT
ejpam-5875	99	13	m1	m1	PROPN
ejpam-5875	100	1	=	=	PUNCT
ejpam-5875	100	2	c	c	PROPN
ejpam-5875	100	3	+	+	SYM
ejpam-5875	100	4	s	s	PROPN
ejpam-5875	100	5	,	,	PUNCT
ejpam-5875	100	6	(	(	PUNCT
ejpam-5875	100	7	4.5	4.5	NUM
ejpam-5875	100	8	)	)	PUNCT
ejpam-5875	100	9	where	where	SCONJ
ejpam-5875	100	10	c	c	NOUN
ejpam-5875	100	11	is	be	AUX
ejpam-5875	100	12	constant	constant	ADJ
ejpam-5875	100	13	.	.	PUNCT
ejpam-5875	101	1	a.	a.	NOUN
ejpam-5875	101	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	101	3	,	,	PUNCT
ejpam-5875	101	4	n.	n.	NOUN
ejpam-5875	101	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	101	6	/	/	SYM
ejpam-5875	101	7	eur	eur	PROPN
ejpam-5875	101	8	.	.	PUNCT
ejpam-5875	102	1	j.	j.	PROPN
ejpam-5875	102	2	pure	pure	PROPN
ejpam-5875	102	3	appl	appl	PROPN
ejpam-5875	102	4	.	.	PROPN
ejpam-5875	102	5	math	math	PROPN
ejpam-5875	102	6	,	,	PUNCT
ejpam-5875	102	7	18	18	NUM
ejpam-5875	102	8	(	(	PUNCT
ejpam-5875	102	9	2	2	NUM
ejpam-5875	102	10	)	)	PUNCT
ejpam-5875	102	11	(	(	PUNCT
ejpam-5875	102	12	2025	2025	NUM
ejpam-5875	102	13	)	)	PUNCT
ejpam-5875	102	14	,	,	PUNCT
ejpam-5875	102	15	5875	5875	NUM
ejpam-5875	102	16	6	6	NUM
ejpam-5875	102	17	of	of	ADP
ejpam-5875	102	18	15	15	NUM
ejpam-5875	102	19	by	by	ADP
ejpam-5875	102	20	substitution	substitution	NOUN
ejpam-5875	102	21	from	from	ADP
ejpam-5875	102	22	equation	equation	NOUN
ejpam-5875	102	23	(	(	PUNCT
ejpam-5875	102	24	4.5	4.5	NUM
ejpam-5875	102	25	)	)	PUNCT
ejpam-5875	102	26	into	into	ADP
ejpam-5875	102	27	equation	equation	NOUN
ejpam-5875	102	28	(	(	PUNCT
ejpam-5875	102	29	4.3	4.3	NUM
ejpam-5875	102	30	)	)	PUNCT
ejpam-5875	102	31	,	,	PUNCT
ejpam-5875	102	32	we	we	PRON
ejpam-5875	102	33	obtain	obtain	VERB
ejpam-5875	102	34	:	:	PUNCT
ejpam-5875	102	35	m3	m3	PROPN
ejpam-5875	102	36	=	=	PUNCT
ejpam-5875	102	37	(	(	PUNCT
ejpam-5875	102	38	c	c	NOUN
ejpam-5875	102	39	+	+	PUNCT
ejpam-5875	102	40	s)k1	s)k1	PROPN
ejpam-5875	102	41	+	+	PROPN
ejpam-5875	102	42	m′	m′	ADJ
ejpam-5875	102	43	2	2	NUM
ejpam-5875	102	44	k3	k3	ADJ
ejpam-5875	102	45	.	.	PUNCT
ejpam-5875	103	1	(	(	PUNCT
ejpam-5875	103	2	4.6	4.6	NUM
ejpam-5875	103	3	)	)	PUNCT
ejpam-5875	103	4	by	by	ADP
ejpam-5875	103	5	differentiate	differentiate	ADJ
ejpam-5875	103	6	equation	equation	NOUN
ejpam-5875	103	7	(	(	PUNCT
ejpam-5875	103	8	4.6	4.6	NUM
ejpam-5875	103	9	)	)	PUNCT
ejpam-5875	103	10	,	,	PUNCT
ejpam-5875	103	11	we	we	PRON
ejpam-5875	103	12	have	have	VERB
ejpam-5875	103	13	m′	m′	NUM
ejpam-5875	103	14	3	3	NUM
ejpam-5875	103	15	=	=	SYM
ejpam-5875	103	16	k3[sk	k3[sk	NOUN
ejpam-5875	103	17	′	′	NOUN
ejpam-5875	103	18	1	1	NUM
ejpam-5875	104	1	+	+	NOUN
ejpam-5875	104	2	k1	k1	NOUN
ejpam-5875	104	3	+	+	CCONJ
ejpam-5875	104	4	ck	ck	INTJ
ejpam-5875	104	5	′	′	NOUN
ejpam-5875	104	6	1	1	NUM
ejpam-5875	104	7	+	+	CCONJ
ejpam-5875	104	8	m′′	m′′	ADJ
ejpam-5875	104	9	2]−k	2]−k	NUM
ejpam-5875	105	1	′	′	NUM
ejpam-5875	105	2	3[sk1	3[sk1	NUM
ejpam-5875	106	1	+	+	CCONJ
ejpam-5875	106	2	ck1	ck1	NOUN
ejpam-5875	107	1	+	+	NOUN
ejpam-5875	107	2	m′	m′	NOUN
ejpam-5875	107	3	2	2	NUM
ejpam-5875	107	4	]	]	PUNCT
ejpam-5875	107	5	(	(	PUNCT
ejpam-5875	107	6	k3)2	k3)2	PROPN
ejpam-5875	107	7	.	.	PUNCT
ejpam-5875	108	1	(	(	PUNCT
ejpam-5875	108	2	4.7	4.7	NUM
ejpam-5875	108	3	)	)	PUNCT
ejpam-5875	108	4	no	no	DET
ejpam-5875	108	5	general	general	ADJ
ejpam-5875	108	6	solution	solution	NOUN
ejpam-5875	108	7	has	have	AUX
ejpam-5875	108	8	been	be	AUX
ejpam-5875	108	9	found	find	VERB
ejpam-5875	108	10	for	for	ADP
ejpam-5875	108	11	this	this	DET
ejpam-5875	108	12	system	system	NOUN
ejpam-5875	108	13	.	.	PUNCT
ejpam-5875	109	1	because	because	SCONJ
ejpam-5875	109	2	of	of	ADP
ejpam-5875	109	3	this	this	PRON
ejpam-5875	109	4	,	,	PUNCT
ejpam-5875	109	5	we	we	PRON
ejpam-5875	109	6	give	give	VERB
ejpam-5875	109	7	the	the	DET
ejpam-5875	109	8	solution	solution	NOUN
ejpam-5875	109	9	in	in	ADP
ejpam-5875	109	10	some	some	DET
ejpam-5875	109	11	special	special	ADJ
ejpam-5875	109	12	cases	case	NOUN
ejpam-5875	109	13	.	.	PUNCT
ejpam-5875	110	1	case	case	NOUN
ejpam-5875	110	2	1	1	NUM
ejpam-5875	110	3	:	:	PUNCT
ejpam-5875	110	4	by	by	ADP
ejpam-5875	110	5	substituting	substitute	VERB
ejpam-5875	110	6	from	from	ADP
ejpam-5875	110	7	equation	equation	NOUN
ejpam-5875	110	8	(	(	PUNCT
ejpam-5875	110	9	4.7	4.7	NUM
ejpam-5875	110	10	)	)	PUNCT
ejpam-5875	110	11	into	into	ADP
ejpam-5875	110	12	equation	equation	NOUN
ejpam-5875	110	13	(	(	PUNCT
ejpam-5875	110	14	4.4	4.4	NUM
ejpam-5875	110	15	)	)	PUNCT
ejpam-5875	110	16	,	,	PUNCT
ejpam-5875	110	17	and	and	CCONJ
ejpam-5875	110	18	substituting	substitute	VERB
ejpam-5875	110	19	k1	k1	NOUN
ejpam-5875	110	20	=	=	SYM
ejpam-5875	110	21	k2	k2	PROPN
ejpam-5875	110	22	=	=	SYM
ejpam-5875	111	1	0,k3	0,k3	NOUN
ejpam-5875	111	2	=	=	NOUN
ejpam-5875	112	1	constant	constant	ADJ
ejpam-5875	112	2	=	=	PUNCT
ejpam-5875	112	3	a	a	DET
ejpam-5875	112	4	̸=	̸=	PROPN
ejpam-5875	112	5	0	0	NUM
ejpam-5875	112	6	,	,	PUNCT
ejpam-5875	112	7	we	we	PRON
ejpam-5875	112	8	obtain	obtain	VERB
ejpam-5875	112	9	a	a	DET
ejpam-5875	112	10	non	non	ADJ
ejpam-5875	112	11	-	-	ADJ
ejpam-5875	112	12	homogeneous	homogeneous	ADJ
ejpam-5875	112	13	second	second	ADJ
ejpam-5875	112	14	linear	linear	NOUN
ejpam-5875	112	15	differential	differential	NOUN
ejpam-5875	112	16	equation	equation	NOUN
ejpam-5875	112	17	k3	k3	VERB
ejpam-5875	112	18	m	m	PROPN
ejpam-5875	112	19	′′	′′	NOUN
ejpam-5875	112	20	2	2	NUM
ejpam-5875	113	1	+	+	NOUN
ejpam-5875	113	2	k3	k3	ADJ
ejpam-5875	113	3	3m2	3m2	NOUN
ejpam-5875	113	4	=	=	SYM
ejpam-5875	113	5	0	0	X
ejpam-5875	113	6	.	.	PUNCT
ejpam-5875	114	1	so	so	ADV
ejpam-5875	114	2	,	,	PUNCT
ejpam-5875	114	3	m2	m2	PROPN
ejpam-5875	114	4	=	=	PROPN
ejpam-5875	114	5	c1	c1	PROPN
ejpam-5875	114	6	cos	cos	PROPN
ejpam-5875	114	7	as+	as+	PROPN
ejpam-5875	114	8	c2	c2	PROPN
ejpam-5875	114	9	sin	sin	VERB
ejpam-5875	114	10	as	as	ADP
ejpam-5875	114	11	,	,	PUNCT
ejpam-5875	114	12	(	(	PUNCT
ejpam-5875	114	13	4.8	4.8	NUM
ejpam-5875	114	14	)	)	PUNCT
ejpam-5875	114	15	by	by	ADP
ejpam-5875	114	16	taking	take	VERB
ejpam-5875	114	17	the	the	DET
ejpam-5875	114	18	derivative	derivative	NOUN
ejpam-5875	114	19	of	of	ADP
ejpam-5875	114	20	(	(	PUNCT
ejpam-5875	114	21	4.8	4.8	NUM
ejpam-5875	114	22	)	)	PUNCT
ejpam-5875	114	23	and	and	CCONJ
ejpam-5875	114	24	substituting	substitute	VERB
ejpam-5875	114	25	k1	k1	NOUN
ejpam-5875	114	26	=	=	SYM
ejpam-5875	114	27	0,k3	0,k3	PROPN
ejpam-5875	114	28	=	=	PUNCT
ejpam-5875	114	29	a	a	X
ejpam-5875	114	30	,	,	PUNCT
ejpam-5875	114	31	into	into	ADP
ejpam-5875	114	32	equation	equation	NOUN
ejpam-5875	114	33	(	(	PUNCT
ejpam-5875	114	34	4.6	4.6	NUM
ejpam-5875	114	35	)	)	PUNCT
ejpam-5875	114	36	,	,	PUNCT
ejpam-5875	114	37	we	we	PRON
ejpam-5875	114	38	obtain	obtain	VERB
ejpam-5875	114	39	:	:	PUNCT
ejpam-5875	115	1	m3	m3	PROPN
ejpam-5875	115	2	=	=	SYM
ejpam-5875	115	3	−c1	−c1	ADJ
ejpam-5875	115	4	sin	sin	NOUN
ejpam-5875	115	5	as+	as+	PROPN
ejpam-5875	115	6	c2	c2	PROPN
ejpam-5875	115	7	cos	cos	PROPN
ejpam-5875	115	8	as	as	ADP
ejpam-5875	115	9	.	.	PUNCT
ejpam-5875	116	1	(	(	PUNCT
ejpam-5875	116	2	4.9	4.9	NUM
ejpam-5875	116	3	)	)	PUNCT
ejpam-5875	116	4	therefore	therefore	ADV
ejpam-5875	116	5	,	,	PUNCT
ejpam-5875	116	6	we	we	PRON
ejpam-5875	116	7	can	can	AUX
ejpam-5875	116	8	write	write	VERB
ejpam-5875	116	9	the	the	DET
ejpam-5875	116	10	position	position	NOUN
ejpam-5875	116	11	vector	vector	NOUN
ejpam-5875	116	12	as	as	ADP
ejpam-5875	116	13	α(s	α(s	PROPN
ejpam-5875	116	14	)	)	PUNCT
ejpam-5875	117	1	=	=	PUNCT
ejpam-5875	117	2	(	(	PUNCT
ejpam-5875	117	3	c	c	NOUN
ejpam-5875	117	4	+	+	NOUN
ejpam-5875	117	5	s)t	s)t	X
ejpam-5875	118	1	+	+	CCONJ
ejpam-5875	118	2	(	(	PUNCT
ejpam-5875	118	3	c1	c1	PROPN
ejpam-5875	118	4	cos	cos	PROPN
ejpam-5875	118	5	as+	as+	PROPN
ejpam-5875	118	6	c2	c2	PROPN
ejpam-5875	118	7	sin	sin	VERB
ejpam-5875	118	8	as)nq	as)nq	PROPN
ejpam-5875	118	9	+	+	CCONJ
ejpam-5875	118	10	(	(	PUNCT
ejpam-5875	118	11	−c1	−c1	PROPN
ejpam-5875	118	12	cos	cos	PROPN
ejpam-5875	118	13	as+	as+	PROPN
ejpam-5875	118	14	c2	c2	PROPN
ejpam-5875	118	15	sin	sin	VERB
ejpam-5875	118	16	as)bq	as)bq	PROPN
ejpam-5875	118	17	,	,	PUNCT
ejpam-5875	118	18	where	where	SCONJ
ejpam-5875	118	19	c	c	NOUN
ejpam-5875	118	20	,	,	PUNCT
ejpam-5875	118	21	c1	c1	PROPN
ejpam-5875	118	22	,	,	PUNCT
ejpam-5875	118	23	c2	c2	PROPN
ejpam-5875	118	24	,	,	PUNCT
ejpam-5875	118	25	a	a	PRON
ejpam-5875	118	26	and	and	CCONJ
ejpam-5875	118	27	a	a	PRON
ejpam-5875	118	28	are	be	AUX
ejpam-5875	118	29	constants	constant	NOUN
ejpam-5875	118	30	.	.	PUNCT
ejpam-5875	119	1	case	case	NOUN
ejpam-5875	119	2	2	2	NUM
ejpam-5875	119	3	:	:	PUNCT
ejpam-5875	119	4	let	let	VERB
ejpam-5875	119	5	m2	m2	PROPN
ejpam-5875	119	6	=	=	PROPN
ejpam-5875	119	7	c3	c3	PROPN
ejpam-5875	119	8	̸=	̸=	PROPN
ejpam-5875	119	9	0	0	NUM
ejpam-5875	119	10	,	,	PUNCT
ejpam-5875	119	11	from	from	ADP
ejpam-5875	119	12	equation	equation	NOUN
ejpam-5875	119	13	(	(	PUNCT
ejpam-5875	119	14	4.3	4.3	NUM
ejpam-5875	119	15	)	)	PUNCT
ejpam-5875	119	16	,	,	PUNCT
ejpam-5875	119	17	we	we	PRON
ejpam-5875	119	18	obtain	obtain	VERB
ejpam-5875	119	19	:	:	PUNCT
ejpam-5875	119	20	m3	m3	PROPN
ejpam-5875	119	21	=	=	PUNCT
ejpam-5875	119	22	(	(	PUNCT
ejpam-5875	119	23	c	c	X
ejpam-5875	119	24	+	+	SYM
ejpam-5875	119	25	s	s	X
ejpam-5875	119	26	)	)	PUNCT
ejpam-5875	119	27	k1	k1	NOUN
ejpam-5875	119	28	k3	k3	NOUN
ejpam-5875	119	29	.	.	PUNCT
ejpam-5875	120	1	therefore	therefore	ADV
ejpam-5875	120	2	,	,	PUNCT
ejpam-5875	120	3	in	in	ADP
ejpam-5875	120	4	this	this	DET
ejpam-5875	120	5	case	case	NOUN
ejpam-5875	120	6	,	,	PUNCT
ejpam-5875	120	7	we	we	PRON
ejpam-5875	120	8	can	can	AUX
ejpam-5875	120	9	write	write	VERB
ejpam-5875	120	10	the	the	DET
ejpam-5875	120	11	position	position	NOUN
ejpam-5875	120	12	vector	vector	NOUN
ejpam-5875	120	13	as	as	ADP
ejpam-5875	120	14	α(s	α(s	PROPN
ejpam-5875	120	15	)	)	PUNCT
ejpam-5875	121	1	=	=	PUNCT
ejpam-5875	121	2	(	(	PUNCT
ejpam-5875	121	3	c	c	X
ejpam-5875	121	4	+	+	X
ejpam-5875	121	5	s)t	s)t	X
ejpam-5875	121	6	+	+	X
ejpam-5875	121	7	c3nq	c3nq	PUNCT
ejpam-5875	121	8	+	+	CCONJ
ejpam-5875	121	9	(	(	PUNCT
ejpam-5875	121	10	c	c	X
ejpam-5875	121	11	+	+	NUM
ejpam-5875	121	12	s	s	X
ejpam-5875	121	13	)	)	PUNCT
ejpam-5875	121	14	k1	k1	NOUN
ejpam-5875	121	15	k3	k3	PROPN
ejpam-5875	121	16	bq	bq	INTJ
ejpam-5875	121	17	.	.	PROPN
ejpam-5875	122	1	from	from	ADP
ejpam-5875	122	2	equation	equation	NOUN
ejpam-5875	122	3	(	(	PUNCT
ejpam-5875	122	4	4.4	4.4	NUM
ejpam-5875	122	5	)	)	PUNCT
ejpam-5875	122	6	,	,	PUNCT
ejpam-5875	122	7	we	we	PRON
ejpam-5875	122	8	have	have	VERB
ejpam-5875	122	9	a	a	DET
ejpam-5875	122	10	linear	linear	ADJ
ejpam-5875	122	11	first	first	ADJ
ejpam-5875	122	12	-	-	PUNCT
ejpam-5875	122	13	order	order	NOUN
ejpam-5875	122	14	differential	differential	ADJ
ejpam-5875	122	15	equation	equation	NOUN
ejpam-5875	122	16	(	(	PUNCT
ejpam-5875	122	17	k1	k1	NOUN
ejpam-5875	122	18	k3	k3	PROPN
ejpam-5875	122	19	)	)	PUNCT
ejpam-5875	122	20	′	′	PUNCT
ejpam-5875	123	1	+	+	CCONJ
ejpam-5875	123	2	1	1	NUM
ejpam-5875	123	3	c	c	NOUN
ejpam-5875	123	4	+	+	SYM
ejpam-5875	123	5	s	s	X
ejpam-5875	123	6	(	(	PUNCT
ejpam-5875	123	7	k1	k1	NOUN
ejpam-5875	123	8	k3	k3	ADJ
ejpam-5875	123	9	)	)	PUNCT
ejpam-5875	123	10	=	=	SYM
ejpam-5875	123	11	k2	k2	PROPN
ejpam-5875	123	12	−	−	PROPN
ejpam-5875	124	1	c3	c3	PROPN
ejpam-5875	124	2	c	c	PROPN
ejpam-5875	125	1	+	+	X
ejpam-5875	125	2	s	s	VERB
ejpam-5875	125	3	k3	k3	ADJ
ejpam-5875	125	4	.	.	PUNCT
ejpam-5875	126	1	(	(	PUNCT
ejpam-5875	126	2	4.10	4.10	NUM
ejpam-5875	126	3	)	)	PUNCT
ejpam-5875	126	4	the	the	DET
ejpam-5875	126	5	integrating	integrate	VERB
ejpam-5875	126	6	factor	factor	NOUN
ejpam-5875	126	7	is	be	AUX
ejpam-5875	126	8	given	give	VERB
ejpam-5875	126	9	by	by	ADP
ejpam-5875	126	10	µ	µ	NOUN
ejpam-5875	126	11	=	=	SYM
ejpam-5875	126	12	e	e	NOUN
ejpam-5875	126	13	∫	∫	PROPN
ejpam-5875	126	14	1	1	NUM
ejpam-5875	126	15	c+s	c+s	NOUN
ejpam-5875	126	16	ds	ds	NOUN
ejpam-5875	126	17	=	=	SYM
ejpam-5875	126	18	(	(	PUNCT
ejpam-5875	126	19	c	c	X
ejpam-5875	126	20	+	+	NUM
ejpam-5875	126	21	s	s	NOUN
ejpam-5875	126	22	)	)	PUNCT
ejpam-5875	126	23	,	,	PUNCT
ejpam-5875	126	24	therefore	therefore	ADV
ejpam-5875	126	25	the	the	DET
ejpam-5875	126	26	solution	solution	NOUN
ejpam-5875	126	27	of	of	ADP
ejpam-5875	126	28	equation(4.10	equation(4.10	PRON
ejpam-5875	126	29	)	)	PUNCT
ejpam-5875	126	30	is	be	AUX
ejpam-5875	126	31	given	give	VERB
ejpam-5875	126	32	by	by	ADP
ejpam-5875	126	33	k1	k1	NOUN
ejpam-5875	126	34	k3	k3	NOUN
ejpam-5875	126	35	=	=	SYM
ejpam-5875	126	36	1	1	NUM
ejpam-5875	126	37	c	c	NOUN
ejpam-5875	126	38	+	+	SYM
ejpam-5875	126	39	s	s	PART
ejpam-5875	126	40	∫	∫	PROPN
ejpam-5875	127	1	[	[	X
ejpam-5875	127	2	(	(	PUNCT
ejpam-5875	127	3	c	c	NOUN
ejpam-5875	127	4	+	+	CCONJ
ejpam-5875	127	5	s)k2	s)k2	PROPN
ejpam-5875	127	6	−	−	PROPN
ejpam-5875	127	7	c3k3]ds+	c3k3]ds+	PROPN
ejpam-5875	127	8	c4	c4	NOUN
ejpam-5875	127	9	.	.	PUNCT
ejpam-5875	128	1	(	(	PUNCT
ejpam-5875	128	2	4.11	4.11	NUM
ejpam-5875	128	3	)	)	PUNCT
ejpam-5875	128	4	a.	a.	NOUN
ejpam-5875	128	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	128	6	,	,	PUNCT
ejpam-5875	128	7	n.	n.	NOUN
ejpam-5875	128	8	elsharkawy	elsharkawy	PROPN
ejpam-5875	128	9	/	/	SYM
ejpam-5875	128	10	eur	eur	PROPN
ejpam-5875	128	11	.	.	PUNCT
ejpam-5875	129	1	j.	j.	PROPN
ejpam-5875	129	2	pure	pure	PROPN
ejpam-5875	129	3	appl	appl	PROPN
ejpam-5875	129	4	.	.	PROPN
ejpam-5875	129	5	math	math	PROPN
ejpam-5875	129	6	,	,	PUNCT
ejpam-5875	129	7	18	18	NUM
ejpam-5875	129	8	(	(	PUNCT
ejpam-5875	129	9	2	2	NUM
ejpam-5875	129	10	)	)	PUNCT
ejpam-5875	129	11	(	(	PUNCT
ejpam-5875	129	12	2025	2025	NUM
ejpam-5875	129	13	)	)	PUNCT
ejpam-5875	129	14	,	,	PUNCT
ejpam-5875	129	15	5875	5875	NUM
ejpam-5875	129	16	7	7	NUM
ejpam-5875	129	17	of	of	ADP
ejpam-5875	129	18	15	15	NUM
ejpam-5875	129	19	case	case	NOUN
ejpam-5875	129	20	3	3	NUM
ejpam-5875	129	21	:	:	PUNCT
ejpam-5875	129	22	let	let	VERB
ejpam-5875	129	23	m3	m3	PROPN
ejpam-5875	129	24	=	=	SYM
ejpam-5875	129	25	c5	c5	PROPN
ejpam-5875	129	26	̸=	̸=	PROPN
ejpam-5875	129	27	0	0	NUM
ejpam-5875	129	28	from	from	ADP
ejpam-5875	129	29	equation	equation	NOUN
ejpam-5875	129	30	(	(	PUNCT
ejpam-5875	129	31	4.4	4.4	NUM
ejpam-5875	129	32	)	)	PUNCT
ejpam-5875	129	33	,	,	PUNCT
ejpam-5875	129	34	we	we	PRON
ejpam-5875	129	35	obtain	obtain	VERB
ejpam-5875	129	36	:	:	PUNCT
ejpam-5875	129	37	m2	m2	PROPN
ejpam-5875	129	38	=	=	PUNCT
ejpam-5875	129	39	(	(	PUNCT
ejpam-5875	129	40	c	c	PROPN
ejpam-5875	129	41	+	+	SYM
ejpam-5875	129	42	s	s	X
ejpam-5875	129	43	)	)	PUNCT
ejpam-5875	129	44	k2	k2	NOUN
ejpam-5875	129	45	k3	k3	NOUN
ejpam-5875	129	46	.	.	PUNCT
ejpam-5875	130	1	(	(	PUNCT
ejpam-5875	130	2	4.12	4.12	NUM
ejpam-5875	130	3	)	)	PUNCT
ejpam-5875	130	4	therefore	therefore	ADV
ejpam-5875	130	5	,	,	PUNCT
ejpam-5875	130	6	in	in	ADP
ejpam-5875	130	7	this	this	DET
ejpam-5875	130	8	case	case	NOUN
ejpam-5875	130	9	,	,	PUNCT
ejpam-5875	130	10	we	we	PRON
ejpam-5875	130	11	can	can	AUX
ejpam-5875	130	12	write	write	VERB
ejpam-5875	130	13	the	the	DET
ejpam-5875	130	14	position	position	NOUN
ejpam-5875	130	15	vector	vector	NOUN
ejpam-5875	130	16	as	as	ADP
ejpam-5875	130	17	α(s	α(s	PROPN
ejpam-5875	130	18	)	)	PUNCT
ejpam-5875	131	1	=	=	PUNCT
ejpam-5875	131	2	(	(	PUNCT
ejpam-5875	131	3	c	c	NOUN
ejpam-5875	131	4	+	+	NOUN
ejpam-5875	131	5	s)t	s)t	X
ejpam-5875	132	1	+	+	CCONJ
ejpam-5875	132	2	(	(	PUNCT
ejpam-5875	132	3	c	c	X
ejpam-5875	132	4	+	+	NOUN
ejpam-5875	132	5	s	s	X
ejpam-5875	132	6	)	)	PUNCT
ejpam-5875	132	7	k2	k2	NOUN
ejpam-5875	132	8	k3	k3	VERB
ejpam-5875	132	9	nq	nq	PROPN
ejpam-5875	132	10	+	+	PROPN
ejpam-5875	132	11	c5bq	c5bq	X
ejpam-5875	132	12	.	.	PUNCT
ejpam-5875	133	1	substituting	substitute	VERB
ejpam-5875	133	2	equation	equation	NOUN
ejpam-5875	133	3	(	(	PUNCT
ejpam-5875	133	4	4.12	4.12	NUM
ejpam-5875	133	5	)	)	PUNCT
ejpam-5875	133	6	into	into	ADP
ejpam-5875	133	7	equation	equation	NOUN
ejpam-5875	133	8	(	(	PUNCT
ejpam-5875	133	9	4.3	4.3	NUM
ejpam-5875	133	10	)	)	PUNCT
ejpam-5875	133	11	yields	yield	VERB
ejpam-5875	133	12	a	a	DET
ejpam-5875	133	13	linear	linear	ADJ
ejpam-5875	133	14	first	first	ADJ
ejpam-5875	133	15	-	-	PUNCT
ejpam-5875	133	16	order	order	NOUN
ejpam-5875	133	17	differential	differential	ADJ
ejpam-5875	133	18	equation	equation	NOUN
ejpam-5875	133	19	:	:	PUNCT
ejpam-5875	133	20	(	(	PUNCT
ejpam-5875	133	21	k2	k2	NOUN
ejpam-5875	133	22	k3	k3	PROPN
ejpam-5875	133	23	)	)	PUNCT
ejpam-5875	133	24	′	′	PUNCT
ejpam-5875	134	1	+	+	CCONJ
ejpam-5875	134	2	1	1	NUM
ejpam-5875	134	3	c	c	NOUN
ejpam-5875	134	4	+	+	SYM
ejpam-5875	134	5	s	s	X
ejpam-5875	134	6	(	(	PUNCT
ejpam-5875	134	7	k2	k2	NOUN
ejpam-5875	134	8	k3	k3	ADJ
ejpam-5875	134	9	)	)	PUNCT
ejpam-5875	134	10	=	=	SYM
ejpam-5875	135	1	c5	c5	PROPN
ejpam-5875	135	2	c	c	PROPN
ejpam-5875	135	3	+	+	SYM
ejpam-5875	135	4	s	s	X
ejpam-5875	135	5	k3	k3	ADJ
ejpam-5875	135	6	+	+	NOUN
ejpam-5875	135	7	k1	k1	NOUN
ejpam-5875	135	8	.	.	PUNCT
ejpam-5875	136	1	so	so	ADV
ejpam-5875	136	2	,	,	PUNCT
ejpam-5875	136	3	k2	k2	ADJ
ejpam-5875	136	4	k3	k3	NOUN
ejpam-5875	136	5	=	=	SYM
ejpam-5875	136	6	1	1	NUM
ejpam-5875	136	7	c	c	NOUN
ejpam-5875	136	8	+	+	SYM
ejpam-5875	136	9	s	s	PART
ejpam-5875	136	10	∫	∫	PROPN
ejpam-5875	136	11	(	(	PUNCT
ejpam-5875	136	12	c5k3	c5k3	SYM
ejpam-5875	136	13	+	+	CCONJ
ejpam-5875	136	14	(	(	PUNCT
ejpam-5875	136	15	c	c	NOUN
ejpam-5875	136	16	+	+	CCONJ
ejpam-5875	136	17	s)k1)ds+	s)k1)ds+	ADJ
ejpam-5875	136	18	c6	c6	PROPN
ejpam-5875	136	19	.	.	PUNCT
ejpam-5875	137	1	(	(	PUNCT
ejpam-5875	137	2	4.13	4.13	NUM
ejpam-5875	137	3	)	)	PUNCT
ejpam-5875	137	4	corollary	corollary	ADJ
ejpam-5875	137	5	3	3	NUM
ejpam-5875	137	6	.	.	PUNCT
ejpam-5875	138	1	in	in	ADP
ejpam-5875	138	2	the	the	DET
ejpam-5875	138	3	case	case	NOUN
ejpam-5875	138	4	of	of	ADP
ejpam-5875	138	5	the	the	DET
ejpam-5875	138	6	frenet	frenet	NOUN
ejpam-5875	138	7	curve	curve	NOUN
ejpam-5875	138	8	,	,	PUNCT
ejpam-5875	138	9	we	we	PRON
ejpam-5875	138	10	can	can	AUX
ejpam-5875	138	11	put	put	VERB
ejpam-5875	138	12	k2	k2	NOUN
ejpam-5875	138	13	=	=	PUNCT
ejpam-5875	139	1	0,k3	0,k3	PROPN
ejpam-5875	139	2	=	=	SYM
ejpam-5875	139	3	τ	τ	X
ejpam-5875	139	4	,	,	PUNCT
ejpam-5875	139	5	and	and	CCONJ
ejpam-5875	139	6	k1	k1	NOUN
ejpam-5875	139	7	=	=	SYM
ejpam-5875	139	8	κ	κ	NOUN
ejpam-5875	139	9	.	.	PUNCT
ejpam-5875	140	1	then	then	ADV
ejpam-5875	140	2	,	,	PUNCT
ejpam-5875	140	3	equations	equation	NOUN
ejpam-5875	140	4	(	(	PUNCT
ejpam-5875	140	5	4.2	4.2	NUM
ejpam-5875	140	6	)	)	PUNCT
ejpam-5875	140	7	,	,	PUNCT
ejpam-5875	140	8	(	(	PUNCT
ejpam-5875	140	9	4.3	4.3	NUM
ejpam-5875	140	10	)	)	PUNCT
ejpam-5875	140	11	,	,	PUNCT
ejpam-5875	140	12	and	and	CCONJ
ejpam-5875	140	13	(	(	PUNCT
ejpam-5875	140	14	4.4	4.4	NUM
ejpam-5875	140	15	)	)	PUNCT
ejpam-5875	140	16	become	become	VERB
ejpam-5875	140	17	m′	m′	NOUN
ejpam-5875	140	18	1	1	NUM
ejpam-5875	140	19	=	=	SYM
ejpam-5875	140	20	1	1	NUM
ejpam-5875	140	21	,	,	PUNCT
ejpam-5875	140	22	(	(	PUNCT
ejpam-5875	140	23	4.14	4.14	NUM
ejpam-5875	140	24	)	)	PUNCT
ejpam-5875	140	25	m1κ+m′	m1κ+m′	NOUN
ejpam-5875	140	26	2	2	X
ejpam-5875	140	27	−m3τ	−m3τ	PUNCT
ejpam-5875	140	28	=	=	SYM
ejpam-5875	140	29	0	0	NUM
ejpam-5875	140	30	,	,	PUNCT
ejpam-5875	140	31	(	(	PUNCT
ejpam-5875	140	32	4.15	4.15	NUM
ejpam-5875	140	33	)	)	PUNCT
ejpam-5875	140	34	m2τ	m2τ	NOUN
ejpam-5875	141	1	+	+	SCONJ
ejpam-5875	141	2	m′	m′	NOUN
ejpam-5875	141	3	3	3	NUM
ejpam-5875	141	4	=	=	SYM
ejpam-5875	141	5	0	0	NUM
ejpam-5875	141	6	.	.	PUNCT
ejpam-5875	142	1	(	(	PUNCT
ejpam-5875	142	2	4.16	4.16	NUM
ejpam-5875	142	3	)	)	PUNCT
ejpam-5875	142	4	therefore	therefore	ADV
ejpam-5875	142	5	,	,	PUNCT
ejpam-5875	142	6	m1	m1	PROPN
ejpam-5875	142	7	=	=	PUNCT
ejpam-5875	142	8	c	c	PROPN
ejpam-5875	142	9	+	+	CCONJ
ejpam-5875	142	10	s.	s.	PROPN
ejpam-5875	142	11	in	in	ADP
ejpam-5875	142	12	case	case	NOUN
ejpam-5875	142	13	2	2	NUM
ejpam-5875	142	14	,	,	PUNCT
ejpam-5875	142	15	if	if	SCONJ
ejpam-5875	142	16	m2	m2	PROPN
ejpam-5875	142	17	=	=	PROPN
ejpam-5875	142	18	c3	c3	PROPN
ejpam-5875	142	19	,	,	PUNCT
ejpam-5875	142	20	thus	thus	ADV
ejpam-5875	142	21	m3	m3	PROPN
ejpam-5875	142	22	=	=	PUNCT
ejpam-5875	142	23	(	(	PUNCT
ejpam-5875	142	24	c	c	NOUN
ejpam-5875	142	25	+	+	CCONJ
ejpam-5875	143	1	s)κτ	s)κτ	PROPN
ejpam-5875	143	2	.	.	PUNCT
ejpam-5875	144	1	also	also	ADV
ejpam-5875	144	2	,	,	PUNCT
ejpam-5875	144	3	equation	equation	NOUN
ejpam-5875	144	4	(	(	PUNCT
ejpam-5875	144	5	4.11	4.11	NUM
ejpam-5875	144	6	)	)	PUNCT
ejpam-5875	144	7	,	,	PUNCT
ejpam-5875	144	8	becomes	become	VERB
ejpam-5875	144	9	−c3	−c3	PROPN
ejpam-5875	144	10	c	c	NOUN
ejpam-5875	144	11	+	+	SYM
ejpam-5875	144	12	s	s	PART
ejpam-5875	144	13	∫	∫	PROPN
ejpam-5875	145	1	[	[	X
ejpam-5875	145	2	τds+	τds+	NOUN
ejpam-5875	145	3	c4	c4	NOUN
ejpam-5875	145	4	]	]	X
ejpam-5875	145	5	=	=	SYM
ejpam-5875	145	6	κ	κ	X
ejpam-5875	145	7	τ	τ	PROPN
ejpam-5875	145	8	.	.	PUNCT
ejpam-5875	146	1	in	in	ADP
ejpam-5875	146	2	case	case	NOUN
ejpam-5875	146	3	3	3	NUM
ejpam-5875	146	4	,	,	PUNCT
ejpam-5875	146	5	from	from	ADP
ejpam-5875	146	6	equation	equation	NOUN
ejpam-5875	146	7	(	(	PUNCT
ejpam-5875	146	8	4.16	4.16	NUM
ejpam-5875	146	9	)	)	PUNCT
ejpam-5875	146	10	,	,	PUNCT
ejpam-5875	146	11	we	we	PRON
ejpam-5875	146	12	obtain	obtain	VERB
ejpam-5875	146	13	τ	τ	X
ejpam-5875	146	14	=	=	NOUN
ejpam-5875	146	15	0	0	PROPN
ejpam-5875	146	16	.	.	PUNCT
ejpam-5875	146	17	from	from	ADP
ejpam-5875	146	18	equation	equation	NOUN
ejpam-5875	146	19	(	(	PUNCT
ejpam-5875	146	20	4.15	4.15	NUM
ejpam-5875	146	21	)	)	PUNCT
ejpam-5875	146	22	,	,	PUNCT
ejpam-5875	146	23	we	we	PRON
ejpam-5875	146	24	obtain	obtain	VERB
ejpam-5875	146	25	m2	m2	PROPN
ejpam-5875	146	26	=	=	SYM
ejpam-5875	146	27	−	−	PROPN
ejpam-5875	146	28	∫	∫	PROPN
ejpam-5875	146	29	(	(	PUNCT
ejpam-5875	146	30	c+	c+	NOUN
ejpam-5875	146	31	s)κds	s)κds	PROPN
ejpam-5875	146	32	.	.	PUNCT
ejpam-5875	147	1	also	also	ADV
ejpam-5875	147	2	,	,	PUNCT
ejpam-5875	147	3	equation	equation	NOUN
ejpam-5875	147	4	(	(	PUNCT
ejpam-5875	147	5	4.13	4.13	NUM
ejpam-5875	147	6	)	)	PUNCT
ejpam-5875	147	7	becomes	become	VERB
ejpam-5875	147	8	c6	c6	NOUN
ejpam-5875	147	9	=	=	PUNCT
ejpam-5875	148	1	−	−	PROPN
ejpam-5875	148	2	∫	∫	PROPN
ejpam-5875	148	3	(	(	PUNCT
ejpam-5875	148	4	(	(	PUNCT
ejpam-5875	148	5	c	c	NOUN
ejpam-5875	148	6	+	+	CCONJ
ejpam-5875	148	7	s)κ)ds	s)κ)d	VERB
ejpam-5875	148	8	.	.	PUNCT
ejpam-5875	149	1	these	these	DET
ejpam-5875	149	2	results	result	NOUN
ejpam-5875	149	3	are	be	AUX
ejpam-5875	149	4	consistent	consistent	ADJ
ejpam-5875	149	5	with	with	ADP
ejpam-5875	149	6	those	those	PRON
ejpam-5875	149	7	in	in	ADP
ejpam-5875	149	8	[	[	X
ejpam-5875	149	9	34	34	NUM
ejpam-5875	149	10	]	]	PUNCT
ejpam-5875	149	11	.	.	PUNCT
ejpam-5875	150	1	a.	a.	NOUN
ejpam-5875	150	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	150	3	,	,	PUNCT
ejpam-5875	150	4	n.	n.	NOUN
ejpam-5875	150	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	150	6	/	/	SYM
ejpam-5875	150	7	eur	eur	PROPN
ejpam-5875	150	8	.	.	PUNCT
ejpam-5875	151	1	j.	j.	PROPN
ejpam-5875	151	2	pure	pure	PROPN
ejpam-5875	151	3	appl	appl	PROPN
ejpam-5875	151	4	.	.	PROPN
ejpam-5875	151	5	math	math	PROPN
ejpam-5875	151	6	,	,	PUNCT
ejpam-5875	151	7	18	18	NUM
ejpam-5875	151	8	(	(	PUNCT
ejpam-5875	151	9	2	2	NUM
ejpam-5875	151	10	)	)	PUNCT
ejpam-5875	151	11	(	(	PUNCT
ejpam-5875	151	12	2025	2025	NUM
ejpam-5875	151	13	)	)	PUNCT
ejpam-5875	151	14	,	,	PUNCT
ejpam-5875	151	15	5875	5875	NUM
ejpam-5875	151	16	8	8	NUM
ejpam-5875	151	17	of	of	ADP
ejpam-5875	151	18	15	15	NUM
ejpam-5875	151	19	5	5	NUM
ejpam-5875	151	20	.	.	PUNCT
ejpam-5875	152	1	q	q	ADJ
ejpam-5875	152	2	-	-	ADJ
ejpam-5875	152	3	normal	normal	ADJ
ejpam-5875	152	4	curves	curve	NOUN
ejpam-5875	152	5	in	in	ADP
ejpam-5875	152	6	this	this	DET
ejpam-5875	152	7	section	section	NOUN
ejpam-5875	152	8	,	,	PUNCT
ejpam-5875	152	9	we	we	PRON
ejpam-5875	152	10	demonstrate	demonstrate	VERB
ejpam-5875	152	11	the	the	DET
ejpam-5875	152	12	absence	absence	NOUN
ejpam-5875	152	13	of	of	ADP
ejpam-5875	152	14	normal	normal	ADJ
ejpam-5875	152	15	curves	curve	NOUN
ejpam-5875	152	16	within	within	ADP
ejpam-5875	152	17	the	the	DET
ejpam-5875	152	18	context	context	NOUN
ejpam-5875	152	19	of	of	ADP
ejpam-5875	152	20	galilean	galilean	PROPN
ejpam-5875	152	21	3	3	NUM
ejpam-5875	152	22	-	-	PUNCT
ejpam-5875	152	23	space	space	NOUN
ejpam-5875	152	24	,	,	PUNCT
ejpam-5875	152	25	as	as	SCONJ
ejpam-5875	152	26	defined	define	VERB
ejpam-5875	152	27	by	by	ADP
ejpam-5875	152	28	both	both	CCONJ
ejpam-5875	152	29	the	the	DET
ejpam-5875	152	30	frenet	frenet	ADJ
ejpam-5875	152	31	frame	frame	NOUN
ejpam-5875	152	32	and	and	CCONJ
ejpam-5875	152	33	the	the	DET
ejpam-5875	152	34	q	q	NOUN
ejpam-5875	152	35	-	-	PUNCT
ejpam-5875	152	36	frame	frame	NOUN
ejpam-5875	152	37	.	.	PUNCT
ejpam-5875	153	1	a	a	DET
ejpam-5875	153	2	curve	curve	NOUN
ejpam-5875	153	3	α(s	α(s	PROPN
ejpam-5875	153	4	)	)	PUNCT
ejpam-5875	153	5	is	be	AUX
ejpam-5875	153	6	classified	classify	VERB
ejpam-5875	153	7	as	as	ADP
ejpam-5875	153	8	a	a	DET
ejpam-5875	153	9	q	q	ADJ
ejpam-5875	153	10	-	-	ADJ
ejpam-5875	153	11	normal	normal	ADJ
ejpam-5875	153	12	curve	curve	NOUN
ejpam-5875	153	13	in	in	ADP
ejpam-5875	153	14	g3	g3	PROPN
ejpam-5875	153	15	if	if	SCONJ
ejpam-5875	153	16	it	it	PRON
ejpam-5875	153	17	resides	reside	VERB
ejpam-5875	153	18	entirely	entirely	ADV
ejpam-5875	153	19	within	within	ADP
ejpam-5875	153	20	its	its	PRON
ejpam-5875	153	21	q	q	ADJ
ejpam-5875	153	22	-	-	ADJ
ejpam-5875	153	23	normal	normal	ADJ
ejpam-5875	153	24	plane	plane	NOUN
ejpam-5875	153	25	.	.	PUNCT
ejpam-5875	154	1	formally	formally	ADV
ejpam-5875	154	2	,	,	PUNCT
ejpam-5875	154	3	this	this	PRON
ejpam-5875	154	4	means	mean	VERB
ejpam-5875	154	5	that	that	SCONJ
ejpam-5875	154	6	the	the	DET
ejpam-5875	154	7	curve	curve	NOUN
ejpam-5875	154	8	α	α	PROPN
ejpam-5875	154	9	satisfies	satisfy	VERB
ejpam-5875	154	10	the	the	DET
ejpam-5875	154	11	equation	equation	NOUN
ejpam-5875	154	12	α(s	α(s	PROPN
ejpam-5875	154	13	)	)	PUNCT
ejpam-5875	154	14	=	=	PUNCT
ejpam-5875	155	1	λ(s)nq(s	λ(s)nq(s	X
ejpam-5875	155	2	)	)	PUNCT
ejpam-5875	156	1	+	+	PUNCT
ejpam-5875	156	2	π(s)bq(s	π(s)bq(s	X
ejpam-5875	156	3	)	)	PUNCT
ejpam-5875	156	4	,	,	PUNCT
ejpam-5875	156	5	where	where	SCONJ
ejpam-5875	156	6	nq(s	nq(s	PUNCT
ejpam-5875	156	7	)	)	PUNCT
ejpam-5875	156	8	and	and	CCONJ
ejpam-5875	156	9	bq(s	bq(	VERB
ejpam-5875	156	10	)	)	PUNCT
ejpam-5875	156	11	represent	represent	VERB
ejpam-5875	156	12	the	the	DET
ejpam-5875	156	13	q	q	NOUN
ejpam-5875	156	14	-	-	ADJ
ejpam-5875	156	15	normal	normal	ADJ
ejpam-5875	156	16	and	and	CCONJ
ejpam-5875	156	17	q	q	ADJ
ejpam-5875	156	18	-	-	PUNCT
ejpam-5875	156	19	binormal	binormal	ADJ
ejpam-5875	156	20	vectors	vector	NOUN
ejpam-5875	156	21	,	,	PUNCT
ejpam-5875	156	22	respectively	respectively	ADV
ejpam-5875	156	23	.	.	PUNCT
ejpam-5875	157	1	theorem	theorem	NOUN
ejpam-5875	157	2	1	1	NUM
ejpam-5875	157	3	.	.	X
ejpam-5875	158	1	for	for	ADP
ejpam-5875	158	2	any	any	DET
ejpam-5875	158	3	admissible	admissible	ADJ
ejpam-5875	158	4	differentiable	differentiable	ADJ
ejpam-5875	158	5	curve	curve	NOUN
ejpam-5875	158	6	parameterized	parameterize	VERB
ejpam-5875	158	7	by	by	ADP
ejpam-5875	158	8	s	s	PRON
ejpam-5875	158	9	,	,	PUNCT
ejpam-5875	158	10	there	there	PRON
ejpam-5875	158	11	do	do	AUX
ejpam-5875	158	12	not	not	PART
ejpam-5875	158	13	exist	exist	VERB
ejpam-5875	158	14	quasi	quasi	ADJ
ejpam-5875	158	15	-	-	ADJ
ejpam-5875	158	16	normal	normal	ADJ
ejpam-5875	158	17	curves	curve	NOUN
ejpam-5875	158	18	in	in	ADP
ejpam-5875	158	19	g3	g3	PROPN
ejpam-5875	158	20	.	.	PUNCT
ejpam-5875	159	1	proof	proof	NOUN
ejpam-5875	159	2	.	.	PUNCT
ejpam-5875	160	1	let	let	VERB
ejpam-5875	160	2	β(s	β(	NOUN
ejpam-5875	160	3	)	)	PUNCT
ejpam-5875	160	4	denote	denote	VERB
ejpam-5875	160	5	an	an	DET
ejpam-5875	160	6	admissible	admissible	ADJ
ejpam-5875	160	7	differentiable	differentiable	ADJ
ejpam-5875	160	8	curve	curve	NOUN
ejpam-5875	160	9	parameterized	parameterize	VERB
ejpam-5875	160	10	by	by	ADP
ejpam-5875	160	11	the	the	DET
ejpam-5875	160	12	galilean	galilean	PROPN
ejpam-5875	160	13	invariant	invariant	PROPN
ejpam-5875	160	14	arc	arc	NOUN
ejpam-5875	160	15	length	length	NOUN
ejpam-5875	160	16	s	s	PROPN
ejpam-5875	160	17	in	in	ADP
ejpam-5875	160	18	g3	g3	PROPN
ejpam-5875	160	19	.	.	PUNCT
ejpam-5875	161	1	we	we	PRON
ejpam-5875	161	2	can	can	AUX
ejpam-5875	161	3	express	express	VERB
ejpam-5875	161	4	β(s	β(	NOUN
ejpam-5875	161	5	)	)	PUNCT
ejpam-5875	161	6	in	in	ADP
ejpam-5875	161	7	the	the	DET
ejpam-5875	161	8	following	follow	VERB
ejpam-5875	161	9	form	form	NOUN
ejpam-5875	161	10	:	:	PUNCT
ejpam-5875	161	11	β(s	β(	NOUN
ejpam-5875	161	12	)	)	PUNCT
ejpam-5875	162	1	=	=	PRON
ejpam-5875	162	2	(	(	PUNCT
ejpam-5875	162	3	s	s	PROPN
ejpam-5875	162	4	,	,	PUNCT
ejpam-5875	162	5	y(s	y(s	PROPN
ejpam-5875	162	6	)	)	PUNCT
ejpam-5875	162	7	,	,	PUNCT
ejpam-5875	162	8	z(s	z(s	PROPN
ejpam-5875	162	9	)	)	PUNCT
ejpam-5875	162	10	)	)	PUNCT
ejpam-5875	162	11	.	.	PUNCT
ejpam-5875	163	1	differentiating	differentiate	VERB
ejpam-5875	163	2	with	with	ADP
ejpam-5875	163	3	respect	respect	NOUN
ejpam-5875	163	4	to	to	ADP
ejpam-5875	163	5	s	s	PRON
ejpam-5875	163	6	,	,	PUNCT
ejpam-5875	163	7	we	we	PRON
ejpam-5875	163	8	obtain	obtain	VERB
ejpam-5875	163	9	the	the	DET
ejpam-5875	163	10	tangent	tangent	NOUN
ejpam-5875	163	11	vector	vector	NOUN
ejpam-5875	163	12	:	:	PUNCT
ejpam-5875	163	13	t	t	NOUN
ejpam-5875	163	14	=	=	SYM
ejpam-5875	163	15	(	(	PUNCT
ejpam-5875	163	16	1	1	NUM
ejpam-5875	163	17	,	,	PUNCT
ejpam-5875	163	18	y′	y′	NUM
ejpam-5875	163	19	,	,	PUNCT
ejpam-5875	163	20	z′	z′	NUM
ejpam-5875	163	21	)	)	PUNCT
ejpam-5875	163	22	.	.	PUNCT
ejpam-5875	164	1	from	from	ADP
ejpam-5875	164	2	the	the	DET
ejpam-5875	164	3	galilean	galilean	PROPN
ejpam-5875	164	4	metric	metric	NOUN
ejpam-5875	164	5	,	,	PUNCT
ejpam-5875	164	6	we	we	PRON
ejpam-5875	164	7	find	find	VERB
ejpam-5875	164	8	that	that	SCONJ
ejpam-5875	164	9	g(β	g(β	NOUN
ejpam-5875	164	10	,	,	PUNCT
ejpam-5875	164	11	t	t	PROPN
ejpam-5875	164	12	)	)	PUNCT
ejpam-5875	165	1	=	=	SYM
ejpam-5875	165	2	s	s	VERB
ejpam-5875	165	3	̸=	̸=	PROPN
ejpam-5875	165	4	0	0	NUM
ejpam-5875	165	5	,	,	PUNCT
ejpam-5875	165	6	which	which	PRON
ejpam-5875	165	7	implies	imply	VERB
ejpam-5875	165	8	that	that	SCONJ
ejpam-5875	165	9	β	β	NOUN
ejpam-5875	165	10	can	can	AUX
ejpam-5875	165	11	not	not	PART
ejpam-5875	165	12	be	be	AUX
ejpam-5875	165	13	classified	classify	VERB
ejpam-5875	165	14	as	as	ADP
ejpam-5875	165	15	a	a	DET
ejpam-5875	165	16	q	q	ADJ
ejpam-5875	165	17	-	-	ADJ
ejpam-5875	165	18	normal	normal	ADJ
ejpam-5875	165	19	curve	curve	NOUN
ejpam-5875	165	20	.	.	PUNCT
ejpam-5875	166	1	corollary	corollary	ADJ
ejpam-5875	166	2	4	4	NUM
ejpam-5875	166	3	.	.	PUNCT
ejpam-5875	167	1	if	if	SCONJ
ejpam-5875	167	2	β(s	β(	NOUN
ejpam-5875	167	3	)	)	PUNCT
ejpam-5875	167	4	is	be	AUX
ejpam-5875	167	5	an	an	DET
ejpam-5875	167	6	admissible	admissible	ADJ
ejpam-5875	167	7	differentiable	differentiable	ADJ
ejpam-5875	167	8	curve	curve	NOUN
ejpam-5875	167	9	parameterized	parameterize	VERB
ejpam-5875	167	10	by	by	ADP
ejpam-5875	167	11	the	the	DET
ejpam-5875	167	12	galilean	galilean	PROPN
ejpam-5875	167	13	invariant	invariant	PROPN
ejpam-5875	167	14	arc	arc	NOUN
ejpam-5875	167	15	length	length	NOUN
ejpam-5875	167	16	s	s	PROPN
ejpam-5875	167	17	,	,	PUNCT
ejpam-5875	167	18	then	then	ADV
ejpam-5875	167	19	β(s	β(	NOUN
ejpam-5875	167	20	)	)	PUNCT
ejpam-5875	167	21	does	do	AUX
ejpam-5875	167	22	not	not	PART
ejpam-5875	167	23	constitute	constitute	VERB
ejpam-5875	167	24	a	a	DET
ejpam-5875	167	25	normal	normal	ADJ
ejpam-5875	167	26	curve	curve	NOUN
ejpam-5875	167	27	in	in	ADP
ejpam-5875	167	28	g3	g3	PROPN
ejpam-5875	167	29	or	or	CCONJ
ejpam-5875	167	30	gn	gn	PROPN
ejpam-5875	167	31	.	.	PUNCT
ejpam-5875	168	1	consequently	consequently	ADV
ejpam-5875	168	2	,	,	PUNCT
ejpam-5875	168	3	the	the	DET
ejpam-5875	168	4	results	result	NOUN
ejpam-5875	168	5	presented	present	VERB
ejpam-5875	168	6	in	in	ADP
ejpam-5875	168	7	[	[	X
ejpam-5875	168	8	23	23	NUM
ejpam-5875	168	9	]	]	PUNCT
ejpam-5875	168	10	are	be	AUX
ejpam-5875	168	11	invalid	invalid	ADJ
ejpam-5875	168	12	.	.	PUNCT
ejpam-5875	169	1	6	6	NUM
ejpam-5875	169	2	.	.	X
ejpam-5875	169	3	quasi	quasi	ADJ
ejpam-5875	169	4	-	-	ADJ
ejpam-5875	169	5	rectifying	rectifying	ADJ
ejpam-5875	169	6	curves	curve	NOUN
ejpam-5875	169	7	in	in	ADP
ejpam-5875	169	8	this	this	DET
ejpam-5875	169	9	subsection	subsection	NOUN
ejpam-5875	169	10	,	,	PUNCT
ejpam-5875	169	11	we	we	PRON
ejpam-5875	169	12	establish	establish	VERB
ejpam-5875	169	13	the	the	DET
ejpam-5875	169	14	fundamental	fundamental	ADJ
ejpam-5875	169	15	criteria	criterion	NOUN
ejpam-5875	169	16	for	for	ADP
ejpam-5875	169	17	characterizing	characterize	VERB
ejpam-5875	169	18	a	a	DET
ejpam-5875	169	19	q	q	NOUN
ejpam-5875	169	20	-	-	PUNCT
ejpam-5875	169	21	curve	curve	NOUN
ejpam-5875	169	22	with	with	ADP
ejpam-5875	169	23	position	position	NOUN
ejpam-5875	169	24	vector	vector	NOUN
ejpam-5875	169	25	γ(s	γ(	NOUN
ejpam-5875	169	26	)	)	PUNCT
ejpam-5875	169	27	as	as	ADP
ejpam-5875	169	28	a	a	DET
ejpam-5875	169	29	q	q	ADJ
ejpam-5875	169	30	-	-	PUNCT
ejpam-5875	169	31	rectifying	rectifying	ADJ
ejpam-5875	169	32	curve	curve	NOUN
ejpam-5875	169	33	within	within	ADP
ejpam-5875	169	34	g3	g3	PROPN
ejpam-5875	169	35	.	.	PUNCT
ejpam-5875	170	1	the	the	DET
ejpam-5875	170	2	definition	definition	NOUN
ejpam-5875	170	3	of	of	ADP
ejpam-5875	170	4	a	a	DET
ejpam-5875	170	5	qrectifying	qrectifying	NOUN
ejpam-5875	170	6	curve	curve	NOUN
ejpam-5875	170	7	is	be	AUX
ejpam-5875	170	8	predicated	predicate	VERB
ejpam-5875	170	9	on	on	ADP
ejpam-5875	170	10	its	its	PRON
ejpam-5875	170	11	geometric	geometric	ADJ
ejpam-5875	170	12	relationship	relationship	NOUN
ejpam-5875	170	13	to	to	ADP
ejpam-5875	170	14	the	the	DET
ejpam-5875	170	15	q	q	ADJ
ejpam-5875	170	16	-	-	PUNCT
ejpam-5875	170	17	rectifying	rectifying	ADJ
ejpam-5875	170	18	plane	plane	NOUN
ejpam-5875	170	19	.	.	PUNCT
ejpam-5875	171	1	specifically	specifically	ADV
ejpam-5875	171	2	,	,	PUNCT
ejpam-5875	171	3	a	a	DET
ejpam-5875	171	4	curve	curve	NOUN
ejpam-5875	171	5	γ	γ	NOUN
ejpam-5875	171	6	is	be	AUX
ejpam-5875	171	7	classified	classify	VERB
ejpam-5875	171	8	as	as	ADP
ejpam-5875	171	9	q	q	NOUN
ejpam-5875	171	10	-	-	PUNCT
ejpam-5875	171	11	rectifying	rectifying	ADJ
ejpam-5875	171	12	if	if	SCONJ
ejpam-5875	171	13	and	and	CCONJ
ejpam-5875	171	14	only	only	ADV
ejpam-5875	171	15	if	if	SCONJ
ejpam-5875	171	16	its	its	PRON
ejpam-5875	171	17	position	position	NOUN
ejpam-5875	171	18	vector	vector	NOUN
ejpam-5875	171	19	can	can	AUX
ejpam-5875	171	20	be	be	AUX
ejpam-5875	171	21	expressed	express	VERB
ejpam-5875	171	22	as	as	ADP
ejpam-5875	171	23	a	a	DET
ejpam-5875	171	24	linear	linear	ADJ
ejpam-5875	171	25	combination	combination	NOUN
ejpam-5875	171	26	of	of	ADP
ejpam-5875	171	27	its	its	PRON
ejpam-5875	171	28	tangent	tangent	NOUN
ejpam-5875	171	29	vector	vector	PROPN
ejpam-5875	171	30	t	t	PROPN
ejpam-5875	171	31	(	(	PUNCT
ejpam-5875	171	32	s	s	NOUN
ejpam-5875	171	33	)	)	PUNCT
ejpam-5875	171	34	and	and	CCONJ
ejpam-5875	171	35	q	q	ADJ
ejpam-5875	171	36	-	-	PUNCT
ejpam-5875	171	37	binormal	binormal	ADJ
ejpam-5875	171	38	vector	vector	NOUN
ejpam-5875	171	39	bq(s	bq(s	NOUN
ejpam-5875	171	40	)	)	PUNCT
ejpam-5875	171	41	,	,	PUNCT
ejpam-5875	171	42	such	such	ADJ
ejpam-5875	171	43	that	that	SCONJ
ejpam-5875	171	44	γ(s	γ(	NOUN
ejpam-5875	171	45	)	)	PUNCT
ejpam-5875	172	1	=	=	PUNCT
ejpam-5875	172	2	γ(s)t	γ(s)t	PROPN
ejpam-5875	172	3	(	(	PUNCT
ejpam-5875	172	4	s	s	NOUN
ejpam-5875	172	5	)	)	PUNCT
ejpam-5875	173	1	+	+	NOUN
ejpam-5875	174	1	ϵ(s)bq(s	ϵ(s)bq(s	NUM
ejpam-5875	174	2	)	)	PUNCT
ejpam-5875	174	3	,	,	PUNCT
ejpam-5875	174	4	where	where	SCONJ
ejpam-5875	174	5	γ(s	γ(	NOUN
ejpam-5875	174	6	)	)	PUNCT
ejpam-5875	174	7	and	and	CCONJ
ejpam-5875	174	8	ϵ(s	ϵ(s	PROPN
ejpam-5875	174	9	)	)	PUNCT
ejpam-5875	174	10	are	be	AUX
ejpam-5875	174	11	scalar	scalar	ADJ
ejpam-5875	174	12	functions	function	NOUN
ejpam-5875	174	13	.	.	PUNCT
ejpam-5875	175	1	theorem	theorem	NOUN
ejpam-5875	175	2	2	2	NUM
ejpam-5875	175	3	.	.	PUNCT
ejpam-5875	176	1	let	let	VERB
ejpam-5875	176	2	γ(s	γ(	NOUN
ejpam-5875	176	3	)	)	PUNCT
ejpam-5875	176	4	be	be	AUX
ejpam-5875	176	5	a	a	DET
ejpam-5875	176	6	q	q	ADJ
ejpam-5875	176	7	-	-	PUNCT
ejpam-5875	176	8	rectifying	rectifying	ADJ
ejpam-5875	176	9	curve	curve	NOUN
ejpam-5875	176	10	in	in	ADP
ejpam-5875	176	11	g3	g3	PROPN
ejpam-5875	176	12	.	.	PUNCT
ejpam-5875	177	1	then	then	ADV
ejpam-5875	177	2	the	the	DET
ejpam-5875	177	3	tangential	tangential	ADJ
ejpam-5875	177	4	and	and	CCONJ
ejpam-5875	177	5	the	the	DET
ejpam-5875	177	6	binormal	binormal	ADJ
ejpam-5875	177	7	components	component	NOUN
ejpam-5875	177	8	of	of	ADP
ejpam-5875	177	9	the	the	DET
ejpam-5875	177	10	position	position	NOUN
ejpam-5875	177	11	vector	vector	NOUN
ejpam-5875	177	12	γ(s	γ(	NOUN
ejpam-5875	177	13	)	)	PUNCT
ejpam-5875	177	14	are	be	AUX
ejpam-5875	177	15	given	give	VERB
ejpam-5875	177	16	,	,	PUNCT
ejpam-5875	177	17	respectively	respectively	ADV
ejpam-5875	177	18	,	,	PUNCT
ejpam-5875	177	19	by	by	ADP
ejpam-5875	177	20	g(γ	g(γ	PROPN
ejpam-5875	177	21	,	,	PUNCT
ejpam-5875	177	22	t	t	NOUN
ejpam-5875	177	23	)	)	PUNCT
ejpam-5875	178	1	=	=	PUNCT
ejpam-5875	179	1	c	c	X
ejpam-5875	179	2	+	+	SYM
ejpam-5875	179	3	s	s	X
ejpam-5875	179	4	,	,	PUNCT
ejpam-5875	179	5	g(γ	g(γ	PROPN
ejpam-5875	179	6	,	,	PUNCT
ejpam-5875	179	7	bq	bq	NOUN
ejpam-5875	179	8	)	)	PUNCT
ejpam-5875	179	9	=	=	PUNCT
ejpam-5875	180	1	(	(	PUNCT
ejpam-5875	180	2	c	c	X
ejpam-5875	180	3	+	+	SYM
ejpam-5875	180	4	s	s	X
ejpam-5875	180	5	)	)	PUNCT
ejpam-5875	180	6	k1	k1	NOUN
ejpam-5875	180	7	k3	k3	NOUN
ejpam-5875	180	8	.	.	PUNCT
ejpam-5875	181	1	a.	a.	NOUN
ejpam-5875	181	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	181	3	,	,	PUNCT
ejpam-5875	181	4	n.	n.	NOUN
ejpam-5875	181	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	181	6	/	/	SYM
ejpam-5875	181	7	eur	eur	PROPN
ejpam-5875	181	8	.	.	PUNCT
ejpam-5875	182	1	j.	j.	PROPN
ejpam-5875	182	2	pure	pure	PROPN
ejpam-5875	182	3	appl	appl	PROPN
ejpam-5875	182	4	.	.	PROPN
ejpam-5875	182	5	math	math	PROPN
ejpam-5875	182	6	,	,	PUNCT
ejpam-5875	182	7	18	18	NUM
ejpam-5875	182	8	(	(	PUNCT
ejpam-5875	182	9	2	2	NUM
ejpam-5875	182	10	)	)	PUNCT
ejpam-5875	182	11	(	(	PUNCT
ejpam-5875	182	12	2025	2025	NUM
ejpam-5875	182	13	)	)	PUNCT
ejpam-5875	182	14	,	,	PUNCT
ejpam-5875	182	15	5875	5875	NUM
ejpam-5875	182	16	9	9	NUM
ejpam-5875	182	17	of	of	ADP
ejpam-5875	182	18	15	15	NUM
ejpam-5875	182	19	proof	proof	NOUN
ejpam-5875	182	20	.	.	PUNCT
ejpam-5875	182	21	suppose	suppose	VERB
ejpam-5875	182	22	that	that	SCONJ
ejpam-5875	182	23	γ(s	γ(	NOUN
ejpam-5875	182	24	)	)	PUNCT
ejpam-5875	182	25	is	be	AUX
ejpam-5875	182	26	a	a	DET
ejpam-5875	182	27	q	q	ADJ
ejpam-5875	182	28	-	-	PUNCT
ejpam-5875	182	29	rectifying	rectifying	ADJ
ejpam-5875	182	30	curve	curve	NOUN
ejpam-5875	182	31	,	,	PUNCT
ejpam-5875	182	32	then	then	ADV
ejpam-5875	182	33	γ(s	γ(	NOUN
ejpam-5875	182	34	)	)	PUNCT
ejpam-5875	183	1	=	=	PUNCT
ejpam-5875	183	2	γ(s)t	γ(s)t	PROPN
ejpam-5875	183	3	(	(	PUNCT
ejpam-5875	183	4	s	s	NOUN
ejpam-5875	183	5	)	)	PUNCT
ejpam-5875	183	6	+	+	NOUN
ejpam-5875	183	7	ϵ(s)bq(s	ϵ(s)bq(s	NUM
ejpam-5875	183	8	)	)	PUNCT
ejpam-5875	183	9	,	,	PUNCT
ejpam-5875	183	10	(	(	PUNCT
ejpam-5875	183	11	6.1	6.1	NUM
ejpam-5875	183	12	)	)	PUNCT
ejpam-5875	183	13	for	for	ADP
ejpam-5875	183	14	some	some	DET
ejpam-5875	183	15	differentiable	differentiable	ADJ
ejpam-5875	183	16	functions	function	NOUN
ejpam-5875	183	17	γ(s	γ(	NOUN
ejpam-5875	183	18	)	)	PUNCT
ejpam-5875	183	19	and	and	CCONJ
ejpam-5875	183	20	ϵ(s	ϵ(s	PROPN
ejpam-5875	183	21	)	)	PUNCT
ejpam-5875	183	22	.	.	PUNCT
ejpam-5875	184	1	we	we	PRON
ejpam-5875	184	2	can	can	AUX
ejpam-5875	184	3	deduce	deduce	VERB
ejpam-5875	184	4	that	that	SCONJ
ejpam-5875	184	5	γ(s	γ(	NOUN
ejpam-5875	184	6	)	)	PUNCT
ejpam-5875	185	1	=	=	PUNCT
ejpam-5875	185	2	c	c	X
ejpam-5875	185	3	+	+	NUM
ejpam-5875	185	4	s	s	X
ejpam-5875	185	5	,	,	PUNCT
ejpam-5875	185	6	ϵ(s	ϵ(s	PROPN
ejpam-5875	185	7	)	)	PUNCT
ejpam-5875	185	8	=	=	PUNCT
ejpam-5875	186	1	(	(	PUNCT
ejpam-5875	186	2	c	c	X
ejpam-5875	186	3	+	+	SYM
ejpam-5875	186	4	s	s	X
ejpam-5875	186	5	)	)	PUNCT
ejpam-5875	186	6	k1	k1	NOUN
ejpam-5875	186	7	k3	k3	NOUN
ejpam-5875	186	8	.	.	PUNCT
ejpam-5875	187	1	therefore	therefore	ADV
ejpam-5875	187	2	,	,	PUNCT
ejpam-5875	187	3	g(γ	g(γ	PROPN
ejpam-5875	187	4	,	,	PUNCT
ejpam-5875	187	5	t	t	NOUN
ejpam-5875	187	6	)	)	PUNCT
ejpam-5875	188	1	=	=	PUNCT
ejpam-5875	189	1	c	c	X
ejpam-5875	189	2	+	+	SYM
ejpam-5875	189	3	s	s	X
ejpam-5875	189	4	,	,	PUNCT
ejpam-5875	189	5	g(γ	g(γ	PROPN
ejpam-5875	189	6	,	,	PUNCT
ejpam-5875	189	7	bq	bq	NOUN
ejpam-5875	189	8	)	)	PUNCT
ejpam-5875	189	9	=	=	PUNCT
ejpam-5875	190	1	(	(	PUNCT
ejpam-5875	190	2	c	c	X
ejpam-5875	190	3	+	+	SYM
ejpam-5875	190	4	s	s	X
ejpam-5875	190	5	)	)	PUNCT
ejpam-5875	190	6	k1	k1	NOUN
ejpam-5875	190	7	k3	k3	NOUN
ejpam-5875	190	8	.	.	PUNCT
ejpam-5875	191	1	thus	thus	ADV
ejpam-5875	191	2	,	,	PUNCT
ejpam-5875	191	3	γ(s	γ(	NOUN
ejpam-5875	191	4	)	)	PUNCT
ejpam-5875	191	5	=	=	PUNCT
ejpam-5875	192	1	(	(	PUNCT
ejpam-5875	192	2	c	c	NOUN
ejpam-5875	192	3	+	+	NOUN
ejpam-5875	192	4	s)t	s)t	X
ejpam-5875	193	1	+	+	CCONJ
ejpam-5875	193	2	(	(	PUNCT
ejpam-5875	193	3	c	c	X
ejpam-5875	193	4	+	+	NUM
ejpam-5875	193	5	s	s	X
ejpam-5875	193	6	)	)	PUNCT
ejpam-5875	193	7	k1	k1	NOUN
ejpam-5875	193	8	k3	k3	PROPN
ejpam-5875	193	9	bq	bq	PROPN
ejpam-5875	193	10	.	.	PUNCT
ejpam-5875	193	11	corollary	corollary	ADJ
ejpam-5875	193	12	5	5	NUM
ejpam-5875	193	13	.	.	PUNCT
ejpam-5875	193	14	let	let	VERB
ejpam-5875	193	15	γ(s	γ(	NOUN
ejpam-5875	193	16	)	)	PUNCT
ejpam-5875	193	17	be	be	AUX
ejpam-5875	193	18	a	a	DET
ejpam-5875	193	19	frenet	frenet	ADJ
ejpam-5875	193	20	rectifying	rectifying	NOUN
ejpam-5875	193	21	curve	curve	NOUN
ejpam-5875	193	22	in	in	ADP
ejpam-5875	193	23	g3	g3	PROPN
ejpam-5875	193	24	.	.	PUNCT
ejpam-5875	194	1	then	then	ADV
ejpam-5875	194	2	the	the	DET
ejpam-5875	194	3	frenet	frenet	ADJ
ejpam-5875	194	4	tangential	tangential	ADJ
ejpam-5875	194	5	and	and	CCONJ
ejpam-5875	194	6	the	the	DET
ejpam-5875	194	7	binormal	binormal	ADJ
ejpam-5875	194	8	components	component	NOUN
ejpam-5875	194	9	of	of	ADP
ejpam-5875	194	10	the	the	DET
ejpam-5875	194	11	position	position	NOUN
ejpam-5875	194	12	vector	vector	NOUN
ejpam-5875	194	13	are	be	AUX
ejpam-5875	194	14	given	give	VERB
ejpam-5875	194	15	,	,	PUNCT
ejpam-5875	194	16	respectively	respectively	ADV
ejpam-5875	194	17	,	,	PUNCT
ejpam-5875	194	18	by	by	ADP
ejpam-5875	194	19	g(γ	g(γ	PROPN
ejpam-5875	194	20	,	,	PUNCT
ejpam-5875	194	21	t	t	NOUN
ejpam-5875	194	22	)	)	PUNCT
ejpam-5875	195	1	=	=	PUNCT
ejpam-5875	196	1	c	c	X
ejpam-5875	196	2	+	+	SYM
ejpam-5875	196	3	s	s	X
ejpam-5875	196	4	,	,	PUNCT
ejpam-5875	196	5	g(γ	g(γ	PROPN
ejpam-5875	196	6	,	,	PUNCT
ejpam-5875	196	7	b	b	NOUN
ejpam-5875	196	8	)	)	PUNCT
ejpam-5875	196	9	=	=	PUNCT
ejpam-5875	197	1	(	(	PUNCT
ejpam-5875	197	2	c	c	X
ejpam-5875	197	3	+	+	NUM
ejpam-5875	197	4	s	s	X
ejpam-5875	197	5	)	)	PUNCT
ejpam-5875	197	6	κ	κ	PROPN
ejpam-5875	197	7	τ	τ	PROPN
ejpam-5875	197	8	.	.	PUNCT
ejpam-5875	198	1	theorem	theorem	NOUN
ejpam-5875	198	2	3	3	X
ejpam-5875	198	3	.	.	PUNCT
ejpam-5875	199	1	let	let	AUX
ejpam-5875	199	2	γ(s	γ(	NOUN
ejpam-5875	199	3	)	)	PUNCT
ejpam-5875	199	4	be	be	AUX
ejpam-5875	199	5	a	a	DET
ejpam-5875	199	6	quasi	quasi	ADJ
ejpam-5875	199	7	curve	curve	NOUN
ejpam-5875	199	8	with	with	ADP
ejpam-5875	199	9	position	position	NOUN
ejpam-5875	199	10	vector	vector	NOUN
ejpam-5875	199	11	in	in	ADP
ejpam-5875	199	12	g3	g3	PROPN
ejpam-5875	199	13	.	.	PUNCT
ejpam-5875	200	1	the	the	DET
ejpam-5875	200	2	curve	curve	NOUN
ejpam-5875	200	3	is	be	AUX
ejpam-5875	200	4	qrectifying	qrectifye	VERB
ejpam-5875	200	5	if	if	SCONJ
ejpam-5875	200	6	and	and	CCONJ
ejpam-5875	200	7	only	only	ADV
ejpam-5875	200	8	if	if	SCONJ
ejpam-5875	200	9	k1	k1	NOUN
ejpam-5875	200	10	k3	k3	NOUN
ejpam-5875	200	11	=	=	SYM
ejpam-5875	201	1	1	1	NUM
ejpam-5875	201	2	c	c	NOUN
ejpam-5875	201	3	+	+	SYM
ejpam-5875	201	4	s	s	PART
ejpam-5875	201	5	∫	∫	PROPN
ejpam-5875	201	6	(	(	PUNCT
ejpam-5875	201	7	c	c	PROPN
ejpam-5875	201	8	+	+	CCONJ
ejpam-5875	201	9	s)k2ds+	s)k2ds+	PROPN
ejpam-5875	201	10	c7	c7	PROPN
ejpam-5875	201	11	,	,	PUNCT
ejpam-5875	201	12	where	where	SCONJ
ejpam-5875	201	13	c7	c7	PROPN
ejpam-5875	201	14	is	be	AUX
ejpam-5875	201	15	constant	constant	ADJ
ejpam-5875	201	16	.	.	PUNCT
ejpam-5875	202	1	proof	proof	NOUN
ejpam-5875	202	2	.	.	PUNCT
ejpam-5875	203	1	suppose	suppose	VERB
ejpam-5875	203	2	that	that	SCONJ
ejpam-5875	203	3	γ(s	γ(	NOUN
ejpam-5875	203	4	)	)	PUNCT
ejpam-5875	203	5	is	be	AUX
ejpam-5875	203	6	a	a	DET
ejpam-5875	203	7	q	q	ADJ
ejpam-5875	203	8	-	-	PUNCT
ejpam-5875	203	9	rectifying	rectifying	NOUN
ejpam-5875	203	10	curve	curve	NOUN
ejpam-5875	203	11	.	.	PUNCT
ejpam-5875	204	1	so	so	ADV
ejpam-5875	204	2	,	,	PUNCT
ejpam-5875	204	3	we	we	PRON
ejpam-5875	204	4	can	can	AUX
ejpam-5875	204	5	put	put	VERB
ejpam-5875	204	6	m2	m2	PROPN
ejpam-5875	204	7	=	=	PROPN
ejpam-5875	204	8	0	0	NUM
ejpam-5875	204	9	in	in	ADP
ejpam-5875	204	10	equation	equation	NOUN
ejpam-5875	204	11	(	(	PUNCT
ejpam-5875	204	12	4.3	4.3	NUM
ejpam-5875	204	13	)	)	PUNCT
ejpam-5875	204	14	and	and	CCONJ
ejpam-5875	204	15	equation	equation	NOUN
ejpam-5875	204	16	(	(	PUNCT
ejpam-5875	204	17	4.4	4.4	NUM
ejpam-5875	204	18	)	)	PUNCT
ejpam-5875	204	19	,	,	PUNCT
ejpam-5875	204	20	we	we	PRON
ejpam-5875	204	21	obtain	obtain	VERB
ejpam-5875	204	22	:	:	PUNCT
ejpam-5875	204	23	(	(	PUNCT
ejpam-5875	204	24	c	c	X
ejpam-5875	204	25	+	+	PUNCT
ejpam-5875	204	26	s)k1	s)k1	PROPN
ejpam-5875	204	27	−	−	NOUN
ejpam-5875	204	28	ϵk3	ϵk3	NOUN
ejpam-5875	204	29	=	=	SYM
ejpam-5875	204	30	0	0	PROPN
ejpam-5875	204	31	,	,	PUNCT
ejpam-5875	204	32	−(c	−(c	NOUN
ejpam-5875	204	33	+	+	CCONJ
ejpam-5875	204	34	s)k2	s)k2	PROPN
ejpam-5875	204	35	+	+	CCONJ
ejpam-5875	204	36	ϵ′	ϵ′	NUM
ejpam-5875	204	37	=	=	NOUN
ejpam-5875	204	38	0	0	X
ejpam-5875	204	39	.	.	PUNCT
ejpam-5875	205	1	by	by	ADP
ejpam-5875	205	2	solving	solve	VERB
ejpam-5875	205	3	these	these	DET
ejpam-5875	205	4	equations	equation	NOUN
ejpam-5875	205	5	we	we	PRON
ejpam-5875	205	6	get	get	VERB
ejpam-5875	205	7	a	a	DET
ejpam-5875	205	8	linear	linear	ADJ
ejpam-5875	205	9	first	first	ADJ
ejpam-5875	205	10	-	-	PUNCT
ejpam-5875	205	11	order	order	NOUN
ejpam-5875	205	12	differential	differential	ADJ
ejpam-5875	205	13	equation	equation	NOUN
ejpam-5875	205	14	(	(	PUNCT
ejpam-5875	205	15	k1	k1	NOUN
ejpam-5875	205	16	k3	k3	PROPN
ejpam-5875	205	17	)	)	PUNCT
ejpam-5875	205	18	′	′	PUNCT
ejpam-5875	206	1	+	+	CCONJ
ejpam-5875	206	2	1	1	NUM
ejpam-5875	206	3	c	c	NOUN
ejpam-5875	206	4	+	+	SYM
ejpam-5875	206	5	s	s	NOUN
ejpam-5875	206	6	k1	k1	NOUN
ejpam-5875	206	7	k3	k3	NOUN
ejpam-5875	206	8	=	=	PROPN
ejpam-5875	206	9	k2	k2	PROPN
ejpam-5875	206	10	.	.	PUNCT
ejpam-5875	207	1	so	so	ADV
ejpam-5875	207	2	,	,	PUNCT
ejpam-5875	207	3	k1	k1	PROPN
ejpam-5875	207	4	k3	k3	NOUN
ejpam-5875	207	5	=	=	NOUN
ejpam-5875	207	6	1	1	NUM
ejpam-5875	207	7	c	c	NOUN
ejpam-5875	207	8	+	+	SYM
ejpam-5875	207	9	s	s	PART
ejpam-5875	207	10	∫	∫	PROPN
ejpam-5875	207	11	(	(	PUNCT
ejpam-5875	207	12	c	c	PROPN
ejpam-5875	207	13	+	+	CCONJ
ejpam-5875	207	14	s)k2ds+	s)k2ds+	PROPN
ejpam-5875	207	15	c7	c7	PROPN
ejpam-5875	207	16	.	.	PUNCT
ejpam-5875	208	1	(	(	PUNCT
ejpam-5875	208	2	6.2	6.2	NUM
ejpam-5875	208	3	)	)	PUNCT
ejpam-5875	208	4	conversely	conversely	ADV
ejpam-5875	208	5	,	,	PUNCT
ejpam-5875	208	6	let	let	VERB
ejpam-5875	208	7	the	the	DET
ejpam-5875	208	8	condition	condition	NOUN
ejpam-5875	208	9	(	(	PUNCT
ejpam-5875	208	10	6.2	6.2	NUM
ejpam-5875	208	11	)	)	PUNCT
ejpam-5875	208	12	be	be	AUX
ejpam-5875	208	13	satisfied	satisfied	ADJ
ejpam-5875	208	14	.	.	PUNCT
ejpam-5875	209	1	let	let	VERB
ejpam-5875	209	2	a	a	DET
ejpam-5875	209	3	vector	vector	NOUN
ejpam-5875	209	4	r	r	NOUN
ejpam-5875	209	5	be	be	VERB
ejpam-5875	209	6	as	as	SCONJ
ejpam-5875	209	7	follows	follow	VERB
ejpam-5875	209	8	r	r	NOUN
ejpam-5875	209	9	=	=	SYM
ejpam-5875	209	10	γ(s)−	γ(s)−	INTJ
ejpam-5875	209	11	(	(	PUNCT
ejpam-5875	209	12	c	c	NOUN
ejpam-5875	209	13	+	+	PUNCT
ejpam-5875	210	1	s)tq	s)tq	PROPN
ejpam-5875	210	2	−	−	PROPN
ejpam-5875	210	3	(	(	PUNCT
ejpam-5875	210	4	c	c	NOUN
ejpam-5875	210	5	+	+	PROPN
ejpam-5875	210	6	s	s	X
ejpam-5875	210	7	)	)	PUNCT
ejpam-5875	210	8	k1	k1	NOUN
ejpam-5875	210	9	k3	k3	PROPN
ejpam-5875	210	10	bq	bq	INTJ
ejpam-5875	210	11	.	.	PUNCT
ejpam-5875	211	1	(	(	PUNCT
ejpam-5875	211	2	6.3	6.3	NUM
ejpam-5875	211	3	)	)	PUNCT
ejpam-5875	211	4	a.	a.	NOUN
ejpam-5875	211	5	elsharkawy	elsharkawy	NOUN
ejpam-5875	211	6	,	,	PUNCT
ejpam-5875	211	7	n.	n.	NOUN
ejpam-5875	211	8	elsharkawy	elsharkawy	PROPN
ejpam-5875	211	9	/	/	SYM
ejpam-5875	211	10	eur	eur	PROPN
ejpam-5875	211	11	.	.	PUNCT
ejpam-5875	212	1	j.	j.	PROPN
ejpam-5875	212	2	pure	pure	PROPN
ejpam-5875	212	3	appl	appl	PROPN
ejpam-5875	212	4	.	.	PROPN
ejpam-5875	212	5	math	math	PROPN
ejpam-5875	212	6	,	,	PUNCT
ejpam-5875	212	7	18	18	NUM
ejpam-5875	212	8	(	(	PUNCT
ejpam-5875	212	9	2	2	NUM
ejpam-5875	212	10	)	)	PUNCT
ejpam-5875	212	11	(	(	PUNCT
ejpam-5875	212	12	2025	2025	NUM
ejpam-5875	212	13	)	)	PUNCT
ejpam-5875	212	14	,	,	PUNCT
ejpam-5875	212	15	5875	5875	NUM
ejpam-5875	212	16	10	10	NUM
ejpam-5875	212	17	of	of	ADP
ejpam-5875	212	18	15	15	NUM
ejpam-5875	212	19	by	by	ADP
ejpam-5875	212	20	differentiating	differentiate	VERB
ejpam-5875	212	21	equation	equation	NOUN
ejpam-5875	212	22	(	(	PUNCT
ejpam-5875	212	23	6.3	6.3	NUM
ejpam-5875	212	24	)	)	PUNCT
ejpam-5875	212	25	with	with	ADP
ejpam-5875	212	26	respect	respect	NOUN
ejpam-5875	212	27	to	to	ADP
ejpam-5875	212	28	s	s	PRON
ejpam-5875	212	29	and	and	CCONJ
ejpam-5875	212	30	using	use	VERB
ejpam-5875	212	31	equation	equation	NOUN
ejpam-5875	212	32	(	(	PUNCT
ejpam-5875	212	33	6.2	6.2	NUM
ejpam-5875	212	34	)	)	PUNCT
ejpam-5875	213	1	,	,	PUNCT
ejpam-5875	213	2	we	we	PRON
ejpam-5875	213	3	deduce	deduce	VERB
ejpam-5875	213	4	r′	r′	VERB
ejpam-5875	214	1	=	=	SYM
ejpam-5875	214	2	0	0	NUM
ejpam-5875	214	3	.	.	PUNCT
ejpam-5875	215	1	thus	thus	ADV
ejpam-5875	215	2	γ(s)−	γ(s)−	PROPN
ejpam-5875	215	3	r	r	NOUN
ejpam-5875	215	4	=	=	PUNCT
ejpam-5875	215	5	(	(	PUNCT
ejpam-5875	215	6	c	c	NOUN
ejpam-5875	215	7	+	+	PUNCT
ejpam-5875	216	1	s)tq	s)tq	PROPN
ejpam-5875	217	1	+	+	CCONJ
ejpam-5875	218	1	(	(	PUNCT
ejpam-5875	218	2	c	c	X
ejpam-5875	218	3	+	+	NUM
ejpam-5875	218	4	s	s	X
ejpam-5875	218	5	)	)	PUNCT
ejpam-5875	218	6	k1	k1	NOUN
ejpam-5875	218	7	k3	k3	PROPN
ejpam-5875	218	8	bq	bq	PROPN
ejpam-5875	218	9	.	.	PROPN
ejpam-5875	219	1	up	up	ADP
ejpam-5875	219	2	to	to	ADP
ejpam-5875	219	3	a	a	DET
ejpam-5875	219	4	transformation	transformation	NOUN
ejpam-5875	219	5	with	with	ADP
ejpam-5875	219	6	r	r	NOUN
ejpam-5875	219	7	,	,	PUNCT
ejpam-5875	219	8	we	we	PRON
ejpam-5875	219	9	find	find	VERB
ejpam-5875	219	10	γ	γ	NOUN
ejpam-5875	219	11	is	be	AUX
ejpam-5875	219	12	a	a	DET
ejpam-5875	219	13	q	q	ADJ
ejpam-5875	219	14	-	-	PUNCT
ejpam-5875	219	15	rectifying	rectifying	ADJ
ejpam-5875	219	16	curve	curve	NOUN
ejpam-5875	219	17	in	in	ADP
ejpam-5875	219	18	g3	g3	PROPN
ejpam-5875	219	19	.	.	PUNCT
ejpam-5875	220	1	corollary	corollary	ADJ
ejpam-5875	220	2	6	6	NUM
ejpam-5875	220	3	.	.	PUNCT
ejpam-5875	221	1	let	let	VERB
ejpam-5875	221	2	γ(s	γ(	NOUN
ejpam-5875	221	3	)	)	PUNCT
ejpam-5875	222	1	be	be	AUX
ejpam-5875	222	2	curve	curve	ADJ
ejpam-5875	222	3	with	with	ADP
ejpam-5875	222	4	position	position	NOUN
ejpam-5875	222	5	vector	vector	NOUN
ejpam-5875	222	6	in	in	ADP
ejpam-5875	222	7	g3	g3	PROPN
ejpam-5875	222	8	.	.	PUNCT
ejpam-5875	223	1	the	the	DET
ejpam-5875	223	2	curve	curve	NOUN
ejpam-5875	223	3	is	be	AUX
ejpam-5875	223	4	a	a	DET
ejpam-5875	223	5	frenet	frenet	ADJ
ejpam-5875	223	6	rectifying	rectifying	NOUN
ejpam-5875	223	7	curve	curve	NOUN
ejpam-5875	223	8	if	if	SCONJ
ejpam-5875	223	9	and	and	CCONJ
ejpam-5875	223	10	only	only	ADV
ejpam-5875	223	11	if	if	SCONJ
ejpam-5875	223	12	κ	κ	PROPN
ejpam-5875	223	13	τ	τ	PROPN
ejpam-5875	223	14	=	=	SYM
ejpam-5875	223	15	c∗	c∗	PROPN
ejpam-5875	223	16	7	7	NUM
ejpam-5875	223	17	c	c	NOUN
ejpam-5875	223	18	+	+	SYM
ejpam-5875	223	19	s	s	NOUN
ejpam-5875	223	20	,	,	PUNCT
ejpam-5875	223	21	where	where	SCONJ
ejpam-5875	223	22	c∗	c∗	PROPN
ejpam-5875	223	23	7	7	NUM
ejpam-5875	223	24	is	be	AUX
ejpam-5875	223	25	constant	constant	ADJ
ejpam-5875	223	26	.	.	PUNCT
ejpam-5875	224	1	7	7	X
ejpam-5875	224	2	.	.	X
ejpam-5875	224	3	quasi	quasi	ADJ
ejpam-5875	224	4	-	-	ADJ
ejpam-5875	224	5	osculating	osculating	ADJ
ejpam-5875	224	6	curves	curve	NOUN
ejpam-5875	224	7	in	in	ADP
ejpam-5875	224	8	this	this	DET
ejpam-5875	224	9	section	section	NOUN
ejpam-5875	224	10	,	,	PUNCT
ejpam-5875	224	11	we	we	PRON
ejpam-5875	224	12	establish	establish	VERB
ejpam-5875	224	13	the	the	DET
ejpam-5875	224	14	necessary	necessary	ADJ
ejpam-5875	224	15	and	and	CCONJ
ejpam-5875	224	16	sufficient	sufficient	ADJ
ejpam-5875	224	17	conditions	condition	NOUN
ejpam-5875	224	18	for	for	ADP
ejpam-5875	224	19	a	a	DET
ejpam-5875	224	20	curve	curve	NOUN
ejpam-5875	224	21	with	with	ADP
ejpam-5875	224	22	position	position	NOUN
ejpam-5875	224	23	vector	vector	NOUN
ejpam-5875	224	24	η(s	η(	NOUN
ejpam-5875	224	25	)	)	PUNCT
ejpam-5875	224	26	to	to	PART
ejpam-5875	224	27	be	be	AUX
ejpam-5875	224	28	classified	classify	VERB
ejpam-5875	224	29	as	as	ADP
ejpam-5875	224	30	a	a	DET
ejpam-5875	224	31	q	q	ADJ
ejpam-5875	224	32	-	-	PUNCT
ejpam-5875	224	33	osculating	osculate	VERB
ejpam-5875	224	34	curve	curve	NOUN
ejpam-5875	224	35	within	within	ADP
ejpam-5875	224	36	the	the	DET
ejpam-5875	224	37	framework	framework	NOUN
ejpam-5875	224	38	of	of	ADP
ejpam-5875	224	39	g3	g3	PROPN
ejpam-5875	224	40	.	.	PUNCT
ejpam-5875	225	1	a	a	DET
ejpam-5875	225	2	curve	curve	NOUN
ejpam-5875	225	3	η(s	η(s	PROPN
ejpam-5875	225	4	)	)	PUNCT
ejpam-5875	225	5	is	be	AUX
ejpam-5875	225	6	deemed	deem	VERB
ejpam-5875	225	7	a	a	DET
ejpam-5875	225	8	q	q	ADJ
ejpam-5875	225	9	-	-	PUNCT
ejpam-5875	225	10	osculating	osculate	VERB
ejpam-5875	225	11	curve	curve	NOUN
ejpam-5875	225	12	if	if	SCONJ
ejpam-5875	225	13	it	it	PRON
ejpam-5875	225	14	remains	remain	VERB
ejpam-5875	225	15	contained	contain	VERB
ejpam-5875	225	16	within	within	ADP
ejpam-5875	225	17	its	its	PRON
ejpam-5875	225	18	q	q	ADJ
ejpam-5875	225	19	-	-	PUNCT
ejpam-5875	225	20	osculating	osculate	VERB
ejpam-5875	225	21	plane	plane	NOUN
ejpam-5875	225	22	.	.	PUNCT
ejpam-5875	226	1	in	in	ADP
ejpam-5875	226	2	mathematical	mathematical	ADJ
ejpam-5875	226	3	terms	term	NOUN
ejpam-5875	226	4	,	,	PUNCT
ejpam-5875	226	5	the	the	DET
ejpam-5875	226	6	curve	curve	NOUN
ejpam-5875	226	7	η	η	PROPN
ejpam-5875	226	8	satisfies	satisfy	VERB
ejpam-5875	226	9	the	the	DET
ejpam-5875	226	10	following	follow	VERB
ejpam-5875	226	11	representation	representation	NOUN
ejpam-5875	226	12	:	:	PUNCT
ejpam-5875	226	13	η(s	η(s	PROPN
ejpam-5875	226	14	)	)	PUNCT
ejpam-5875	226	15	=	=	SYM
ejpam-5875	226	16	ϱ(s)t	ϱ(s)t	NOUN
ejpam-5875	226	17	(	(	PUNCT
ejpam-5875	226	18	s	s	NOUN
ejpam-5875	226	19	)	)	PUNCT
ejpam-5875	226	20	+	+	CCONJ
ejpam-5875	226	21	ε(s)nq(s	ε(s)nq(s	ADJ
ejpam-5875	226	22	)	)	PUNCT
ejpam-5875	226	23	,	,	PUNCT
ejpam-5875	226	24	where	where	SCONJ
ejpam-5875	226	25	t	t	PROPN
ejpam-5875	226	26	(	(	PUNCT
ejpam-5875	226	27	s	s	NOUN
ejpam-5875	226	28	)	)	PUNCT
ejpam-5875	226	29	and	and	CCONJ
ejpam-5875	226	30	nq(s	nq(s	NUM
ejpam-5875	226	31	)	)	PUNCT
ejpam-5875	226	32	denote	denote	VERB
ejpam-5875	226	33	the	the	DET
ejpam-5875	226	34	tangent	tangent	NOUN
ejpam-5875	226	35	and	and	CCONJ
ejpam-5875	226	36	q	q	ADJ
ejpam-5875	226	37	-	-	ADJ
ejpam-5875	226	38	normal	normal	ADJ
ejpam-5875	226	39	vectors	vector	NOUN
ejpam-5875	226	40	,	,	PUNCT
ejpam-5875	226	41	respectively	respectively	ADV
ejpam-5875	226	42	.	.	PUNCT
ejpam-5875	227	1	theorem	theorem	VERB
ejpam-5875	227	2	4	4	NUM
ejpam-5875	227	3	.	.	PUNCT
ejpam-5875	228	1	let	let	VERB
ejpam-5875	228	2	η(s	η(s	PROPN
ejpam-5875	228	3	)	)	PUNCT
ejpam-5875	228	4	be	be	VERB
ejpam-5875	228	5	a	a	DET
ejpam-5875	228	6	q	q	ADJ
ejpam-5875	228	7	-	-	PUNCT
ejpam-5875	228	8	osculating	osculate	VERB
ejpam-5875	228	9	curve	curve	NOUN
ejpam-5875	228	10	in	in	ADP
ejpam-5875	228	11	g3	g3	PROPN
ejpam-5875	228	12	.	.	PUNCT
ejpam-5875	229	1	then	then	ADV
ejpam-5875	229	2	the	the	DET
ejpam-5875	229	3	tangential	tangential	ADJ
ejpam-5875	229	4	and	and	CCONJ
ejpam-5875	229	5	q	q	ADJ
ejpam-5875	229	6	-	-	ADJ
ejpam-5875	229	7	normal	normal	ADJ
ejpam-5875	229	8	components	component	NOUN
ejpam-5875	229	9	of	of	ADP
ejpam-5875	229	10	the	the	DET
ejpam-5875	229	11	position	position	NOUN
ejpam-5875	229	12	vector	vector	NOUN
ejpam-5875	229	13	η	η	PROPN
ejpam-5875	229	14	can	can	AUX
ejpam-5875	229	15	be	be	AUX
ejpam-5875	229	16	expressed	express	VERB
ejpam-5875	229	17	as	as	SCONJ
ejpam-5875	229	18	follows	follow	VERB
ejpam-5875	229	19	:	:	PUNCT
ejpam-5875	229	20	g(η	g(η	VERB
ejpam-5875	229	21	,	,	PUNCT
ejpam-5875	229	22	t	t	NOUN
ejpam-5875	229	23	)	)	PUNCT
ejpam-5875	230	1	=	=	PUNCT
ejpam-5875	231	1	c	c	X
ejpam-5875	231	2	+	+	SYM
ejpam-5875	231	3	s	s	X
ejpam-5875	231	4	,	,	PUNCT
ejpam-5875	231	5	g(η	g(η	PROPN
ejpam-5875	231	6	,	,	PUNCT
ejpam-5875	231	7	nq	nq	PROPN
ejpam-5875	231	8	)	)	PUNCT
ejpam-5875	231	9	=	=	PUNCT
ejpam-5875	232	1	−	−	PROPN
ejpam-5875	232	2	∫	∫	PROPN
ejpam-5875	232	3	(	(	PUNCT
ejpam-5875	232	4	c	c	NOUN
ejpam-5875	232	5	+	+	PUNCT
ejpam-5875	232	6	s)k1	s)k1	PROPN
ejpam-5875	232	7	ds+	ds+	ADJ
ejpam-5875	232	8	c8	c8	NOUN
ejpam-5875	232	9	,	,	PUNCT
ejpam-5875	232	10	where	where	SCONJ
ejpam-5875	232	11	c	c	PROPN
ejpam-5875	232	12	and	and	CCONJ
ejpam-5875	232	13	c8	c8	PROPN
ejpam-5875	232	14	are	be	AUX
ejpam-5875	232	15	constants	constant	NOUN
ejpam-5875	232	16	.	.	PUNCT
ejpam-5875	233	1	proof	proof	NOUN
ejpam-5875	233	2	.	.	PUNCT
ejpam-5875	234	1	assuming	assume	VERB
ejpam-5875	234	2	that	that	SCONJ
ejpam-5875	234	3	η(s	η(	NOUN
ejpam-5875	234	4	)	)	PUNCT
ejpam-5875	234	5	is	be	AUX
ejpam-5875	234	6	a	a	DET
ejpam-5875	234	7	q	q	ADJ
ejpam-5875	234	8	-	-	PUNCT
ejpam-5875	234	9	osculating	osculating	NOUN
ejpam-5875	234	10	curve	curve	NOUN
ejpam-5875	234	11	,	,	PUNCT
ejpam-5875	234	12	we	we	PRON
ejpam-5875	234	13	can	can	AUX
ejpam-5875	234	14	write	write	VERB
ejpam-5875	234	15	:	:	PUNCT
ejpam-5875	234	16	η(s	η(s	PROPN
ejpam-5875	234	17	)	)	PUNCT
ejpam-5875	235	1	=	=	SYM
ejpam-5875	235	2	ϱ(s)t	ϱ(s)t	NOUN
ejpam-5875	235	3	(	(	PUNCT
ejpam-5875	235	4	s	s	NOUN
ejpam-5875	235	5	)	)	PUNCT
ejpam-5875	235	6	+	+	X
ejpam-5875	235	7	ε(s)nq(s	ε(s)nq(s	ADJ
ejpam-5875	235	8	)	)	PUNCT
ejpam-5875	235	9	.	.	PUNCT
ejpam-5875	236	1	from	from	ADP
ejpam-5875	236	2	this	this	DET
ejpam-5875	236	3	representation	representation	NOUN
ejpam-5875	236	4	,	,	PUNCT
ejpam-5875	236	5	we	we	PRON
ejpam-5875	236	6	derive	derive	VERB
ejpam-5875	236	7	:	:	PUNCT
ejpam-5875	236	8	ϱ(s	ϱ(s	NUM
ejpam-5875	236	9	)	)	PUNCT
ejpam-5875	237	1	=	=	PUNCT
ejpam-5875	238	1	c	c	X
ejpam-5875	238	2	+	+	NUM
ejpam-5875	238	3	s	s	PROPN
ejpam-5875	238	4	,	,	PUNCT
ejpam-5875	238	5	ε(s	ε(s	NOUN
ejpam-5875	238	6	)	)	PUNCT
ejpam-5875	238	7	=	=	PUNCT
ejpam-5875	239	1	−	−	PROPN
ejpam-5875	239	2	∫	∫	PROPN
ejpam-5875	239	3	(	(	PUNCT
ejpam-5875	239	4	c	c	NOUN
ejpam-5875	239	5	+	+	PUNCT
ejpam-5875	239	6	s)k1	s)k1	PROPN
ejpam-5875	239	7	ds+	ds+	ADJ
ejpam-5875	239	8	c8	c8	NOUN
ejpam-5875	239	9	.	.	PUNCT
ejpam-5875	240	1	thus	thus	ADV
ejpam-5875	240	2	,	,	PUNCT
ejpam-5875	240	3	we	we	PRON
ejpam-5875	240	4	find	find	VERB
ejpam-5875	240	5	that	that	SCONJ
ejpam-5875	240	6	:	:	PUNCT
ejpam-5875	240	7	g(η	g(η	VERB
ejpam-5875	240	8	,	,	PUNCT
ejpam-5875	240	9	t	t	NOUN
ejpam-5875	240	10	)	)	PUNCT
ejpam-5875	240	11	=	=	PUNCT
ejpam-5875	241	1	c	c	X
ejpam-5875	241	2	+	+	SYM
ejpam-5875	241	3	s	s	X
ejpam-5875	241	4	,	,	PUNCT
ejpam-5875	241	5	g(η	g(η	PROPN
ejpam-5875	241	6	,	,	PUNCT
ejpam-5875	241	7	nq	nq	PROPN
ejpam-5875	241	8	)	)	PUNCT
ejpam-5875	241	9	=	=	PUNCT
ejpam-5875	242	1	−	−	PROPN
ejpam-5875	242	2	∫	∫	PROPN
ejpam-5875	242	3	(	(	PUNCT
ejpam-5875	242	4	c	c	NOUN
ejpam-5875	242	5	+	+	PUNCT
ejpam-5875	242	6	s)k1	s)k1	PROPN
ejpam-5875	242	7	ds+	ds+	ADJ
ejpam-5875	242	8	c8	c8	NOUN
ejpam-5875	242	9	.	.	PUNCT
ejpam-5875	243	1	consequently	consequently	ADV
ejpam-5875	243	2	,	,	PUNCT
ejpam-5875	243	3	we	we	PRON
ejpam-5875	243	4	can	can	AUX
ejpam-5875	243	5	express	express	VERB
ejpam-5875	243	6	η(s	η(s	PROPN
ejpam-5875	243	7	)	)	PUNCT
ejpam-5875	243	8	as	as	ADP
ejpam-5875	243	9	:	:	PUNCT
ejpam-5875	243	10	η(s	η(	NOUN
ejpam-5875	243	11	)	)	PUNCT
ejpam-5875	243	12	=	=	PUNCT
ejpam-5875	244	1	(	(	PUNCT
ejpam-5875	244	2	c	c	NOUN
ejpam-5875	244	3	+	+	NOUN
ejpam-5875	244	4	s)t	s)t	X
ejpam-5875	245	1	+	+	CCONJ
ejpam-5875	245	2	[	[	PUNCT
ejpam-5875	245	3	−	−	NUM
ejpam-5875	245	4	∫	∫	PROPN
ejpam-5875	245	5	(	(	PUNCT
ejpam-5875	245	6	c	c	NOUN
ejpam-5875	245	7	+	+	PUNCT
ejpam-5875	245	8	s)k1	s)k1	PROPN
ejpam-5875	245	9	ds+	ds+	ADJ
ejpam-5875	245	10	c8	c8	NOUN
ejpam-5875	245	11	]	]	PUNCT
ejpam-5875	245	12	nq	nq	PROPN
ejpam-5875	245	13	,	,	PUNCT
ejpam-5875	245	14	where	where	SCONJ
ejpam-5875	245	15	c	c	PROPN
ejpam-5875	245	16	and	and	CCONJ
ejpam-5875	245	17	c8	c8	PROPN
ejpam-5875	245	18	are	be	AUX
ejpam-5875	245	19	constants	constant	NOUN
ejpam-5875	245	20	.	.	PUNCT
ejpam-5875	246	1	a.	a.	NOUN
ejpam-5875	246	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	246	3	,	,	PUNCT
ejpam-5875	246	4	n.	n.	NOUN
ejpam-5875	246	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	246	6	/	/	SYM
ejpam-5875	246	7	eur	eur	PROPN
ejpam-5875	246	8	.	.	PUNCT
ejpam-5875	247	1	j.	j.	PROPN
ejpam-5875	247	2	pure	pure	PROPN
ejpam-5875	247	3	appl	appl	PROPN
ejpam-5875	247	4	.	.	PROPN
ejpam-5875	247	5	math	math	PROPN
ejpam-5875	247	6	,	,	PUNCT
ejpam-5875	247	7	18	18	NUM
ejpam-5875	247	8	(	(	PUNCT
ejpam-5875	247	9	2	2	NUM
ejpam-5875	247	10	)	)	PUNCT
ejpam-5875	247	11	(	(	PUNCT
ejpam-5875	247	12	2025	2025	NUM
ejpam-5875	247	13	)	)	PUNCT
ejpam-5875	247	14	,	,	PUNCT
ejpam-5875	247	15	5875	5875	NUM
ejpam-5875	247	16	11	11	NUM
ejpam-5875	247	17	of	of	ADP
ejpam-5875	247	18	15	15	NUM
ejpam-5875	247	19	corollary	corollary	ADJ
ejpam-5875	247	20	7	7	NUM
ejpam-5875	247	21	.	.	PUNCT
ejpam-5875	248	1	if	if	SCONJ
ejpam-5875	248	2	the	the	DET
ejpam-5875	248	3	curve	curve	NOUN
ejpam-5875	248	4	η(s	η(s	PROPN
ejpam-5875	248	5	)	)	PUNCT
ejpam-5875	248	6	is	be	AUX
ejpam-5875	248	7	a	a	DET
ejpam-5875	248	8	frenet	frenet	NOUN
ejpam-5875	248	9	osculating	osculating	NOUN
ejpam-5875	248	10	curve	curve	NOUN
ejpam-5875	248	11	,	,	PUNCT
ejpam-5875	248	12	then	then	ADV
ejpam-5875	248	13	its	its	PRON
ejpam-5875	248	14	tangential	tangential	ADJ
ejpam-5875	248	15	and	and	CCONJ
ejpam-5875	248	16	normal	normal	ADJ
ejpam-5875	248	17	components	component	NOUN
ejpam-5875	248	18	are	be	AUX
ejpam-5875	248	19	given	give	VERB
ejpam-5875	248	20	by	by	ADP
ejpam-5875	248	21	:	:	PUNCT
ejpam-5875	248	22	g(η	g(η	PROPN
ejpam-5875	248	23	,	,	PUNCT
ejpam-5875	248	24	t	t	NOUN
ejpam-5875	248	25	)	)	PUNCT
ejpam-5875	249	1	=	=	PUNCT
ejpam-5875	250	1	c	c	X
ejpam-5875	250	2	+	+	SYM
ejpam-5875	250	3	s	s	PROPN
ejpam-5875	250	4	,	,	PUNCT
ejpam-5875	250	5	g(η	g(η	VERB
ejpam-5875	250	6	,	,	PUNCT
ejpam-5875	250	7	n	n	CCONJ
ejpam-5875	250	8	)	)	PUNCT
ejpam-5875	250	9	=	=	SYM
ejpam-5875	251	1	−	−	PROPN
ejpam-5875	251	2	∫	∫	PROPN
ejpam-5875	251	3	(	(	PUNCT
ejpam-5875	251	4	c	c	NOUN
ejpam-5875	251	5	+	+	CCONJ
ejpam-5875	251	6	s)κ	s)κ	X
ejpam-5875	251	7	ds+	ds+	PROPN
ejpam-5875	251	8	c∗	c∗	PROPN
ejpam-5875	251	9	,	,	PUNCT
ejpam-5875	251	10	where	where	SCONJ
ejpam-5875	251	11	c∗	c∗	PROPN
ejpam-5875	251	12	is	be	AUX
ejpam-5875	251	13	a	a	DET
ejpam-5875	251	14	constant	constant	ADJ
ejpam-5875	251	15	.	.	PUNCT
ejpam-5875	252	1	theorem	theorem	NOUN
ejpam-5875	252	2	5	5	NUM
ejpam-5875	252	3	.	.	PUNCT
ejpam-5875	253	1	consider	consider	VERB
ejpam-5875	253	2	a	a	DET
ejpam-5875	253	3	quasi	quasi	ADJ
ejpam-5875	253	4	curve	curve	NOUN
ejpam-5875	253	5	with	with	ADP
ejpam-5875	253	6	position	position	NOUN
ejpam-5875	253	7	vector	vector	NOUN
ejpam-5875	253	8	η(s	η(s	PROPN
ejpam-5875	253	9	)	)	PUNCT
ejpam-5875	253	10	in	in	ADP
ejpam-5875	253	11	g3	g3	PROPN
ejpam-5875	253	12	.	.	PUNCT
ejpam-5875	254	1	the	the	DET
ejpam-5875	254	2	curve	curve	NOUN
ejpam-5875	254	3	is	be	AUX
ejpam-5875	254	4	classified	classify	VERB
ejpam-5875	254	5	as	as	ADP
ejpam-5875	254	6	q	q	NOUN
ejpam-5875	254	7	-	-	PUNCT
ejpam-5875	254	8	osculating	osculating	NOUN
ejpam-5875	254	9	if	if	SCONJ
ejpam-5875	255	1	and	and	CCONJ
ejpam-5875	255	2	only	only	ADV
ejpam-5875	255	3	if	if	SCONJ
ejpam-5875	255	4	the	the	DET
ejpam-5875	255	5	following	follow	VERB
ejpam-5875	255	6	condition	condition	NOUN
ejpam-5875	255	7	holds	hold	VERB
ejpam-5875	255	8	:	:	PUNCT
ejpam-5875	255	9	k2	k2	ADJ
ejpam-5875	255	10	k3	k3	NOUN
ejpam-5875	255	11	=	=	SYM
ejpam-5875	255	12	1	1	NUM
ejpam-5875	255	13	c	c	NOUN
ejpam-5875	256	1	+	+	SYM
ejpam-5875	256	2	s	s	PART
ejpam-5875	256	3	∫	∫	PROPN
ejpam-5875	257	1	[	[	X
ejpam-5875	257	2	−(c	−(c	NOUN
ejpam-5875	257	3	+	+	CCONJ
ejpam-5875	257	4	s)k1	s)k1	PROPN
ejpam-5875	257	5	]	]	PUNCT
ejpam-5875	257	6	ds+	ds+	ADJ
ejpam-5875	257	7	c9	c9	NOUN
ejpam-5875	257	8	,	,	PUNCT
ejpam-5875	257	9	where	where	SCONJ
ejpam-5875	257	10	c9	c9	NOUN
ejpam-5875	257	11	is	be	AUX
ejpam-5875	257	12	a	a	DET
ejpam-5875	257	13	constant	constant	ADJ
ejpam-5875	257	14	.	.	PUNCT
ejpam-5875	258	1	proof	proof	NOUN
ejpam-5875	258	2	.	.	PUNCT
ejpam-5875	259	1	suppose	suppose	VERB
ejpam-5875	259	2	that	that	SCONJ
ejpam-5875	259	3	η(s	η(s	PROPN
ejpam-5875	259	4	)	)	PUNCT
ejpam-5875	259	5	is	be	AUX
ejpam-5875	259	6	a	a	DET
ejpam-5875	259	7	q	q	ADJ
ejpam-5875	259	8	-	-	PUNCT
ejpam-5875	259	9	osculating	osculating	NOUN
ejpam-5875	259	10	curve	curve	NOUN
ejpam-5875	259	11	.	.	PUNCT
ejpam-5875	260	1	then	then	ADV
ejpam-5875	260	2	,	,	PUNCT
ejpam-5875	260	3	setting	set	VERB
ejpam-5875	260	4	m3	m3	PROPN
ejpam-5875	260	5	=	=	PUNCT
ejpam-5875	260	6	0	0	NUM
ejpam-5875	260	7	in	in	ADP
ejpam-5875	260	8	equations	equation	NOUN
ejpam-5875	260	9	(	(	PUNCT
ejpam-5875	260	10	4.3	4.3	NUM
ejpam-5875	260	11	)	)	PUNCT
ejpam-5875	260	12	and	and	CCONJ
ejpam-5875	260	13	(	(	PUNCT
ejpam-5875	260	14	4.4	4.4	NUM
ejpam-5875	260	15	)	)	PUNCT
ejpam-5875	260	16	,	,	PUNCT
ejpam-5875	260	17	we	we	PRON
ejpam-5875	260	18	obtain	obtain	VERB
ejpam-5875	260	19	:	:	PUNCT
ejpam-5875	260	20	(	(	PUNCT
ejpam-5875	260	21	c	c	X
ejpam-5875	260	22	+	+	PUNCT
ejpam-5875	260	23	s)k1	s)k1	PROPN
ejpam-5875	260	24	+	+	CCONJ
ejpam-5875	260	25	ϵ′	ϵ′	NOUN
ejpam-5875	260	26	=	=	SYM
ejpam-5875	260	27	0	0	NUM
ejpam-5875	260	28	,	,	PUNCT
ejpam-5875	260	29	(	(	PUNCT
ejpam-5875	260	30	7.1	7.1	NUM
ejpam-5875	260	31	)	)	PUNCT
ejpam-5875	260	32	−(c	−(c	NOUN
ejpam-5875	260	33	+	+	CCONJ
ejpam-5875	260	34	s)k2	s)k2	PROPN
ejpam-5875	260	35	+	+	CCONJ
ejpam-5875	260	36	ϵk3	ϵk3	NOUN
ejpam-5875	261	1	=	=	NOUN
ejpam-5875	262	1	0	0	PROPN
ejpam-5875	262	2	.	.	PUNCT
ejpam-5875	263	1	(	(	PUNCT
ejpam-5875	263	2	7.2	7.2	NUM
ejpam-5875	263	3	)	)	PUNCT
ejpam-5875	263	4	from	from	ADP
ejpam-5875	263	5	equation	equation	NOUN
ejpam-5875	263	6	(	(	PUNCT
ejpam-5875	263	7	7.1	7.1	NUM
ejpam-5875	263	8	)	)	PUNCT
ejpam-5875	263	9	,	,	PUNCT
ejpam-5875	263	10	we	we	PRON
ejpam-5875	263	11	deduce	deduce	VERB
ejpam-5875	263	12	m2	m2	PROPN
ejpam-5875	263	13	=	=	SYM
ejpam-5875	264	1	−	−	PROPN
ejpam-5875	264	2	∫	∫	PROPN
ejpam-5875	264	3	(	(	PUNCT
ejpam-5875	264	4	c	c	NOUN
ejpam-5875	264	5	+	+	CCONJ
ejpam-5875	264	6	s)k1ds	s)k1ds	NOUN
ejpam-5875	264	7	+	+	CCONJ
ejpam-5875	264	8	c9	c9	NOUN
ejpam-5875	264	9	,	,	PUNCT
ejpam-5875	264	10	and	and	CCONJ
ejpam-5875	264	11	by	by	ADP
ejpam-5875	264	12	solving	solve	VERB
ejpam-5875	264	13	these	these	DET
ejpam-5875	264	14	equations	equation	NOUN
ejpam-5875	264	15	we	we	PRON
ejpam-5875	264	16	get	get	VERB
ejpam-5875	264	17	a	a	DET
ejpam-5875	264	18	linear	linear	ADJ
ejpam-5875	264	19	first	first	ADJ
ejpam-5875	264	20	-	-	PUNCT
ejpam-5875	264	21	order	order	NOUN
ejpam-5875	264	22	differential	differential	ADJ
ejpam-5875	264	23	equation	equation	NOUN
ejpam-5875	264	24	(	(	PUNCT
ejpam-5875	264	25	k2	k2	NOUN
ejpam-5875	264	26	k3	k3	PROPN
ejpam-5875	264	27	)	)	PUNCT
ejpam-5875	264	28	′	′	PUNCT
ejpam-5875	265	1	+	+	CCONJ
ejpam-5875	265	2	1	1	NUM
ejpam-5875	265	3	c	c	NOUN
ejpam-5875	265	4	+	+	SYM
ejpam-5875	265	5	s	s	X
ejpam-5875	265	6	(	(	PUNCT
ejpam-5875	265	7	k2	k2	NOUN
ejpam-5875	265	8	k3	k3	ADJ
ejpam-5875	265	9	)	)	PUNCT
ejpam-5875	265	10	=	=	SYM
ejpam-5875	265	11	−k1	−k1	NOUN
ejpam-5875	265	12	.	.	PUNCT
ejpam-5875	266	1	so	so	ADV
ejpam-5875	266	2	,	,	PUNCT
ejpam-5875	266	3	k2	k2	ADJ
ejpam-5875	266	4	k3	k3	NOUN
ejpam-5875	266	5	=	=	SYM
ejpam-5875	266	6	1	1	NUM
ejpam-5875	266	7	c	c	NOUN
ejpam-5875	266	8	+	+	SYM
ejpam-5875	266	9	s	s	PART
ejpam-5875	266	10	∫	∫	PROPN
ejpam-5875	266	11	−(c	−(c	NOUN
ejpam-5875	266	12	+	+	CCONJ
ejpam-5875	266	13	s)k1ds+	s)k1ds+	NOUN
ejpam-5875	266	14	c10	c10	VERB
ejpam-5875	266	15	.	.	PUNCT
ejpam-5875	267	1	(	(	PUNCT
ejpam-5875	267	2	7.3	7.3	NUM
ejpam-5875	267	3	)	)	PUNCT
ejpam-5875	267	4	conversely	conversely	ADV
ejpam-5875	267	5	,	,	PUNCT
ejpam-5875	267	6	let	let	VERB
ejpam-5875	267	7	the	the	DET
ejpam-5875	267	8	condition	condition	NOUN
ejpam-5875	267	9	(	(	PUNCT
ejpam-5875	267	10	7.3	7.3	NUM
ejpam-5875	267	11	)	)	PUNCT
ejpam-5875	267	12	be	be	AUX
ejpam-5875	267	13	satisfied	satisfied	ADJ
ejpam-5875	267	14	.	.	PUNCT
ejpam-5875	268	1	let	let	VERB
ejpam-5875	268	2	a	a	DET
ejpam-5875	268	3	vector	vector	NOUN
ejpam-5875	268	4	p	p	NOUN
ejpam-5875	268	5	be	be	AUX
ejpam-5875	268	6	as	as	SCONJ
ejpam-5875	268	7	follows	follow	VERB
ejpam-5875	268	8	p	p	NOUN
ejpam-5875	268	9	=	=	PUNCT
ejpam-5875	268	10	η(s)−	η(s)−	PROPN
ejpam-5875	268	11	(	(	PUNCT
ejpam-5875	268	12	c	c	NOUN
ejpam-5875	268	13	+	+	PUNCT
ejpam-5875	269	1	s)tq	s)tq	PROPN
ejpam-5875	269	2	−	−	PROPN
ejpam-5875	269	3	(	(	PUNCT
ejpam-5875	269	4	c	c	NOUN
ejpam-5875	269	5	+	+	PROPN
ejpam-5875	269	6	s	s	X
ejpam-5875	269	7	)	)	PUNCT
ejpam-5875	269	8	k2	k2	PROPN
ejpam-5875	269	9	k3	k3	PROPN
ejpam-5875	269	10	nq	nq	PROPN
ejpam-5875	269	11	.	.	PROPN
ejpam-5875	269	12	(	(	PUNCT
ejpam-5875	269	13	7.4	7.4	NUM
ejpam-5875	269	14	)	)	PUNCT
ejpam-5875	269	15	by	by	ADP
ejpam-5875	269	16	differentiating	differentiate	VERB
ejpam-5875	269	17	equation(7.4	equation(7.4	NOUN
ejpam-5875	269	18	)	)	PUNCT
ejpam-5875	269	19	,	,	PUNCT
ejpam-5875	269	20	we	we	PRON
ejpam-5875	269	21	deduce	deduce	VERB
ejpam-5875	269	22	p′	p′	NOUN
ejpam-5875	269	23	=	=	SYM
ejpam-5875	269	24	0	0	NUM
ejpam-5875	269	25	,	,	PUNCT
ejpam-5875	269	26	therefore	therefore	ADV
ejpam-5875	269	27	p	p	NOUN
ejpam-5875	269	28	is	be	AUX
ejpam-5875	269	29	constant	constant	ADJ
ejpam-5875	269	30	.	.	PUNCT
ejpam-5875	270	1	thus	thus	ADV
ejpam-5875	270	2	η(s)−	η(s)−	PROPN
ejpam-5875	270	3	p	p	PROPN
ejpam-5875	271	1	=	=	PUNCT
ejpam-5875	272	1	(	(	PUNCT
ejpam-5875	272	2	c	c	NOUN
ejpam-5875	272	3	+	+	PUNCT
ejpam-5875	273	1	s)tq	s)tq	PROPN
ejpam-5875	273	2	+	+	CCONJ
ejpam-5875	273	3	(	(	PUNCT
ejpam-5875	273	4	c	c	X
ejpam-5875	273	5	+	+	NOUN
ejpam-5875	273	6	s	s	X
ejpam-5875	273	7	)	)	PUNCT
ejpam-5875	273	8	k2	k2	PROPN
ejpam-5875	273	9	k3	k3	PROPN
ejpam-5875	273	10	nq	nq	PROPN
ejpam-5875	273	11	.	.	PROPN
ejpam-5875	273	12	up	up	ADP
ejpam-5875	273	13	to	to	ADP
ejpam-5875	273	14	a	a	DET
ejpam-5875	273	15	transformation	transformation	NOUN
ejpam-5875	273	16	with	with	ADP
ejpam-5875	273	17	p	p	NOUN
ejpam-5875	273	18	,	,	PUNCT
ejpam-5875	273	19	we	we	PRON
ejpam-5875	273	20	find	find	VERB
ejpam-5875	273	21	η	η	PROPN
ejpam-5875	273	22	is	be	AUX
ejpam-5875	273	23	a	a	DET
ejpam-5875	273	24	q	q	ADJ
ejpam-5875	273	25	-	-	PUNCT
ejpam-5875	273	26	osculating	osculate	VERB
ejpam-5875	273	27	curve	curve	NOUN
ejpam-5875	273	28	in	in	ADP
ejpam-5875	273	29	g3	g3	PROPN
ejpam-5875	273	30	.	.	PUNCT
ejpam-5875	274	1	corollary	corollary	ADJ
ejpam-5875	274	2	8	8	NUM
ejpam-5875	274	3	.	.	PUNCT
ejpam-5875	275	1	let	let	VERB
ejpam-5875	275	2	η(s	η(	NOUN
ejpam-5875	275	3	)	)	PUNCT
ejpam-5875	275	4	be	be	AUX
ejpam-5875	275	5	curve	curve	ADJ
ejpam-5875	275	6	with	with	ADP
ejpam-5875	275	7	position	position	NOUN
ejpam-5875	275	8	vector	vector	NOUN
ejpam-5875	275	9	in	in	ADP
ejpam-5875	275	10	g3	g3	PROPN
ejpam-5875	275	11	.	.	PUNCT
ejpam-5875	276	1	the	the	DET
ejpam-5875	276	2	curve	curve	NOUN
ejpam-5875	276	3	is	be	AUX
ejpam-5875	276	4	a	a	DET
ejpam-5875	276	5	frenet	frenet	NOUN
ejpam-5875	276	6	osculating	osculating	NOUN
ejpam-5875	276	7	curve	curve	NOUN
ejpam-5875	276	8	if	if	SCONJ
ejpam-5875	276	9	and	and	CCONJ
ejpam-5875	276	10	only	only	ADV
ejpam-5875	276	11	if	if	SCONJ
ejpam-5875	276	12	∫	∫	PROPN
ejpam-5875	276	13	(	(	PUNCT
ejpam-5875	276	14	c	c	NOUN
ejpam-5875	276	15	+	+	NOUN
ejpam-5875	276	16	s)κds	s)κds	NOUN
ejpam-5875	276	17	=	=	PUNCT
ejpam-5875	276	18	c∗∗	c∗∗	NOUN
ejpam-5875	276	19	,	,	PUNCT
ejpam-5875	276	20	where	where	SCONJ
ejpam-5875	276	21	c	c	NOUN
ejpam-5875	276	22	and	and	CCONJ
ejpam-5875	276	23	c∗∗	c∗∗	NOUN
ejpam-5875	276	24	are	be	AUX
ejpam-5875	276	25	constants	constant	NOUN
ejpam-5875	276	26	.	.	PUNCT
ejpam-5875	277	1	a.	a.	NOUN
ejpam-5875	277	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	277	3	,	,	PUNCT
ejpam-5875	277	4	n.	n.	NOUN
ejpam-5875	277	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	277	6	/	/	SYM
ejpam-5875	277	7	eur	eur	PROPN
ejpam-5875	277	8	.	.	PUNCT
ejpam-5875	278	1	j.	j.	PROPN
ejpam-5875	278	2	pure	pure	PROPN
ejpam-5875	278	3	appl	appl	PROPN
ejpam-5875	278	4	.	.	PROPN
ejpam-5875	278	5	math	math	PROPN
ejpam-5875	278	6	,	,	PUNCT
ejpam-5875	278	7	18	18	NUM
ejpam-5875	278	8	(	(	PUNCT
ejpam-5875	278	9	2	2	NUM
ejpam-5875	278	10	)	)	PUNCT
ejpam-5875	278	11	(	(	PUNCT
ejpam-5875	278	12	2025	2025	NUM
ejpam-5875	278	13	)	)	PUNCT
ejpam-5875	278	14	,	,	PUNCT
ejpam-5875	278	15	5875	5875	NUM
ejpam-5875	278	16	12	12	NUM
ejpam-5875	278	17	of	of	ADP
ejpam-5875	278	18	15	15	NUM
ejpam-5875	278	19	8	8	NUM
ejpam-5875	278	20	.	.	PUNCT
ejpam-5875	279	1	conclusion	conclusion	NOUN
ejpam-5875	279	2	in	in	ADP
ejpam-5875	279	3	this	this	DET
ejpam-5875	279	4	paper	paper	NOUN
ejpam-5875	280	1	,	,	PUNCT
ejpam-5875	280	2	we	we	PRON
ejpam-5875	280	3	have	have	AUX
ejpam-5875	280	4	investigated	investigate	VERB
ejpam-5875	280	5	the	the	DET
ejpam-5875	280	6	theoretical	theoretical	ADJ
ejpam-5875	280	7	foundations	foundation	NOUN
ejpam-5875	280	8	of	of	ADP
ejpam-5875	280	9	the	the	DET
ejpam-5875	280	10	quasi	quasi	NOUN
ejpam-5875	280	11	-	-	NOUN
ejpam-5875	280	12	frame	frame	NOUN
ejpam-5875	280	13	(	(	PUNCT
ejpam-5875	280	14	q	q	NOUN
ejpam-5875	280	15	-	-	PUNCT
ejpam-5875	280	16	frame	frame	NOUN
ejpam-5875	280	17	)	)	PUNCT
ejpam-5875	280	18	in	in	ADP
ejpam-5875	280	19	three	three	NUM
ejpam-5875	280	20	-	-	PUNCT
ejpam-5875	280	21	dimensional	dimensional	ADJ
ejpam-5875	280	22	galilean	galilean	PROPN
ejpam-5875	280	23	space	space	NOUN
ejpam-5875	280	24	(	(	PUNCT
ejpam-5875	280	25	g3	g3	NOUN
ejpam-5875	280	26	)	)	PUNCT
ejpam-5875	280	27	and	and	CCONJ
ejpam-5875	280	28	explored	explore	VERB
ejpam-5875	280	29	its	its	PRON
ejpam-5875	280	30	applications	application	NOUN
ejpam-5875	280	31	to	to	ADP
ejpam-5875	280	32	the	the	DET
ejpam-5875	280	33	study	study	NOUN
ejpam-5875	280	34	of	of	ADP
ejpam-5875	280	35	curves	curve	NOUN
ejpam-5875	280	36	in	in	ADP
ejpam-5875	280	37	this	this	DET
ejpam-5875	280	38	geometric	geometric	ADJ
ejpam-5875	280	39	framework	framework	NOUN
ejpam-5875	280	40	.	.	PUNCT
ejpam-5875	281	1	our	our	PRON
ejpam-5875	281	2	work	work	NOUN
ejpam-5875	281	3	has	have	AUX
ejpam-5875	281	4	provided	provide	VERB
ejpam-5875	281	5	a	a	DET
ejpam-5875	281	6	comprehensive	comprehensive	ADJ
ejpam-5875	281	7	analysis	analysis	NOUN
ejpam-5875	281	8	of	of	ADP
ejpam-5875	281	9	the	the	DET
ejpam-5875	281	10	quasi	quasi	NOUN
ejpam-5875	281	11	-	-	NOUN
ejpam-5875	281	12	frame	frame	NOUN
ejpam-5875	281	13	,	,	PUNCT
ejpam-5875	281	14	its	its	PRON
ejpam-5875	281	15	relationship	relationship	NOUN
ejpam-5875	281	16	with	with	ADP
ejpam-5875	281	17	the	the	DET
ejpam-5875	281	18	classical	classical	ADJ
ejpam-5875	281	19	frenet	frenet	NOUN
ejpam-5875	281	20	frame	frame	NOUN
ejpam-5875	281	21	,	,	PUNCT
ejpam-5875	281	22	and	and	CCONJ
ejpam-5875	281	23	the	the	DET
ejpam-5875	281	24	geometric	geometric	ADJ
ejpam-5875	281	25	properties	property	NOUN
ejpam-5875	281	26	of	of	ADP
ejpam-5875	281	27	curves	curve	NOUN
ejpam-5875	281	28	defined	define	VERB
ejpam-5875	281	29	in	in	ADP
ejpam-5875	281	30	relation	relation	NOUN
ejpam-5875	281	31	to	to	ADP
ejpam-5875	281	32	this	this	DET
ejpam-5875	281	33	frame	frame	NOUN
ejpam-5875	281	34	.	.	PUNCT
ejpam-5875	282	1	the	the	DET
ejpam-5875	282	2	quasi	quasi	ADJ
ejpam-5875	282	3	-	-	NOUN
ejpam-5875	282	4	frame	frame	NOUN
ejpam-5875	282	5	serves	serve	VERB
ejpam-5875	282	6	as	as	ADP
ejpam-5875	282	7	a	a	DET
ejpam-5875	282	8	generalization	generalization	NOUN
ejpam-5875	282	9	of	of	ADP
ejpam-5875	282	10	the	the	DET
ejpam-5875	282	11	frenet	frenet	ADJ
ejpam-5875	282	12	frame	frame	NOUN
ejpam-5875	282	13	,	,	PUNCT
ejpam-5875	282	14	particularly	particularly	ADV
ejpam-5875	282	15	useful	useful	ADJ
ejpam-5875	282	16	in	in	ADP
ejpam-5875	282	17	scenarios	scenario	NOUN
ejpam-5875	282	18	where	where	SCONJ
ejpam-5875	282	19	the	the	DET
ejpam-5875	282	20	curvature	curvature	NOUN
ejpam-5875	282	21	vanishes	vanish	VERB
ejpam-5875	282	22	and	and	CCONJ
ejpam-5875	282	23	the	the	DET
ejpam-5875	282	24	frenet	frenet	ADJ
ejpam-5875	282	25	frame	frame	NOUN
ejpam-5875	282	26	becomes	become	VERB
ejpam-5875	282	27	undefined	undefined	ADJ
ejpam-5875	282	28	.	.	PUNCT
ejpam-5875	283	1	by	by	ADP
ejpam-5875	283	2	establishing	establish	VERB
ejpam-5875	283	3	the	the	DET
ejpam-5875	283	4	quasi	quasi	ADJ
ejpam-5875	283	5	equations	equation	NOUN
ejpam-5875	283	6	in	in	ADP
ejpam-5875	283	7	matrix	matrix	NOUN
ejpam-5875	283	8	form	form	NOUN
ejpam-5875	283	9	,	,	PUNCT
ejpam-5875	283	10	we	we	PRON
ejpam-5875	283	11	derived	derive	VERB
ejpam-5875	283	12	the	the	DET
ejpam-5875	283	13	quasi	quasi	ADJ
ejpam-5875	283	14	curvatures	curvature	NOUN
ejpam-5875	283	15	k1	k1	NOUN
ejpam-5875	283	16	,	,	PUNCT
ejpam-5875	283	17	k2	k2	NOUN
ejpam-5875	283	18	,	,	PUNCT
ejpam-5875	283	19	and	and	CCONJ
ejpam-5875	283	20	k3	k3	PROPN
ejpam-5875	283	21	,	,	PUNCT
ejpam-5875	283	22	which	which	PRON
ejpam-5875	283	23	govern	govern	VERB
ejpam-5875	283	24	the	the	DET
ejpam-5875	283	25	behavior	behavior	NOUN
ejpam-5875	283	26	of	of	ADP
ejpam-5875	283	27	curves	curve	NOUN
ejpam-5875	283	28	in	in	ADP
ejpam-5875	283	29	g3	g3	PROPN
ejpam-5875	283	30	.	.	PUNCT
ejpam-5875	284	1	this	this	DET
ejpam-5875	284	2	framework	framework	NOUN
ejpam-5875	284	3	not	not	PART
ejpam-5875	284	4	only	only	ADV
ejpam-5875	284	5	extends	extend	VERB
ejpam-5875	284	6	the	the	DET
ejpam-5875	284	7	applicability	applicability	NOUN
ejpam-5875	284	8	of	of	ADP
ejpam-5875	284	9	the	the	DET
ejpam-5875	284	10	frenet	frenet	ADJ
ejpam-5875	284	11	frame	frame	NOUN
ejpam-5875	284	12	but	but	CCONJ
ejpam-5875	284	13	also	also	ADV
ejpam-5875	284	14	provides	provide	VERB
ejpam-5875	284	15	a	a	DET
ejpam-5875	284	16	more	more	ADV
ejpam-5875	284	17	robust	robust	ADJ
ejpam-5875	284	18	tool	tool	NOUN
ejpam-5875	284	19	for	for	ADP
ejpam-5875	284	20	analyzing	analyze	VERB
ejpam-5875	284	21	curves	curve	NOUN
ejpam-5875	284	22	in	in	ADP
ejpam-5875	284	23	galilean	galilean	PROPN
ejpam-5875	284	24	space	space	NOUN
ejpam-5875	284	25	.	.	PUNCT
ejpam-5875	285	1	we	we	PRON
ejpam-5875	285	2	derived	derive	VERB
ejpam-5875	285	3	explicit	explicit	ADJ
ejpam-5875	285	4	expressions	expression	NOUN
ejpam-5875	285	5	for	for	ADP
ejpam-5875	285	6	the	the	DET
ejpam-5875	285	7	position	position	NOUN
ejpam-5875	285	8	vectors	vector	NOUN
ejpam-5875	285	9	of	of	ADP
ejpam-5875	285	10	curves	curve	NOUN
ejpam-5875	285	11	in	in	ADP
ejpam-5875	285	12	g3	g3	NOUN
ejpam-5875	285	13	with	with	ADP
ejpam-5875	285	14	respect	respect	NOUN
ejpam-5875	285	15	to	to	ADP
ejpam-5875	285	16	the	the	DET
ejpam-5875	285	17	q	q	NOUN
ejpam-5875	285	18	-	-	PUNCT
ejpam-5875	285	19	frame	frame	NOUN
ejpam-5875	285	20	.	.	PUNCT
ejpam-5875	286	1	by	by	ADP
ejpam-5875	286	2	examining	examine	VERB
ejpam-5875	286	3	specific	specific	ADJ
ejpam-5875	286	4	cases	case	NOUN
ejpam-5875	286	5	,	,	PUNCT
ejpam-5875	286	6	we	we	PRON
ejpam-5875	286	7	obtained	obtain	VERB
ejpam-5875	286	8	solutions	solution	NOUN
ejpam-5875	286	9	for	for	ADP
ejpam-5875	286	10	the	the	DET
ejpam-5875	286	11	components	component	NOUN
ejpam-5875	286	12	of	of	ADP
ejpam-5875	286	13	the	the	DET
ejpam-5875	286	14	position	position	NOUN
ejpam-5875	286	15	vector	vector	NOUN
ejpam-5875	286	16	under	under	ADP
ejpam-5875	286	17	various	various	ADJ
ejpam-5875	286	18	conditions	condition	NOUN
ejpam-5875	286	19	.	.	PUNCT
ejpam-5875	287	1	this	this	DET
ejpam-5875	287	2	analysis	analysis	NOUN
ejpam-5875	287	3	allowed	allow	VERB
ejpam-5875	287	4	us	we	PRON
ejpam-5875	287	5	to	to	PART
ejpam-5875	287	6	classify	classify	VERB
ejpam-5875	287	7	curves	curve	NOUN
ejpam-5875	287	8	into	into	ADP
ejpam-5875	287	9	quasi	quasi	ADJ
ejpam-5875	287	10	-	-	ADJ
ejpam-5875	287	11	rectifying	rectifying	ADJ
ejpam-5875	287	12	and	and	CCONJ
ejpam-5875	287	13	quasi	quasi	ADJ
ejpam-5875	287	14	-	-	ADJ
ejpam-5875	287	15	osculating	osculate	VERB
ejpam-5875	287	16	categories	category	NOUN
ejpam-5875	287	17	based	base	VERB
ejpam-5875	287	18	on	on	ADP
ejpam-5875	287	19	their	their	PRON
ejpam-5875	287	20	geometric	geometric	ADJ
ejpam-5875	287	21	properties	property	NOUN
ejpam-5875	287	22	.	.	PUNCT
ejpam-5875	288	1	a	a	DET
ejpam-5875	288	2	curve	curve	NOUN
ejpam-5875	288	3	is	be	AUX
ejpam-5875	288	4	quasi	quasi	ADJ
ejpam-5875	288	5	-	-	ADJ
ejpam-5875	288	6	rectifying	rectifying	ADJ
ejpam-5875	288	7	if	if	SCONJ
ejpam-5875	288	8	its	its	PRON
ejpam-5875	288	9	position	position	NOUN
ejpam-5875	288	10	vector	vector	NOUN
ejpam-5875	288	11	lies	lie	VERB
ejpam-5875	288	12	in	in	ADP
ejpam-5875	288	13	the	the	DET
ejpam-5875	288	14	plane	plane	NOUN
ejpam-5875	288	15	spanned	span	VERB
ejpam-5875	288	16	by	by	ADP
ejpam-5875	288	17	its	its	PRON
ejpam-5875	288	18	tangent	tangent	NOUN
ejpam-5875	288	19	and	and	CCONJ
ejpam-5875	288	20	quasi	quasi	ADJ
ejpam-5875	288	21	-	-	ADJ
ejpam-5875	288	22	binormal	binormal	ADJ
ejpam-5875	288	23	vectors	vector	NOUN
ejpam-5875	288	24	,	,	PUNCT
ejpam-5875	288	25	while	while	SCONJ
ejpam-5875	288	26	a	a	DET
ejpam-5875	288	27	curve	curve	NOUN
ejpam-5875	288	28	is	be	AUX
ejpam-5875	288	29	quasi	quasi	ADJ
ejpam-5875	288	30	-	-	ADJ
ejpam-5875	288	31	osculating	osculate	VERB
ejpam-5875	288	32	if	if	SCONJ
ejpam-5875	288	33	it	it	PRON
ejpam-5875	288	34	remains	remain	VERB
ejpam-5875	288	35	entirely	entirely	ADV
ejpam-5875	288	36	within	within	ADP
ejpam-5875	288	37	its	its	PRON
ejpam-5875	288	38	quasi	quasi	ADJ
ejpam-5875	288	39	-	-	ADJ
ejpam-5875	288	40	osculating	osculating	ADJ
ejpam-5875	288	41	plane	plane	NOUN
ejpam-5875	288	42	,	,	PUNCT
ejpam-5875	288	43	defined	define	VERB
ejpam-5875	288	44	by	by	ADP
ejpam-5875	288	45	the	the	DET
ejpam-5875	288	46	tangent	tangent	NOUN
ejpam-5875	288	47	and	and	CCONJ
ejpam-5875	288	48	quasi	quasi	ADJ
ejpam-5875	288	49	-	-	ADJ
ejpam-5875	288	50	normal	normal	ADJ
ejpam-5875	288	51	vectors	vector	NOUN
ejpam-5875	288	52	.	.	PUNCT
ejpam-5875	289	1	these	these	DET
ejpam-5875	289	2	classifications	classification	NOUN
ejpam-5875	289	3	provide	provide	VERB
ejpam-5875	289	4	a	a	DET
ejpam-5875	289	5	deeper	deep	ADJ
ejpam-5875	289	6	understanding	understanding	NOUN
ejpam-5875	289	7	of	of	ADP
ejpam-5875	289	8	the	the	DET
ejpam-5875	289	9	geometric	geometric	ADJ
ejpam-5875	289	10	behavior	behavior	NOUN
ejpam-5875	289	11	of	of	ADP
ejpam-5875	289	12	curves	curve	NOUN
ejpam-5875	289	13	in	in	ADP
ejpam-5875	289	14	g3	g3	PROPN
ejpam-5875	289	15	.	.	PUNCT
ejpam-5875	290	1	one	one	NUM
ejpam-5875	290	2	of	of	ADP
ejpam-5875	290	3	the	the	DET
ejpam-5875	290	4	most	most	ADV
ejpam-5875	290	5	significant	significant	ADJ
ejpam-5875	290	6	findings	finding	NOUN
ejpam-5875	290	7	of	of	ADP
ejpam-5875	290	8	this	this	DET
ejpam-5875	290	9	study	study	NOUN
ejpam-5875	290	10	is	be	AUX
ejpam-5875	290	11	the	the	DET
ejpam-5875	290	12	demonstration	demonstration	NOUN
ejpam-5875	290	13	that	that	PRON
ejpam-5875	290	14	normal	normal	ADJ
ejpam-5875	290	15	curves	curve	NOUN
ejpam-5875	290	16	do	do	AUX
ejpam-5875	290	17	not	not	PART
ejpam-5875	290	18	exist	exist	VERB
ejpam-5875	290	19	in	in	ADP
ejpam-5875	290	20	g3	g3	PROPN
ejpam-5875	290	21	.	.	PUNCT
ejpam-5875	291	1	this	this	DET
ejpam-5875	291	2	result	result	NOUN
ejpam-5875	291	3	challenges	challenge	VERB
ejpam-5875	291	4	existing	exist	VERB
ejpam-5875	291	5	theories	theory	NOUN
ejpam-5875	291	6	and	and	CCONJ
ejpam-5875	291	7	provides	provide	VERB
ejpam-5875	291	8	new	new	ADJ
ejpam-5875	291	9	insights	insight	NOUN
ejpam-5875	291	10	into	into	ADP
ejpam-5875	291	11	the	the	DET
ejpam-5875	291	12	geometric	geometric	ADJ
ejpam-5875	291	13	structure	structure	NOUN
ejpam-5875	291	14	of	of	ADP
ejpam-5875	291	15	galilean	galilean	PROPN
ejpam-5875	291	16	space	space	NOUN
ejpam-5875	291	17	.	.	PUNCT
ejpam-5875	292	1	by	by	ADP
ejpam-5875	292	2	proving	prove	VERB
ejpam-5875	292	3	the	the	DET
ejpam-5875	292	4	absence	absence	NOUN
ejpam-5875	292	5	of	of	ADP
ejpam-5875	292	6	normal	normal	ADJ
ejpam-5875	292	7	curves	curve	NOUN
ejpam-5875	292	8	,	,	PUNCT
ejpam-5875	292	9	we	we	PRON
ejpam-5875	292	10	have	have	AUX
ejpam-5875	292	11	clarified	clarify	VERB
ejpam-5875	292	12	the	the	DET
ejpam-5875	292	13	limitations	limitation	NOUN
ejpam-5875	292	14	of	of	ADP
ejpam-5875	292	15	certain	certain	ADJ
ejpam-5875	292	16	geometric	geometric	ADJ
ejpam-5875	292	17	classifications	classification	NOUN
ejpam-5875	292	18	in	in	ADP
ejpam-5875	292	19	this	this	DET
ejpam-5875	292	20	context	context	NOUN
ejpam-5875	292	21	.	.	PUNCT
ejpam-5875	293	1	furthermore	furthermore	ADV
ejpam-5875	293	2	,	,	PUNCT
ejpam-5875	293	3	our	our	PRON
ejpam-5875	293	4	results	result	NOUN
ejpam-5875	293	5	generalize	generalize	VERB
ejpam-5875	293	6	several	several	ADJ
ejpam-5875	293	7	well	well	ADV
ejpam-5875	293	8	-	-	PUNCT
ejpam-5875	293	9	known	know	VERB
ejpam-5875	293	10	properties	property	NOUN
ejpam-5875	293	11	of	of	ADP
ejpam-5875	293	12	curves	curve	NOUN
ejpam-5875	293	13	in	in	ADP
ejpam-5875	293	14	euclidean	euclidean	ADJ
ejpam-5875	293	15	and	and	CCONJ
ejpam-5875	293	16	minkowski	minkowski	ADJ
ejpam-5875	293	17	spaces	space	NOUN
ejpam-5875	293	18	to	to	ADP
ejpam-5875	293	19	the	the	DET
ejpam-5875	293	20	galilean	galilean	PROPN
ejpam-5875	293	21	setting	setting	NOUN
ejpam-5875	293	22	.	.	PUNCT
ejpam-5875	294	1	for	for	ADP
ejpam-5875	294	2	instance	instance	NOUN
ejpam-5875	294	3	,	,	PUNCT
ejpam-5875	294	4	the	the	DET
ejpam-5875	294	5	conditions	condition	NOUN
ejpam-5875	294	6	for	for	ADP
ejpam-5875	294	7	quasirectifying	quasirectifying	NOUN
ejpam-5875	294	8	and	and	CCONJ
ejpam-5875	294	9	quasi	quasi	ADJ
ejpam-5875	294	10	-	-	ADJ
ejpam-5875	294	11	osculating	osculating	ADJ
ejpam-5875	294	12	curves	curve	NOUN
ejpam-5875	294	13	reduce	reduce	VERB
ejpam-5875	294	14	to	to	ADP
ejpam-5875	294	15	their	their	PRON
ejpam-5875	294	16	frenet	frenet	NOUN
ejpam-5875	294	17	counterparts	counterpart	NOUN
ejpam-5875	294	18	when	when	SCONJ
ejpam-5875	294	19	the	the	DET
ejpam-5875	294	20	quasi	quasi	ADJ
ejpam-5875	294	21	curvatures	curvature	NOUN
ejpam-5875	294	22	are	be	AUX
ejpam-5875	294	23	appropriately	appropriately	ADV
ejpam-5875	294	24	specialized	specialize	VERB
ejpam-5875	294	25	.	.	PUNCT
ejpam-5875	295	1	this	this	PRON
ejpam-5875	295	2	demonstrates	demonstrate	VERB
ejpam-5875	295	3	the	the	DET
ejpam-5875	295	4	versatility	versatility	NOUN
ejpam-5875	295	5	of	of	ADP
ejpam-5875	295	6	the	the	DET
ejpam-5875	295	7	q	q	NOUN
ejpam-5875	295	8	-	-	PUNCT
ejpam-5875	295	9	frame	frame	NOUN
ejpam-5875	295	10	and	and	CCONJ
ejpam-5875	295	11	its	its	PRON
ejpam-5875	295	12	ability	ability	NOUN
ejpam-5875	295	13	to	to	PART
ejpam-5875	295	14	unify	unify	VERB
ejpam-5875	295	15	various	various	ADJ
ejpam-5875	295	16	geometric	geometric	ADJ
ejpam-5875	295	17	frameworks	framework	NOUN
ejpam-5875	295	18	.	.	PUNCT
ejpam-5875	296	1	the	the	DET
ejpam-5875	296	2	findings	finding	NOUN
ejpam-5875	296	3	of	of	ADP
ejpam-5875	296	4	this	this	DET
ejpam-5875	296	5	study	study	NOUN
ejpam-5875	296	6	have	have	VERB
ejpam-5875	296	7	several	several	ADJ
ejpam-5875	296	8	important	important	ADJ
ejpam-5875	296	9	implications	implication	NOUN
ejpam-5875	296	10	for	for	ADP
ejpam-5875	296	11	both	both	CCONJ
ejpam-5875	296	12	theoretical	theoretical	ADJ
ejpam-5875	296	13	and	and	CCONJ
ejpam-5875	296	14	applied	applied	ADJ
ejpam-5875	296	15	mathematics	mathematic	NOUN
ejpam-5875	296	16	.	.	PUNCT
ejpam-5875	297	1	the	the	DET
ejpam-5875	297	2	quasi	quasi	NOUN
ejpam-5875	297	3	-	-	NOUN
ejpam-5875	297	4	frame	frame	NOUN
ejpam-5875	297	5	provides	provide	VERB
ejpam-5875	297	6	a	a	DET
ejpam-5875	297	7	powerful	powerful	ADJ
ejpam-5875	297	8	tool	tool	NOUN
ejpam-5875	297	9	for	for	ADP
ejpam-5875	297	10	analyzing	analyze	VERB
ejpam-5875	297	11	curves	curve	NOUN
ejpam-5875	297	12	in	in	ADP
ejpam-5875	297	13	galilean	galilean	PROPN
ejpam-5875	297	14	space	space	NOUN
ejpam-5875	297	15	,	,	PUNCT
ejpam-5875	297	16	particularly	particularly	ADV
ejpam-5875	297	17	in	in	ADP
ejpam-5875	297	18	cases	case	NOUN
ejpam-5875	297	19	where	where	SCONJ
ejpam-5875	297	20	the	the	DET
ejpam-5875	297	21	frenet	frenet	ADJ
ejpam-5875	297	22	frame	frame	NOUN
ejpam-5875	297	23	fails	fail	VERB
ejpam-5875	297	24	.	.	PUNCT
ejpam-5875	298	1	this	this	PRON
ejpam-5875	298	2	has	have	VERB
ejpam-5875	298	3	potential	potential	ADJ
ejpam-5875	298	4	applications	application	NOUN
ejpam-5875	298	5	in	in	ADP
ejpam-5875	298	6	physics	physics	NOUN
ejpam-5875	298	7	,	,	PUNCT
ejpam-5875	298	8	engineering	engineering	NOUN
ejpam-5875	298	9	,	,	PUNCT
ejpam-5875	298	10	and	and	CCONJ
ejpam-5875	298	11	computer	computer	NOUN
ejpam-5875	298	12	graphics	graphic	NOUN
ejpam-5875	298	13	,	,	PUNCT
ejpam-5875	298	14	where	where	SCONJ
ejpam-5875	298	15	galilean	galilean	PROPN
ejpam-5875	298	16	geometry	geometry	NOUN
ejpam-5875	298	17	is	be	AUX
ejpam-5875	298	18	often	often	ADV
ejpam-5875	298	19	used	use	VERB
ejpam-5875	298	20	to	to	PART
ejpam-5875	298	21	model	model	VERB
ejpam-5875	298	22	motion	motion	NOUN
ejpam-5875	298	23	and	and	CCONJ
ejpam-5875	298	24	spatial	spatial	ADJ
ejpam-5875	298	25	relationships	relationship	NOUN
ejpam-5875	298	26	.	.	PUNCT
ejpam-5875	299	1	future	future	ADJ
ejpam-5875	299	2	research	research	NOUN
ejpam-5875	299	3	could	could	AUX
ejpam-5875	299	4	explore	explore	VERB
ejpam-5875	299	5	extending	extend	VERB
ejpam-5875	299	6	the	the	DET
ejpam-5875	299	7	quasi	quasi	NOUN
ejpam-5875	299	8	-	-	NOUN
ejpam-5875	299	9	frame	frame	NOUN
ejpam-5875	299	10	to	to	ADP
ejpam-5875	299	11	higher	higher	ADV
ejpam-5875	299	12	-	-	PUNCT
ejpam-5875	299	13	dimensional	dimensional	ADJ
ejpam-5875	299	14	galilean	galilean	PROPN
ejpam-5875	299	15	spaces	space	NOUN
ejpam-5875	299	16	(	(	PUNCT
ejpam-5875	299	17	gn	gn	PROPN
ejpam-5875	299	18	)	)	PUNCT
ejpam-5875	299	19	,	,	PUNCT
ejpam-5875	299	20	providing	provide	VERB
ejpam-5875	299	21	new	new	ADJ
ejpam-5875	299	22	insights	insight	NOUN
ejpam-5875	299	23	into	into	ADP
ejpam-5875	299	24	the	the	DET
ejpam-5875	299	25	geometry	geometry	NOUN
ejpam-5875	299	26	of	of	ADP
ejpam-5875	299	27	curves	curve	NOUN
ejpam-5875	299	28	and	and	CCONJ
ejpam-5875	299	29	surfaces	surface	NOUN
ejpam-5875	299	30	in	in	ADP
ejpam-5875	299	31	these	these	DET
ejpam-5875	299	32	settings	setting	NOUN
ejpam-5875	299	33	.	.	PUNCT
ejpam-5875	300	1	additionally	additionally	ADV
ejpam-5875	300	2	,	,	PUNCT
ejpam-5875	300	3	the	the	DET
ejpam-5875	300	4	quasi	quasi	NOUN
ejpam-5875	300	5	-	-	NOUN
ejpam-5875	300	6	frame	frame	NOUN
ejpam-5875	300	7	could	could	AUX
ejpam-5875	300	8	be	be	AUX
ejpam-5875	300	9	applied	apply	VERB
ejpam-5875	300	10	to	to	ADP
ejpam-5875	300	11	problems	problem	NOUN
ejpam-5875	300	12	in	in	ADP
ejpam-5875	300	13	classical	classical	ADJ
ejpam-5875	300	14	and	and	CCONJ
ejpam-5875	300	15	relativistic	relativistic	ADJ
ejpam-5875	300	16	mechanics	mechanic	NOUN
ejpam-5875	300	17	,	,	PUNCT
ejpam-5875	300	18	where	where	SCONJ
ejpam-5875	300	19	galilean	galilean	PROPN
ejpam-5875	300	20	transformations	transformation	NOUN
ejpam-5875	300	21	play	play	VERB
ejpam-5875	300	22	a	a	DET
ejpam-5875	300	23	central	central	ADJ
ejpam-5875	300	24	role	role	NOUN
ejpam-5875	300	25	.	.	PUNCT
ejpam-5875	301	1	investigating	investigate	VERB
ejpam-5875	301	2	the	the	DET
ejpam-5875	301	3	properties	property	NOUN
ejpam-5875	301	4	of	of	ADP
ejpam-5875	301	5	surfaces	surface	NOUN
ejpam-5875	301	6	generated	generate	VERB
ejpam-5875	301	7	by	by	ADP
ejpam-5875	301	8	quasi	quasi	NOUN
ejpam-5875	301	9	-	-	NOUN
ejpam-5875	301	10	curves	curve	NOUN
ejpam-5875	301	11	,	,	PUNCT
ejpam-5875	301	12	such	such	ADJ
ejpam-5875	301	13	as	as	ADP
ejpam-5875	301	14	quasi	quasi	ADJ
ejpam-5875	301	15	-	-	ADJ
ejpam-5875	301	16	ruled	rule	VERB
ejpam-5875	301	17	surfaces	surface	NOUN
ejpam-5875	301	18	,	,	PUNCT
ejpam-5875	301	19	could	could	AUX
ejpam-5875	301	20	also	also	ADV
ejpam-5875	301	21	lead	lead	VERB
ejpam-5875	301	22	to	to	ADP
ejpam-5875	301	23	new	new	ADJ
ejpam-5875	301	24	geometric	geometric	ADJ
ejpam-5875	301	25	constructions	construction	NOUN
ejpam-5875	301	26	and	and	CCONJ
ejpam-5875	301	27	classifications	classification	NOUN
ejpam-5875	301	28	.	.	PUNCT
ejpam-5875	302	1	a.	a.	NOUN
ejpam-5875	302	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	302	3	,	,	PUNCT
ejpam-5875	302	4	n.	n.	NOUN
ejpam-5875	302	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	302	6	/	/	SYM
ejpam-5875	302	7	eur	eur	PROPN
ejpam-5875	302	8	.	.	PUNCT
ejpam-5875	303	1	j.	j.	PROPN
ejpam-5875	303	2	pure	pure	PROPN
ejpam-5875	303	3	appl	appl	PROPN
ejpam-5875	303	4	.	.	PROPN
ejpam-5875	303	5	math	math	PROPN
ejpam-5875	303	6	,	,	PUNCT
ejpam-5875	303	7	18	18	NUM
ejpam-5875	303	8	(	(	PUNCT
ejpam-5875	303	9	2	2	NUM
ejpam-5875	303	10	)	)	PUNCT
ejpam-5875	303	11	(	(	PUNCT
ejpam-5875	303	12	2025	2025	NUM
ejpam-5875	303	13	)	)	PUNCT
ejpam-5875	303	14	,	,	PUNCT
ejpam-5875	303	15	5875	5875	NUM
ejpam-5875	303	16	13	13	NUM
ejpam-5875	303	17	of	of	ADP
ejpam-5875	303	18	15	15	NUM
ejpam-5875	303	19	references	reference	NOUN
ejpam-5875	303	20	[	[	X
ejpam-5875	303	21	1	1	NUM
ejpam-5875	303	22	]	]	PUNCT
ejpam-5875	303	23	i.	i.	NOUN
ejpam-5875	303	24	m.	m.	PROPN
ejpam-5875	303	25	yaglom	yaglom	PROPN
ejpam-5875	303	26	.	.	PUNCT
ejpam-5875	304	1	a	a	DET
ejpam-5875	304	2	simple	simple	ADJ
ejpam-5875	304	3	non	non	ADJ
ejpam-5875	304	4	-	-	ADJ
ejpam-5875	304	5	euclidean	euclidean	ADJ
ejpam-5875	304	6	geometry	geometry	NOUN
ejpam-5875	304	7	and	and	CCONJ
ejpam-5875	304	8	its	its	PRON
ejpam-5875	304	9	physical	physical	ADJ
ejpam-5875	304	10	basis	basis	NOUN
ejpam-5875	304	11	.	.	PUNCT
ejpam-5875	305	1	springerverlag	springerverlag	NOUN
ejpam-5875	305	2	,	,	PUNCT
ejpam-5875	305	3	new	new	PROPN
ejpam-5875	305	4	york	york	PROPN
ejpam-5875	305	5	,	,	PUNCT
ejpam-5875	305	6	ny	ny	PROPN
ejpam-5875	305	7	,	,	PUNCT
ejpam-5875	305	8	1979	1979	NUM
ejpam-5875	305	9	.	.	PUNCT
ejpam-5875	306	1	[	[	X
ejpam-5875	306	2	2	2	X
ejpam-5875	306	3	]	]	PUNCT
ejpam-5875	306	4	h.	h.	PROPN
ejpam-5875	306	5	k.	k.	PROPN
ejpam-5875	306	6	elsayied	elsayied	PROPN
ejpam-5875	306	7	,	,	PUNCT
ejpam-5875	306	8	a.	a.	NOUN
ejpam-5875	306	9	a.	a.	NOUN
ejpam-5875	306	10	altaha	altaha	NOUN
ejpam-5875	306	11	,	,	PUNCT
ejpam-5875	306	12	and	and	CCONJ
ejpam-5875	306	13	a.	a.	NOUN
ejpam-5875	306	14	elsharkawy	elsharkawy	NOUN
ejpam-5875	306	15	.	.	PUNCT
ejpam-5875	307	1	on	on	ADP
ejpam-5875	307	2	some	some	DET
ejpam-5875	307	3	special	special	ADJ
ejpam-5875	307	4	curves	curve	NOUN
ejpam-5875	307	5	according	accord	VERB
ejpam-5875	307	6	to	to	ADP
ejpam-5875	307	7	the	the	DET
ejpam-5875	307	8	modified	modify	VERB
ejpam-5875	307	9	orthogonal	orthogonal	ADJ
ejpam-5875	307	10	frame	frame	NOUN
ejpam-5875	307	11	in	in	ADP
ejpam-5875	307	12	minkowski	minkowski	ADJ
ejpam-5875	307	13	3	3	NUM
ejpam-5875	307	14	-	-	PUNCT
ejpam-5875	307	15	space	space	NOUN
ejpam-5875	307	16	e3	e3	NOUN
ejpam-5875	307	17	1	1	NUM
ejpam-5875	307	18	.	.	PUNCT
ejpam-5875	308	1	kasmera	kasmera	NOUN
ejpam-5875	308	2	,	,	PUNCT
ejpam-5875	308	3	49(1):2–15	49(1):2–15	PROPN
ejpam-5875	308	4	,	,	PUNCT
ejpam-5875	308	5	2021	2021	NUM
ejpam-5875	308	6	.	.	PUNCT
ejpam-5875	309	1	[	[	X
ejpam-5875	309	2	3	3	X
ejpam-5875	309	3	]	]	X
ejpam-5875	309	4	h.	h.	PROPN
ejpam-5875	309	5	k.	k.	PROPN
ejpam-5875	309	6	elsayied	elsayied	PROPN
ejpam-5875	309	7	,	,	PUNCT
ejpam-5875	309	8	a.	a.	NOUN
ejpam-5875	309	9	a.	a.	NOUN
ejpam-5875	309	10	altaha	altaha	NOUN
ejpam-5875	309	11	,	,	PUNCT
ejpam-5875	309	12	and	and	CCONJ
ejpam-5875	309	13	a.	a.	NOUN
ejpam-5875	309	14	elsharkawy	elsharkawy	PROPN
ejpam-5875	309	15	.	.	PUNCT
ejpam-5875	310	1	bertrand	bertrand	PROPN
ejpam-5875	310	2	curves	curve	VERB
ejpam-5875	310	3	with	with	ADP
ejpam-5875	310	4	the	the	DET
ejpam-5875	310	5	modified	modify	VERB
ejpam-5875	310	6	orthogonal	orthogonal	ADJ
ejpam-5875	310	7	frame	frame	NOUN
ejpam-5875	310	8	in	in	ADP
ejpam-5875	310	9	minkowski	minkowski	ADJ
ejpam-5875	310	10	3	3	NUM
ejpam-5875	310	11	-	-	PUNCT
ejpam-5875	310	12	space	space	NOUN
ejpam-5875	310	13	e3	e3	NOUN
ejpam-5875	310	14	1	1	NUM
ejpam-5875	310	15	.	.	PUNCT
ejpam-5875	310	16	revista	revista	PROPN
ejpam-5875	310	17	de	de	PROPN
ejpam-5875	310	18	educacion	educacion	PROPN
ejpam-5875	310	19	,	,	PUNCT
ejpam-5875	310	20	392(6):43–55	392(6):43–55	ADV
ejpam-5875	310	21	,	,	PUNCT
ejpam-5875	310	22	2022	2022	NUM
ejpam-5875	310	23	.	.	PUNCT
ejpam-5875	311	1	[	[	X
ejpam-5875	311	2	4	4	X
ejpam-5875	311	3	]	]	PUNCT
ejpam-5875	311	4	h.	h.	PROPN
ejpam-5875	311	5	k.	k.	PROPN
ejpam-5875	311	6	elsayied	elsayied	PROPN
ejpam-5875	311	7	,	,	PUNCT
ejpam-5875	311	8	m.	m.	NOUN
ejpam-5875	311	9	elzawy	elzawy	PROPN
ejpam-5875	311	10	,	,	PUNCT
ejpam-5875	311	11	and	and	CCONJ
ejpam-5875	311	12	a.	a.	NOUN
ejpam-5875	311	13	elsharkawy	elsharkawy	PROPN
ejpam-5875	311	14	.	.	PUNCT
ejpam-5875	312	1	equiform	equiform	PROPN
ejpam-5875	312	2	timelike	timelike	PROPN
ejpam-5875	312	3	normal	normal	ADJ
ejpam-5875	312	4	curves	curve	NOUN
ejpam-5875	312	5	in	in	ADP
ejpam-5875	312	6	minkowski	minkowski	ADJ
ejpam-5875	312	7	space	space	NOUN
ejpam-5875	312	8	e3	e3	NOUN
ejpam-5875	312	9	1	1	NUM
ejpam-5875	312	10	.	.	PUNCT
ejpam-5875	313	1	far	far	PROPN
ejpam-5875	313	2	east	east	PROPN
ejpam-5875	313	3	journal	journal	PROPN
ejpam-5875	313	4	of	of	ADP
ejpam-5875	313	5	mathematical	mathematical	ADJ
ejpam-5875	313	6	sciences	science	NOUN
ejpam-5875	313	7	,	,	PUNCT
ejpam-5875	313	8	101:1619–1629	101:1619–1629	NUM
ejpam-5875	313	9	,	,	PUNCT
ejpam-5875	313	10	2017	2017	NUM
ejpam-5875	313	11	.	.	PUNCT
ejpam-5875	314	1	[	[	X
ejpam-5875	314	2	5	5	X
ejpam-5875	314	3	]	]	PUNCT
ejpam-5875	314	4	h.	h.	PROPN
ejpam-5875	314	5	k.	k.	PROPN
ejpam-5875	314	6	elsayied	elsayied	PROPN
ejpam-5875	314	7	,	,	PUNCT
ejpam-5875	314	8	a.	a.	PROPN
ejpam-5875	314	9	m.	m.	PROPN
ejpam-5875	314	10	tawfiq	tawfiq	PROPN
ejpam-5875	314	11	,	,	PUNCT
ejpam-5875	314	12	and	and	CCONJ
ejpam-5875	314	13	a.	a.	NOUN
ejpam-5875	314	14	elsharkawy	elsharkawy	PROPN
ejpam-5875	314	15	.	.	PUNCT
ejpam-5875	315	1	the	the	DET
ejpam-5875	315	2	quasi	quasi	ADJ
ejpam-5875	315	3	frame	frame	NOUN
ejpam-5875	315	4	and	and	CCONJ
ejpam-5875	315	5	equations	equation	NOUN
ejpam-5875	315	6	of	of	ADP
ejpam-5875	315	7	non	non	ADJ
ejpam-5875	315	8	-	-	ADJ
ejpam-5875	315	9	lightlike	lightlike	ADJ
ejpam-5875	315	10	curves	curve	NOUN
ejpam-5875	315	11	in	in	ADP
ejpam-5875	315	12	minkowski	minkowski	ADJ
ejpam-5875	315	13	e3	e3	NOUN
ejpam-5875	315	14	1	1	NUM
ejpam-5875	315	15	and	and	CCONJ
ejpam-5875	315	16	e4	e4	PROPN
ejpam-5875	315	17	1	1	NUM
ejpam-5875	315	18	.	.	PUNCT
ejpam-5875	316	1	italian	italian	ADJ
ejpam-5875	316	2	journal	journal	NOUN
ejpam-5875	316	3	of	of	ADP
ejpam-5875	316	4	pure	pure	ADJ
ejpam-5875	316	5	and	and	CCONJ
ejpam-5875	316	6	applied	applied	ADJ
ejpam-5875	316	7	mathematics	mathematic	NOUN
ejpam-5875	316	8	,	,	PUNCT
ejpam-5875	316	9	49:225–239	49:225–239	NUM
ejpam-5875	316	10	,	,	PUNCT
ejpam-5875	316	11	2023	2023	NUM
ejpam-5875	316	12	.	.	PUNCT
ejpam-5875	317	1	[	[	X
ejpam-5875	317	2	6	6	NUM
ejpam-5875	317	3	]	]	PUNCT
ejpam-5875	317	4	a.	a.	NOUN
ejpam-5875	317	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	317	6	and	and	CCONJ
ejpam-5875	317	7	h.	h.	PROPN
ejpam-5875	317	8	baizeed	baizeed	NOUN
ejpam-5875	317	9	.	.	PUNCT
ejpam-5875	318	1	some	some	DET
ejpam-5875	318	2	integral	integral	ADJ
ejpam-5875	318	3	curves	curve	NOUN
ejpam-5875	318	4	according	accord	VERB
ejpam-5875	318	5	to	to	ADP
ejpam-5875	318	6	quasi	quasi	NOUN
ejpam-5875	318	7	-	-	NOUN
ejpam-5875	318	8	frame	frame	NOUN
ejpam-5875	318	9	in	in	ADP
ejpam-5875	318	10	euclidean	euclidean	ADJ
ejpam-5875	318	11	3	3	NUM
ejpam-5875	318	12	-	-	PUNCT
ejpam-5875	318	13	space	space	NOUN
ejpam-5875	318	14	.	.	PUNCT
ejpam-5875	319	1	scientific	scientific	ADJ
ejpam-5875	319	2	african	african	PROPN
ejpam-5875	319	3	,	,	PUNCT
ejpam-5875	319	4	27	27	NUM
ejpam-5875	319	5	:	:	PUNCT
ejpam-5875	319	6	e02583	e02583	ADJ
ejpam-5875	319	7	,	,	PUNCT
ejpam-5875	319	8	2025	2025	NUM
ejpam-5875	319	9	.	.	PUNCT
ejpam-5875	320	1	[	[	X
ejpam-5875	320	2	7	7	NUM
ejpam-5875	320	3	]	]	PUNCT
ejpam-5875	320	4	a.	a.	NOUN
ejpam-5875	320	5	elsharkawy	elsharkawy	NOUN
ejpam-5875	320	6	.	.	PUNCT
ejpam-5875	321	1	exploring	explore	VERB
ejpam-5875	321	2	hasimoto	hasimoto	NOUN
ejpam-5875	321	3	surfaces	surface	NOUN
ejpam-5875	321	4	within	within	ADP
ejpam-5875	321	5	equiform	equiform	NOUN
ejpam-5875	321	6	geometry	geometry	NOUN
ejpam-5875	321	7	in	in	ADP
ejpam-5875	321	8	minkowski	minkowski	ADJ
ejpam-5875	321	9	space	space	NOUN
ejpam-5875	321	10	.	.	PUNCT
ejpam-5875	322	1	physica	physica	PROPN
ejpam-5875	322	2	scripta	scripta	PROPN
ejpam-5875	322	3	,	,	PUNCT
ejpam-5875	322	4	100(1	100(1	NUM
ejpam-5875	322	5	)	)	PUNCT
ejpam-5875	322	6	,	,	PUNCT
ejpam-5875	322	7	2024	2024	NUM
ejpam-5875	322	8	.	.	PUNCT
ejpam-5875	323	1	[	[	X
ejpam-5875	323	2	8	8	NUM
ejpam-5875	323	3	]	]	PUNCT
ejpam-5875	323	4	a.	a.	NOUN
ejpam-5875	323	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	323	6	,	,	PUNCT
ejpam-5875	323	7	c.	c.	PROPN
ejpam-5875	323	8	cesarano	cesarano	PROPN
ejpam-5875	323	9	,	,	PUNCT
ejpam-5875	323	10	and	and	CCONJ
ejpam-5875	323	11	h.	h.	PROPN
ejpam-5875	323	12	alhazmi	alhazmi	PROPN
ejpam-5875	323	13	.	.	PUNCT
ejpam-5875	324	1	on	on	ADP
ejpam-5875	324	2	the	the	DET
ejpam-5875	324	3	jerk	jerk	NOUN
ejpam-5875	324	4	and	and	CCONJ
ejpam-5875	324	5	snap	snap	VERB
ejpam-5875	324	6	in	in	ADP
ejpam-5875	324	7	motion	motion	NOUN
ejpam-5875	324	8	along	along	ADP
ejpam-5875	324	9	non	non	ADJ
ejpam-5875	324	10	-	-	ADJ
ejpam-5875	324	11	lightlike	lightlike	ADJ
ejpam-5875	324	12	curves	curve	NOUN
ejpam-5875	324	13	in	in	ADP
ejpam-5875	324	14	minkowski	minkowski	ADJ
ejpam-5875	324	15	3	3	NUM
ejpam-5875	324	16	-	-	PUNCT
ejpam-5875	324	17	space	space	NOUN
ejpam-5875	324	18	.	.	PUNCT
ejpam-5875	325	1	mathematical	mathematical	ADJ
ejpam-5875	325	2	methods	method	NOUN
ejpam-5875	325	3	in	in	ADP
ejpam-5875	325	4	the	the	DET
ejpam-5875	325	5	applied	apply	VERB
ejpam-5875	325	6	sciences	science	NOUN
ejpam-5875	325	7	,	,	PUNCT
ejpam-5875	325	8	pages	page	NOUN
ejpam-5875	325	9	1–13	1–13	NOUN
ejpam-5875	325	10	,	,	PUNCT
ejpam-5875	325	11	2024	2024	NUM
ejpam-5875	325	12	.	.	PUNCT
ejpam-5875	326	1	[	[	X
ejpam-5875	326	2	9	9	NUM
ejpam-5875	326	3	]	]	SYM
ejpam-5875	326	4	a.	a.	NOUN
ejpam-5875	326	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	326	6	and	and	CCONJ
ejpam-5875	326	7	a.	a.	NOUN
ejpam-5875	326	8	m.	m.	PROPN
ejpam-5875	326	9	elshenhab	elshenhab	PROPN
ejpam-5875	326	10	.	.	PUNCT
ejpam-5875	327	1	mannheim	mannheim	NOUN
ejpam-5875	327	2	curves	curve	NOUN
ejpam-5875	327	3	and	and	CCONJ
ejpam-5875	327	4	their	their	PRON
ejpam-5875	327	5	partner	partner	NOUN
ejpam-5875	327	6	curves	curve	NOUN
ejpam-5875	327	7	in	in	ADP
ejpam-5875	327	8	minkowski	minkowski	ADJ
ejpam-5875	327	9	3	3	NUM
ejpam-5875	327	10	-	-	PUNCT
ejpam-5875	327	11	space	space	NOUN
ejpam-5875	327	12	e3	e3	NOUN
ejpam-5875	327	13	1	1	NUM
ejpam-5875	327	14	.	.	PUNCT
ejpam-5875	327	15	demonstratio	demonstratio	PROPN
ejpam-5875	327	16	mathematica	mathematica	PROPN
ejpam-5875	327	17	,	,	PUNCT
ejpam-5875	327	18	55(1):798–811	55(1):798–811	PROPN
ejpam-5875	327	19	,	,	PUNCT
ejpam-5875	327	20	2022	2022	NUM
ejpam-5875	327	21	.	.	PUNCT
ejpam-5875	328	1	[	[	X
ejpam-5875	328	2	10	10	NUM
ejpam-5875	328	3	]	]	PUNCT
ejpam-5875	328	4	a.	a.	NOUN
ejpam-5875	328	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	328	6	and	and	CCONJ
ejpam-5875	328	7	n.	n.	PROPN
ejpam-5875	328	8	elsharkawy	elsharkawy	NOUN
ejpam-5875	328	9	.	.	PUNCT
ejpam-5875	329	1	quasi	quasi	ADJ
ejpam-5875	329	2	-	-	NOUN
ejpam-5875	329	3	position	position	ADJ
ejpam-5875	329	4	vector	vector	NOUN
ejpam-5875	329	5	curves	curve	NOUN
ejpam-5875	329	6	in	in	ADP
ejpam-5875	329	7	galilean	galilean	PROPN
ejpam-5875	329	8	4	4	NUM
ejpam-5875	329	9	-	-	PUNCT
ejpam-5875	329	10	space	space	NOUN
ejpam-5875	329	11	.	.	PUNCT
ejpam-5875	330	1	frontiers	frontier	NOUN
ejpam-5875	330	2	in	in	ADP
ejpam-5875	330	3	physics	physics	PROPN
ejpam-5875	330	4	,	,	PUNCT
ejpam-5875	330	5	12	12	NUM
ejpam-5875	330	6	,	,	PUNCT
ejpam-5875	330	7	2024	2024	NUM
ejpam-5875	330	8	.	.	PUNCT
ejpam-5875	331	1	[	[	X
ejpam-5875	331	2	11	11	NUM
ejpam-5875	331	3	]	]	PUNCT
ejpam-5875	331	4	a.	a.	NOUN
ejpam-5875	331	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	331	6	,	,	PUNCT
ejpam-5875	331	7	y.	y.	PROPN
ejpam-5875	331	8	tashkandy	tashkandy	PROPN
ejpam-5875	331	9	,	,	PUNCT
ejpam-5875	331	10	w.	w.	PROPN
ejpam-5875	331	11	emam	emam	PROPN
ejpam-5875	331	12	,	,	PUNCT
ejpam-5875	331	13	c.	c.	PROPN
ejpam-5875	331	14	cesarano	cesarano	PROPN
ejpam-5875	331	15	,	,	PUNCT
ejpam-5875	331	16	and	and	CCONJ
ejpam-5875	331	17	n.	n.	NOUN
ejpam-5875	331	18	elsharkawy	elsharkawy	NOUN
ejpam-5875	331	19	.	.	PUNCT
ejpam-5875	332	1	on	on	ADP
ejpam-5875	332	2	some	some	DET
ejpam-5875	332	3	quasi	quasi	NOUN
ejpam-5875	332	4	-	-	NOUN
ejpam-5875	332	5	curves	curve	NOUN
ejpam-5875	332	6	in	in	ADP
ejpam-5875	332	7	galilean	galilean	PROPN
ejpam-5875	332	8	three	three	NUM
ejpam-5875	332	9	-	-	PUNCT
ejpam-5875	332	10	space	space	NOUN
ejpam-5875	332	11	.	.	PUNCT
ejpam-5875	333	1	axioms	axiom	NOUN
ejpam-5875	333	2	,	,	PUNCT
ejpam-5875	333	3	12(9):823	12(9):823	NUM
ejpam-5875	333	4	,	,	PUNCT
ejpam-5875	333	5	2023	2023	NUM
ejpam-5875	333	6	.	.	PUNCT
ejpam-5875	334	1	[	[	X
ejpam-5875	334	2	12	12	NUM
ejpam-5875	334	3	]	]	X
ejpam-5875	334	4	n.	n.	NOUN
ejpam-5875	334	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	334	6	,	,	PUNCT
ejpam-5875	334	7	c.	c.	PROPN
ejpam-5875	334	8	cesarano	cesarano	PROPN
ejpam-5875	334	9	,	,	PUNCT
ejpam-5875	334	10	r.	r.	PROPN
ejpam-5875	334	11	dmytryshyn	dmytryshyn	PROPN
ejpam-5875	334	12	,	,	PUNCT
ejpam-5875	334	13	and	and	CCONJ
ejpam-5875	334	14	a.	a.	NOUN
ejpam-5875	334	15	elsharkawy	elsharkawy	PROPN
ejpam-5875	334	16	.	.	PUNCT
ejpam-5875	335	1	timelike	timelike	PROPN
ejpam-5875	335	2	spherical	spherical	ADJ
ejpam-5875	335	3	curves	curve	NOUN
ejpam-5875	335	4	according	accord	VERB
ejpam-5875	335	5	to	to	ADP
ejpam-5875	335	6	equiform	equiform	NOUN
ejpam-5875	335	7	bishop	bishop	PROPN
ejpam-5875	335	8	frame	frame	NOUN
ejpam-5875	335	9	in	in	ADP
ejpam-5875	335	10	3	3	NUM
ejpam-5875	335	11	-	-	PUNCT
ejpam-5875	335	12	dimensional	dimensional	ADJ
ejpam-5875	335	13	minkowski	minkowski	ADJ
ejpam-5875	335	14	space	space	NOUN
ejpam-5875	335	15	.	.	PUNCT
ejpam-5875	336	1	carpathian	carpathian	ADJ
ejpam-5875	336	2	mathematical	mathematical	ADJ
ejpam-5875	336	3	publications	publication	NOUN
ejpam-5875	336	4	,	,	PUNCT
ejpam-5875	336	5	15(2):88–95	15(2):88–95	NUM
ejpam-5875	336	6	,	,	PUNCT
ejpam-5875	336	7	2023	2023	NUM
ejpam-5875	336	8	.	.	PUNCT
ejpam-5875	337	1	[	[	X
ejpam-5875	337	2	13	13	NUM
ejpam-5875	337	3	]	]	X
ejpam-5875	337	4	s.	s.	PROPN
ejpam-5875	337	5	kiziltug	kiziltug	PROPN
ejpam-5875	337	6	,	,	PUNCT
ejpam-5875	337	7	a.	a.	NOUN
ejpam-5875	337	8	cakmak	cakmak	PROPN
ejpam-5875	337	9	,	,	PUNCT
ejpam-5875	337	10	t.	t.	PROPN
ejpam-5875	337	11	erisir	erisir	PROPN
ejpam-5875	337	12	,	,	PUNCT
ejpam-5875	337	13	and	and	CCONJ
ejpam-5875	337	14	g.	g.	PROPN
ejpam-5875	337	15	mumcu	mumcu	PROPN
ejpam-5875	337	16	.	.	PUNCT
ejpam-5875	338	1	on	on	ADP
ejpam-5875	338	2	tubular	tubular	ADJ
ejpam-5875	338	3	surfaces	surface	NOUN
ejpam-5875	338	4	with	with	ADP
ejpam-5875	338	5	modified	modified	ADJ
ejpam-5875	338	6	orthogonal	orthogonal	ADJ
ejpam-5875	338	7	frame	frame	NOUN
ejpam-5875	338	8	in	in	ADP
ejpam-5875	338	9	galilean	galilean	PROPN
ejpam-5875	338	10	space	space	PROPN
ejpam-5875	338	11	g3	g3	PROPN
ejpam-5875	338	12	.	.	PUNCT
ejpam-5875	338	13	thermal	thermal	ADJ
ejpam-5875	338	14	science	science	NOUN
ejpam-5875	338	15	,	,	PUNCT
ejpam-5875	338	16	26(spec	26(spec	NUM
ejpam-5875	338	17	.	.	PUNCT
ejpam-5875	339	1	issue	issue	NOUN
ejpam-5875	339	2	2):571–581	2):571–581	NUM
ejpam-5875	339	3	,	,	PUNCT
ejpam-5875	339	4	2022	2022	NUM
ejpam-5875	339	5	.	.	PUNCT
ejpam-5875	340	1	[	[	X
ejpam-5875	340	2	14	14	NUM
ejpam-5875	340	3	]	]	X
ejpam-5875	340	4	d.	d.	PROPN
ejpam-5875	340	5	w.	w.	PROPN
ejpam-5875	340	6	yoon	yoon	PROPN
ejpam-5875	340	7	.	.	PUNCT
ejpam-5875	341	1	inelastic	inelastic	ADJ
ejpam-5875	341	2	flows	flow	NOUN
ejpam-5875	341	3	of	of	ADP
ejpam-5875	341	4	curves	curve	NOUN
ejpam-5875	341	5	according	accord	VERB
ejpam-5875	341	6	to	to	ADP
ejpam-5875	341	7	equiform	equiform	NOUN
ejpam-5875	341	8	in	in	ADP
ejpam-5875	341	9	galilean	galilean	PROPN
ejpam-5875	341	10	space	space	NOUN
ejpam-5875	341	11	.	.	PUNCT
ejpam-5875	342	1	journal	journal	NOUN
ejpam-5875	342	2	of	of	ADP
ejpam-5875	342	3	the	the	DET
ejpam-5875	342	4	chungcheong	chungcheong	PROPN
ejpam-5875	342	5	mathematical	mathematical	ADJ
ejpam-5875	342	6	society	society	NOUN
ejpam-5875	342	7	,	,	PUNCT
ejpam-5875	342	8	24(4):665	24(4):665	NUM
ejpam-5875	342	9	,	,	PUNCT
ejpam-5875	342	10	2011	2011	NUM
ejpam-5875	342	11	.	.	PUNCT
ejpam-5875	343	1	[	[	X
ejpam-5875	343	2	15	15	NUM
ejpam-5875	343	3	]	]	X
ejpam-5875	343	4	h.	h.	PROPN
ejpam-5875	343	5	k.	k.	PROPN
ejpam-5875	343	6	elsayied	elsayied	PROPN
ejpam-5875	343	7	,	,	PUNCT
ejpam-5875	343	8	m.	m.	NOUN
ejpam-5875	343	9	elzawy	elzawy	PROPN
ejpam-5875	343	10	,	,	PUNCT
ejpam-5875	343	11	and	and	CCONJ
ejpam-5875	343	12	a.	a.	NOUN
ejpam-5875	343	13	elsharkawy	elsharkawy	PROPN
ejpam-5875	343	14	.	.	PUNCT
ejpam-5875	344	1	equiform	equiform	PROPN
ejpam-5875	344	2	spacelike	spacelike	VERB
ejpam-5875	344	3	normal	normal	ADJ
ejpam-5875	344	4	curves	curve	NOUN
ejpam-5875	344	5	according	accord	VERB
ejpam-5875	344	6	to	to	ADP
ejpam-5875	344	7	equiform	equiform	NOUN
ejpam-5875	344	8	-	-	PUNCT
ejpam-5875	344	9	bishop	bishop	PROPN
ejpam-5875	344	10	frame	frame	NOUN
ejpam-5875	344	11	in	in	ADP
ejpam-5875	344	12	e3	e3	PROPN
ejpam-5875	344	13	1	1	NUM
ejpam-5875	344	14	.	.	PUNCT
ejpam-5875	345	1	mathematical	mathematical	ADJ
ejpam-5875	345	2	methods	method	NOUN
ejpam-5875	345	3	in	in	ADP
ejpam-5875	345	4	the	the	DET
ejpam-5875	345	5	applied	apply	VERB
ejpam-5875	345	6	sciences	science	NOUN
ejpam-5875	345	7	,	,	PUNCT
ejpam-5875	345	8	41(15):5754–5760	41(15):5754–5760	NUM
ejpam-5875	345	9	,	,	PUNCT
ejpam-5875	345	10	2018	2018	NUM
ejpam-5875	345	11	.	.	PUNCT
ejpam-5875	346	1	[	[	X
ejpam-5875	346	2	16	16	NUM
ejpam-5875	346	3	]	]	PUNCT
ejpam-5875	346	4	a.	a.	NOUN
ejpam-5875	346	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	346	6	.	.	PUNCT
ejpam-5875	347	1	generalized	generalized	ADJ
ejpam-5875	347	2	involute	involute	NOUN
ejpam-5875	347	3	and	and	CCONJ
ejpam-5875	347	4	evolute	evolute	PROPN
ejpam-5875	347	5	curves	curve	NOUN
ejpam-5875	347	6	of	of	ADP
ejpam-5875	347	7	equiform	equiform	NOUN
ejpam-5875	347	8	spacelike	spacelike	PROPN
ejpam-5875	347	9	curves	curve	NOUN
ejpam-5875	347	10	with	with	ADP
ejpam-5875	347	11	a	a	DET
ejpam-5875	347	12	timelike	timelike	PROPN
ejpam-5875	347	13	equiform	equiform	NOUN
ejpam-5875	347	14	principal	principal	NOUN
ejpam-5875	347	15	normal	normal	ADJ
ejpam-5875	347	16	in	in	ADP
ejpam-5875	347	17	e3	e3	PROPN
ejpam-5875	347	18	1	1	NUM
ejpam-5875	347	19	.	.	PUNCT
ejpam-5875	348	1	journal	journal	NOUN
ejpam-5875	348	2	of	of	ADP
ejpam-5875	348	3	the	the	DET
ejpam-5875	348	4	egyptian	egyptian	PROPN
ejpam-5875	348	5	mathematical	mathematical	PROPN
ejpam-5875	348	6	society	society	NOUN
ejpam-5875	348	7	,	,	PUNCT
ejpam-5875	348	8	28(1):26	28(1):26	NUM
ejpam-5875	348	9	,	,	PUNCT
ejpam-5875	348	10	2020	2020	NUM
ejpam-5875	348	11	.	.	PUNCT
ejpam-5875	349	1	[	[	X
ejpam-5875	349	2	17	17	NUM
ejpam-5875	349	3	]	]	PUNCT
ejpam-5875	349	4	t.	t.	PROPN
ejpam-5875	349	5	sahin	sahin	PROPN
ejpam-5875	349	6	,	,	PUNCT
ejpam-5875	349	7	f.	f.	PROPN
ejpam-5875	349	8	karakus	karakus	PROPN
ejpam-5875	349	9	,	,	PUNCT
ejpam-5875	349	10	and	and	CCONJ
ejpam-5875	349	11	k.	k.	PROPN
ejpam-5875	349	12	orbay	orbay	PROPN
ejpam-5875	349	13	.	.	PUNCT
ejpam-5875	350	1	parallel	parallel	ADJ
ejpam-5875	350	2	transports	transport	NOUN
ejpam-5875	350	3	with	with	ADP
ejpam-5875	350	4	respect	respect	NOUN
ejpam-5875	350	5	to	to	ADP
ejpam-5875	350	6	frenet	frenet	NOUN
ejpam-5875	350	7	and	and	CCONJ
ejpam-5875	350	8	a.	a.	NOUN
ejpam-5875	350	9	elsharkawy	elsharkawy	PROPN
ejpam-5875	350	10	,	,	PUNCT
ejpam-5875	350	11	n.	n.	NOUN
ejpam-5875	350	12	elsharkawy	elsharkawy	PROPN
ejpam-5875	350	13	/	/	SYM
ejpam-5875	350	14	eur	eur	PROPN
ejpam-5875	350	15	.	.	PUNCT
ejpam-5875	351	1	j.	j.	PROPN
ejpam-5875	351	2	pure	pure	PROPN
ejpam-5875	351	3	appl	appl	PROPN
ejpam-5875	351	4	.	.	PROPN
ejpam-5875	351	5	math	math	PROPN
ejpam-5875	351	6	,	,	PUNCT
ejpam-5875	351	7	18	18	NUM
ejpam-5875	351	8	(	(	PUNCT
ejpam-5875	351	9	2	2	NUM
ejpam-5875	351	10	)	)	PUNCT
ejpam-5875	351	11	(	(	PUNCT
ejpam-5875	351	12	2025	2025	NUM
ejpam-5875	351	13	)	)	PUNCT
ejpam-5875	351	14	,	,	PUNCT
ejpam-5875	351	15	5875	5875	NUM
ejpam-5875	351	16	14	14	NUM
ejpam-5875	351	17	of	of	ADP
ejpam-5875	351	18	15	15	NUM
ejpam-5875	351	19	darboux	darboux	VERB
ejpam-5875	351	20	frames	frame	NOUN
ejpam-5875	351	21	in	in	ADP
ejpam-5875	351	22	the	the	DET
ejpam-5875	351	23	galilean	galilean	PROPN
ejpam-5875	351	24	space	space	NOUN
ejpam-5875	351	25	.	.	PUNCT
ejpam-5875	352	1	journal	journal	PROPN
ejpam-5875	352	2	of	of	ADP
ejpam-5875	352	3	science	science	NOUN
ejpam-5875	352	4	and	and	CCONJ
ejpam-5875	352	5	arts	art	NOUN
ejpam-5875	352	6	,	,	PUNCT
ejpam-5875	352	7	1(50):13–24	1(50):13–24	NUM
ejpam-5875	352	8	,	,	PUNCT
ejpam-5875	352	9	2020	2020	NUM
ejpam-5875	352	10	.	.	PUNCT
ejpam-5875	353	1	[	[	X
ejpam-5875	353	2	18	18	NUM
ejpam-5875	353	3	]	]	PUNCT
ejpam-5875	353	4	t.	t.	PROPN
ejpam-5875	353	5	sahin	sahin	PROPN
ejpam-5875	353	6	and	and	CCONJ
ejpam-5875	353	7	m.	m.	NOUN
ejpam-5875	353	8	okur	okur	PROPN
ejpam-5875	353	9	.	.	PUNCT
ejpam-5875	354	1	special	special	ADJ
ejpam-5875	354	2	smarandache	smarandache	NOUN
ejpam-5875	354	3	curves	curve	VERB
ejpam-5875	354	4	with	with	ADP
ejpam-5875	354	5	respect	respect	NOUN
ejpam-5875	354	6	to	to	ADP
ejpam-5875	354	7	darboux	darboux	VERB
ejpam-5875	354	8	frame	frame	NOUN
ejpam-5875	354	9	in	in	ADP
ejpam-5875	354	10	galilean	galilean	PROPN
ejpam-5875	354	11	3	3	NUM
ejpam-5875	354	12	-	-	PUNCT
ejpam-5875	354	13	space	space	NOUN
ejpam-5875	354	14	.	.	PUNCT
ejpam-5875	355	1	infinite	infinite	ADJ
ejpam-5875	355	2	study	study	NOUN
ejpam-5875	355	3	,	,	PUNCT
ejpam-5875	355	4	2017	2017	NUM
ejpam-5875	355	5	.	.	PUNCT
ejpam-5875	356	1	[	[	X
ejpam-5875	356	2	19	19	NUM
ejpam-5875	356	3	]	]	X
ejpam-5875	356	4	e.	e.	PROPN
ejpam-5875	356	5	d.	d.	PROPN
ejpam-5875	356	6	cetin	cetin	PROPN
ejpam-5875	356	7	,	,	PUNCT
ejpam-5875	356	8	i.	i.	PROPN
ejpam-5875	356	9	gök	gök	NOUN
ejpam-5875	356	10	,	,	PUNCT
ejpam-5875	356	11	and	and	CCONJ
ejpam-5875	356	12	y.	y.	PROPN
ejpam-5875	356	13	yayli	yayli	PROPN
ejpam-5875	356	14	.	.	PUNCT
ejpam-5875	357	1	a	a	DET
ejpam-5875	357	2	new	new	ADJ
ejpam-5875	357	3	aspect	aspect	NOUN
ejpam-5875	357	4	of	of	ADP
ejpam-5875	357	5	rectifying	rectifying	NOUN
ejpam-5875	357	6	curves	curve	NOUN
ejpam-5875	357	7	and	and	CCONJ
ejpam-5875	357	8	ruled	rule	VERB
ejpam-5875	357	9	surfaces	surface	NOUN
ejpam-5875	357	10	in	in	ADP
ejpam-5875	357	11	galilean	galilean	PROPN
ejpam-5875	357	12	3	3	NUM
ejpam-5875	357	13	-	-	PUNCT
ejpam-5875	357	14	space	space	NOUN
ejpam-5875	357	15	.	.	PUNCT
ejpam-5875	358	1	filomat	filomat	NOUN
ejpam-5875	358	2	,	,	PUNCT
ejpam-5875	358	3	32(8	32(8	NUM
ejpam-5875	358	4	)	)	PUNCT
ejpam-5875	358	5	,	,	PUNCT
ejpam-5875	358	6	2018	2018	NUM
ejpam-5875	358	7	.	.	PUNCT
ejpam-5875	359	1	[	[	X
ejpam-5875	359	2	20	20	NUM
ejpam-5875	359	3	]	]	PUNCT
ejpam-5875	359	4	a.	a.	NOUN
ejpam-5875	359	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	359	6	,	,	PUNCT
ejpam-5875	359	7	h.	h.	PROPN
ejpam-5875	359	8	elsayied	elsayied	PROPN
ejpam-5875	359	9	,	,	PUNCT
ejpam-5875	359	10	and	and	CCONJ
ejpam-5875	359	11	a.	a.	NOUN
ejpam-5875	359	12	refaat	refaat	PROPN
ejpam-5875	359	13	.	.	PUNCT
ejpam-5875	360	1	quasi	quasi	PROPN
ejpam-5875	360	2	ruled	rule	VERB
ejpam-5875	360	3	surfaces	surface	NOUN
ejpam-5875	360	4	in	in	ADP
ejpam-5875	360	5	euclidean	euclidean	ADJ
ejpam-5875	360	6	3space	3space	NUM
ejpam-5875	360	7	.	.	PUNCT
ejpam-5875	361	1	european	european	ADJ
ejpam-5875	361	2	journal	journal	PROPN
ejpam-5875	361	3	of	of	ADP
ejpam-5875	361	4	pure	pure	ADJ
ejpam-5875	361	5	and	and	CCONJ
ejpam-5875	361	6	applied	applied	ADJ
ejpam-5875	361	7	mathematics	mathematic	NOUN
ejpam-5875	361	8	,	,	PUNCT
ejpam-5875	361	9	18(1):5710–5710	18(1):5710–5710	NUM
ejpam-5875	361	10	,	,	PUNCT
ejpam-5875	361	11	2025	2025	NUM
ejpam-5875	361	12	.	.	PUNCT
ejpam-5875	362	1	[	[	X
ejpam-5875	362	2	21	21	NUM
ejpam-5875	362	3	]	]	X
ejpam-5875	362	4	m.	m.	NOUN
ejpam-5875	362	5	a.	a.	NOUN
ejpam-5875	362	6	kulahci	kulahci	PROPN
ejpam-5875	362	7	,	,	PUNCT
ejpam-5875	362	8	m.	m.	NOUN
ejpam-5875	362	9	bektaş	bektaş	NOUN
ejpam-5875	362	10	,	,	PUNCT
ejpam-5875	362	11	and	and	CCONJ
ejpam-5875	362	12	a.	a.	NOUN
ejpam-5875	362	13	bilici	bilici	PROPN
ejpam-5875	362	14	.	.	PUNCT
ejpam-5875	363	1	on	on	ADP
ejpam-5875	363	2	classification	classification	NOUN
ejpam-5875	363	3	of	of	ADP
ejpam-5875	363	4	normal	normal	ADJ
ejpam-5875	363	5	and	and	CCONJ
ejpam-5875	363	6	osculating	osculating	NOUN
ejpam-5875	363	7	curve	curve	NOUN
ejpam-5875	363	8	in	in	ADP
ejpam-5875	363	9	3	3	NUM
ejpam-5875	363	10	-	-	PUNCT
ejpam-5875	363	11	dimensional	dimensional	ADJ
ejpam-5875	363	12	sasakian	sasakian	ADJ
ejpam-5875	363	13	space	space	NOUN
ejpam-5875	363	14	.	.	PUNCT
ejpam-5875	364	1	mathematical	mathematical	ADJ
ejpam-5875	364	2	sciences	sciences	PROPN
ejpam-5875	364	3	and	and	CCONJ
ejpam-5875	364	4	applications	application	NOUN
ejpam-5875	364	5	enotes	enote	NOUN
ejpam-5875	364	6	,	,	PUNCT
ejpam-5875	364	7	7:120–127	7:120–127	NOUN
ejpam-5875	364	8	,	,	PUNCT
ejpam-5875	364	9	2019	2019	NUM
ejpam-5875	364	10	.	.	PUNCT
ejpam-5875	365	1	[	[	X
ejpam-5875	365	2	22	22	NUM
ejpam-5875	365	3	]	]	PUNCT
ejpam-5875	365	4	m.	m.	NOUN
ejpam-5875	365	5	s.	s.	PROPN
ejpam-5875	365	6	lone	lone	PROPN
ejpam-5875	365	7	.	.	PUNCT
ejpam-5875	366	1	some	some	DET
ejpam-5875	366	2	characterizations	characterization	NOUN
ejpam-5875	366	3	of	of	ADP
ejpam-5875	366	4	rectifying	rectifying	NOUN
ejpam-5875	366	5	curves	curve	NOUN
ejpam-5875	366	6	in	in	ADP
ejpam-5875	366	7	four	four	NUM
ejpam-5875	366	8	-	-	PUNCT
ejpam-5875	366	9	dimensional	dimensional	ADJ
ejpam-5875	366	10	galilean	galilean	PROPN
ejpam-5875	366	11	space	space	NOUN
ejpam-5875	366	12	g4	g4	NOUN
ejpam-5875	366	13	.	.	PUNCT
ejpam-5875	367	1	global	global	ADJ
ejpam-5875	367	2	journal	journal	NOUN
ejpam-5875	367	3	of	of	ADP
ejpam-5875	367	4	pure	pure	ADJ
ejpam-5875	367	5	and	and	CCONJ
ejpam-5875	367	6	applied	applied	ADJ
ejpam-5875	367	7	mathematics	mathematic	NOUN
ejpam-5875	367	8	,	,	PUNCT
ejpam-5875	367	9	13:579–587	13:579–587	NUM
ejpam-5875	367	10	,	,	PUNCT
ejpam-5875	367	11	2017	2017	NUM
ejpam-5875	367	12	.	.	PUNCT
ejpam-5875	368	1	[	[	X
ejpam-5875	368	2	23	23	NUM
ejpam-5875	368	3	]	]	X
ejpam-5875	368	4	s.	s.	PROPN
ejpam-5875	368	5	mosa	mosa	PROPN
ejpam-5875	368	6	,	,	PUNCT
ejpam-5875	368	7	m.	m.	PROPN
ejpam-5875	368	8	el	el	PROPN
ejpam-5875	368	9	-	-	NOUN
ejpam-5875	368	10	fakharany	fakharany	ADJ
ejpam-5875	368	11	,	,	PUNCT
ejpam-5875	368	12	and	and	CCONJ
ejpam-5875	368	13	m.	m.	PROPN
ejpam-5875	368	14	elzawy	elzawy	PROPN
ejpam-5875	368	15	.	.	PUNCT
ejpam-5875	369	1	normal	normal	ADJ
ejpam-5875	369	2	curves	curve	NOUN
ejpam-5875	369	3	in	in	ADP
ejpam-5875	369	4	4	4	NUM
ejpam-5875	369	5	-	-	PUNCT
ejpam-5875	369	6	dimensional	dimensional	ADJ
ejpam-5875	369	7	galilean	galilean	PROPN
ejpam-5875	369	8	space	space	NOUN
ejpam-5875	369	9	g4	g4	NOUN
ejpam-5875	369	10	.	.	PUNCT
ejpam-5875	370	1	frontiers	frontier	NOUN
ejpam-5875	370	2	in	in	ADP
ejpam-5875	370	3	physics	physics	PROPN
ejpam-5875	370	4	,	,	PUNCT
ejpam-5875	370	5	9	9	NUM
ejpam-5875	370	6	,	,	PUNCT
ejpam-5875	370	7	2021	2021	NUM
ejpam-5875	370	8	.	.	PUNCT
ejpam-5875	371	1	[	[	X
ejpam-5875	371	2	24	24	NUM
ejpam-5875	371	3	]	]	PUNCT
ejpam-5875	371	4	h.	h.	NOUN
ejpam-5875	371	5	oztekin	oztekin	PROPN
ejpam-5875	371	6	.	.	PUNCT
ejpam-5875	372	1	normal	normal	ADJ
ejpam-5875	372	2	and	and	CCONJ
ejpam-5875	372	3	rectifying	rectifying	NOUN
ejpam-5875	372	4	curves	curve	NOUN
ejpam-5875	372	5	in	in	ADP
ejpam-5875	372	6	galilean	galilean	PROPN
ejpam-5875	372	7	space	space	PROPN
ejpam-5875	372	8	g3	g3	PROPN
ejpam-5875	372	9	.	.	PUNCT
ejpam-5875	373	1	proc	proc	PROPN
ejpam-5875	373	2	.	.	PROPN
ejpam-5875	374	1	of	of	ADP
ejpam-5875	374	2	iam	iam	PROPN
ejpam-5875	374	3	,	,	PUNCT
ejpam-5875	374	4	5(1):98–109	5(1):98–109	NUM
ejpam-5875	374	5	,	,	PUNCT
ejpam-5875	374	6	2016	2016	NUM
ejpam-5875	374	7	.	.	PUNCT
ejpam-5875	375	1	[	[	X
ejpam-5875	375	2	25	25	NUM
ejpam-5875	375	3	]	]	PUNCT
ejpam-5875	375	4	m.	m.	NOUN
ejpam-5875	375	5	dede	dede	NOUN
ejpam-5875	375	6	,	,	PUNCT
ejpam-5875	375	7	c.	c.	PROPN
ejpam-5875	375	8	ekici	ekici	NOUN
ejpam-5875	375	9	,	,	PUNCT
ejpam-5875	375	10	and	and	CCONJ
ejpam-5875	375	11	a.	a.	NOUN
ejpam-5875	375	12	görgülü	görgülü	PROPN
ejpam-5875	375	13	.	.	PUNCT
ejpam-5875	376	1	directional	directional	ADJ
ejpam-5875	376	2	q	q	NOUN
ejpam-5875	376	3	-	-	PUNCT
ejpam-5875	376	4	frame	frame	NOUN
ejpam-5875	376	5	along	along	ADP
ejpam-5875	376	6	a	a	DET
ejpam-5875	376	7	space	space	NOUN
ejpam-5875	376	8	curve	curve	NOUN
ejpam-5875	376	9	.	.	PUNCT
ejpam-5875	377	1	international	international	ADJ
ejpam-5875	377	2	journal	journal	NOUN
ejpam-5875	377	3	of	of	ADP
ejpam-5875	377	4	advanced	advanced	ADJ
ejpam-5875	377	5	computer	computer	NOUN
ejpam-5875	377	6	science	science	NOUN
ejpam-5875	377	7	and	and	CCONJ
ejpam-5875	377	8	applications	application	NOUN
ejpam-5875	377	9	,	,	PUNCT
ejpam-5875	377	10	5:775–780	5:775–780	NUM
ejpam-5875	377	11	,	,	PUNCT
ejpam-5875	377	12	2015	2015	NUM
ejpam-5875	377	13	.	.	PUNCT
ejpam-5875	378	1	[	[	X
ejpam-5875	378	2	26	26	NUM
ejpam-5875	378	3	]	]	PUNCT
ejpam-5875	378	4	h.	h.	PROPN
ejpam-5875	378	5	k.	k.	PROPN
ejpam-5875	378	6	elsayied	elsayied	PROPN
ejpam-5875	378	7	,	,	PUNCT
ejpam-5875	378	8	a.	a.	PROPN
ejpam-5875	378	9	m.	m.	PROPN
ejpam-5875	378	10	tawfiq	tawfiq	PROPN
ejpam-5875	378	11	,	,	PUNCT
ejpam-5875	378	12	and	and	CCONJ
ejpam-5875	378	13	a.	a.	NOUN
ejpam-5875	378	14	elsharkawy	elsharkawy	PROPN
ejpam-5875	378	15	.	.	PUNCT
ejpam-5875	379	1	special	special	ADJ
ejpam-5875	379	2	smarandach	smarandach	ADJ
ejpam-5875	379	3	curves	curve	NOUN
ejpam-5875	379	4	according	accord	VERB
ejpam-5875	379	5	to	to	ADP
ejpam-5875	379	6	the	the	DET
ejpam-5875	379	7	quasi	quasi	ADJ
ejpam-5875	379	8	frame	frame	NOUN
ejpam-5875	379	9	in	in	ADP
ejpam-5875	379	10	4	4	NUM
ejpam-5875	379	11	-	-	PUNCT
ejpam-5875	379	12	dimensional	dimensional	ADJ
ejpam-5875	379	13	euclidean	euclidean	ADJ
ejpam-5875	379	14	space	space	NOUN
ejpam-5875	379	15	e4	e4	PROPN
ejpam-5875	379	16	.	.	PUNCT
ejpam-5875	380	1	houston	houston	PROPN
ejpam-5875	380	2	journal	journal	PROPN
ejpam-5875	380	3	of	of	ADP
ejpam-5875	380	4	mathematics	mathematic	NOUN
ejpam-5875	380	5	,	,	PUNCT
ejpam-5875	380	6	74(2):467–482	74(2):467–482	PROPN
ejpam-5875	380	7	,	,	PUNCT
ejpam-5875	380	8	2021	2021	NUM
ejpam-5875	380	9	.	.	PUNCT
ejpam-5875	381	1	[	[	X
ejpam-5875	381	2	27	27	NUM
ejpam-5875	381	3	]	]	PUNCT
ejpam-5875	381	4	a.	a.	NOUN
ejpam-5875	381	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	381	6	,	,	PUNCT
ejpam-5875	381	7	c.	c.	PROPN
ejpam-5875	381	8	cesarano	cesarano	PROPN
ejpam-5875	381	9	,	,	PUNCT
ejpam-5875	381	10	a.	a.	PROPN
ejpam-5875	381	11	tawfiq	tawfiq	PROPN
ejpam-5875	381	12	,	,	PUNCT
ejpam-5875	381	13	and	and	CCONJ
ejpam-5875	381	14	a.	a.	NOUN
ejpam-5875	381	15	a.	a.	PROPN
ejpam-5875	381	16	ismail	ismail	PROPN
ejpam-5875	381	17	.	.	PUNCT
ejpam-5875	382	1	the	the	DET
ejpam-5875	382	2	non	non	ADJ
ejpam-5875	382	3	-	-	ADJ
ejpam-5875	382	4	linear	linear	ADJ
ejpam-5875	382	5	schrodinger	schrodinger	NOUN
ejpam-5875	382	6	equation	equation	NOUN
ejpam-5875	382	7	associated	associate	VERB
ejpam-5875	382	8	with	with	ADP
ejpam-5875	382	9	the	the	DET
ejpam-5875	382	10	soliton	soliton	NOUN
ejpam-5875	382	11	surfaces	surface	NOUN
ejpam-5875	382	12	in	in	ADP
ejpam-5875	382	13	minkowski	minkowski	ADJ
ejpam-5875	382	14	3	3	NUM
ejpam-5875	382	15	-	-	PUNCT
ejpam-5875	382	16	space	space	NOUN
ejpam-5875	382	17	.	.	PUNCT
ejpam-5875	383	1	aims	aim	VERB
ejpam-5875	383	2	mathematics	mathematic	NOUN
ejpam-5875	383	3	,	,	PUNCT
ejpam-5875	383	4	7(10):17879–17893	7(10):17879–17893	NUM
ejpam-5875	383	5	,	,	PUNCT
ejpam-5875	383	6	2022	2022	NUM
ejpam-5875	383	7	.	.	PUNCT
ejpam-5875	384	1	[	[	X
ejpam-5875	384	2	28	28	NUM
ejpam-5875	384	3	]	]	X
ejpam-5875	384	4	a.	a.	NOUN
ejpam-5875	384	5	m.	m.	PROPN
ejpam-5875	384	6	elshenhab	elshenhab	PROPN
ejpam-5875	384	7	,	,	PUNCT
ejpam-5875	384	8	o.	o.	PROPN
ejpam-5875	384	9	moaaz	moaaz	PROPN
ejpam-5875	384	10	,	,	PUNCT
ejpam-5875	384	11	i.	i.	NOUN
ejpam-5875	384	12	dassios	dassios	PROPN
ejpam-5875	384	13	,	,	PUNCT
ejpam-5875	384	14	and	and	CCONJ
ejpam-5875	384	15	a.	a.	NOUN
ejpam-5875	384	16	elsharkawy	elsharkawy	PROPN
ejpam-5875	384	17	.	.	PUNCT
ejpam-5875	385	1	motion	motion	NOUN
ejpam-5875	385	2	along	along	ADP
ejpam-5875	385	3	a	a	DET
ejpam-5875	385	4	space	space	NOUN
ejpam-5875	385	5	curve	curve	NOUN
ejpam-5875	385	6	with	with	ADP
ejpam-5875	385	7	a	a	DET
ejpam-5875	385	8	quasi	quasi	NOUN
ejpam-5875	385	9	-	-	NOUN
ejpam-5875	385	10	frame	frame	NOUN
ejpam-5875	385	11	in	in	ADP
ejpam-5875	385	12	euclidean	euclidean	ADJ
ejpam-5875	385	13	3	3	NUM
ejpam-5875	385	14	-	-	PUNCT
ejpam-5875	385	15	space	space	NOUN
ejpam-5875	385	16	:	:	PUNCT
ejpam-5875	385	17	acceleration	acceleration	NOUN
ejpam-5875	385	18	and	and	CCONJ
ejpam-5875	385	19	jerk	jerk	NOUN
ejpam-5875	385	20	.	.	PUNCT
ejpam-5875	386	1	symmetry	symmetry	PROPN
ejpam-5875	386	2	,	,	PUNCT
ejpam-5875	386	3	14(8):1610	14(8):1610	NUM
ejpam-5875	386	4	,	,	PUNCT
ejpam-5875	386	5	2022	2022	NUM
ejpam-5875	386	6	.	.	PUNCT
ejpam-5875	387	1	[	[	X
ejpam-5875	387	2	29	29	NUM
ejpam-5875	387	3	]	]	PUNCT
ejpam-5875	387	4	e.	e.	PROPN
ejpam-5875	387	5	hamouda	hamouda	PROPN
ejpam-5875	387	6	,	,	PUNCT
ejpam-5875	387	7	c.	c.	PROPN
ejpam-5875	387	8	cesarano	cesarano	PROPN
ejpam-5875	387	9	,	,	PUNCT
ejpam-5875	387	10	s.	s.	PROPN
ejpam-5875	387	11	askar	askar	PROPN
ejpam-5875	387	12	,	,	PUNCT
ejpam-5875	387	13	and	and	CCONJ
ejpam-5875	387	14	a.	a.	NOUN
ejpam-5875	387	15	elsharkawy	elsharkawy	NOUN
ejpam-5875	387	16	.	.	PUNCT
ejpam-5875	388	1	resolutions	resolution	NOUN
ejpam-5875	388	2	of	of	ADP
ejpam-5875	388	3	the	the	DET
ejpam-5875	388	4	jerk	jerk	NOUN
ejpam-5875	388	5	and	and	CCONJ
ejpam-5875	388	6	snap	snap	VERB
ejpam-5875	388	7	vectors	vector	NOUN
ejpam-5875	388	8	for	for	ADP
ejpam-5875	388	9	a	a	DET
ejpam-5875	388	10	quasi	quasi	ADJ
ejpam-5875	388	11	curve	curve	NOUN
ejpam-5875	388	12	in	in	ADP
ejpam-5875	388	13	euclidean	euclidean	ADJ
ejpam-5875	388	14	3	3	NUM
ejpam-5875	388	15	-	-	PUNCT
ejpam-5875	388	16	space	space	NOUN
ejpam-5875	388	17	.	.	PUNCT
ejpam-5875	389	1	mathematics	mathematic	NOUN
ejpam-5875	389	2	,	,	PUNCT
ejpam-5875	389	3	9(23):3128	9(23):3128	NOUN
ejpam-5875	389	4	,	,	PUNCT
ejpam-5875	389	5	2021	2021	NUM
ejpam-5875	389	6	.	.	PUNCT
ejpam-5875	390	1	[	[	X
ejpam-5875	390	2	30	30	NUM
ejpam-5875	390	3	]	]	X
ejpam-5875	390	4	e.	e.	PROPN
ejpam-5875	390	5	hamouda	hamouda	PROPN
ejpam-5875	390	6	,	,	PUNCT
ejpam-5875	390	7	o.	o.	PROPN
ejpam-5875	390	8	moaaz	moaaz	PROPN
ejpam-5875	390	9	,	,	PUNCT
ejpam-5875	390	10	c.	c.	PROPN
ejpam-5875	390	11	cesarano	cesarano	PROPN
ejpam-5875	390	12	,	,	PUNCT
ejpam-5875	390	13	s.	s.	PROPN
ejpam-5875	390	14	askar	askar	PROPN
ejpam-5875	390	15	,	,	PUNCT
ejpam-5875	390	16	and	and	CCONJ
ejpam-5875	390	17	a.	a.	NOUN
ejpam-5875	390	18	elsharkawy	elsharkawy	PROPN
ejpam-5875	390	19	.	.	PUNCT
ejpam-5875	391	1	geometry	geometry	NOUN
ejpam-5875	391	2	of	of	ADP
ejpam-5875	391	3	solutions	solution	NOUN
ejpam-5875	391	4	of	of	ADP
ejpam-5875	391	5	the	the	DET
ejpam-5875	391	6	quasi	quasi	ADJ
ejpam-5875	391	7	-	-	ADJ
ejpam-5875	391	8	vortex	vortex	ADJ
ejpam-5875	391	9	filament	filament	NOUN
ejpam-5875	391	10	equation	equation	NOUN
ejpam-5875	391	11	in	in	ADP
ejpam-5875	391	12	euclidean	euclidean	ADJ
ejpam-5875	391	13	3	3	NUM
ejpam-5875	391	14	-	-	PUNCT
ejpam-5875	391	15	space	space	NOUN
ejpam-5875	391	16	e3	e3	NOUN
ejpam-5875	391	17	.	.	PUNCT
ejpam-5875	392	1	mathematics	mathematic	NOUN
ejpam-5875	392	2	,	,	PUNCT
ejpam-5875	392	3	10(6):891	10(6):891	NUM
ejpam-5875	392	4	,	,	PUNCT
ejpam-5875	392	5	2022	2022	NUM
ejpam-5875	392	6	.	.	PUNCT
ejpam-5875	393	1	[	[	X
ejpam-5875	393	2	31	31	NUM
ejpam-5875	393	3	]	]	PUNCT
ejpam-5875	393	4	a.	a.	NOUN
ejpam-5875	393	5	t.	t.	PROPN
ejpam-5875	393	6	ali	ali	PROPN
ejpam-5875	393	7	.	.	PUNCT
ejpam-5875	394	1	position	position	NOUN
ejpam-5875	394	2	vectors	vector	NOUN
ejpam-5875	394	3	of	of	ADP
ejpam-5875	394	4	curves	curve	NOUN
ejpam-5875	394	5	in	in	ADP
ejpam-5875	394	6	the	the	DET
ejpam-5875	394	7	galilean	galilean	PROPN
ejpam-5875	394	8	space	space	PROPN
ejpam-5875	394	9	g3	g3	PROPN
ejpam-5875	394	10	.	.	PUNCT
ejpam-5875	395	1	matematicki	matematicki	PROPN
ejpam-5875	395	2	vesnik	vesnik	PROPN
ejpam-5875	395	3	,	,	PUNCT
ejpam-5875	395	4	64(249):200–210	64(249):200–210	PROPN
ejpam-5875	395	5	,	,	PUNCT
ejpam-5875	395	6	2012	2012	NUM
ejpam-5875	395	7	.	.	PUNCT
ejpam-5875	396	1	[	[	X
ejpam-5875	396	2	32	32	NUM
ejpam-5875	396	3	]	]	X
ejpam-5875	396	4	s.	s.	PROPN
ejpam-5875	396	5	büyükkütük	büyükkütük	PROPN
ejpam-5875	396	6	,	,	PUNCT
ejpam-5875	396	7	i.	i.	PROPN
ejpam-5875	396	8	kisi	kisi	PROPN
ejpam-5875	396	9	,	,	PUNCT
ejpam-5875	396	10	v.	v.	PROPN
ejpam-5875	396	11	mishra	mishra	PROPN
ejpam-5875	396	12	,	,	PUNCT
ejpam-5875	396	13	and	and	CCONJ
ejpam-5875	396	14	g.	g.	PROPN
ejpam-5875	396	15	oztürk	oztürk	PROPN
ejpam-5875	396	16	.	.	PUNCT
ejpam-5875	397	1	some	some	DET
ejpam-5875	397	2	characterizations	characterization	NOUN
ejpam-5875	397	3	of	of	ADP
ejpam-5875	397	4	curves	curve	NOUN
ejpam-5875	397	5	in	in	ADP
ejpam-5875	397	6	galilean	galilean	PROPN
ejpam-5875	397	7	3	3	NUM
ejpam-5875	397	8	-	-	PUNCT
ejpam-5875	397	9	space	space	NOUN
ejpam-5875	397	10	g3	g3	NOUN
ejpam-5875	397	11	.	.	PUNCT
ejpam-5875	397	12	facta	facta	PROPN
ejpam-5875	397	13	universitatis	universitatis	PROPN
ejpam-5875	397	14	,	,	PUNCT
ejpam-5875	397	15	series	series	NOUN
ejpam-5875	397	16	:	:	PUNCT
ejpam-5875	397	17	mathematics	mathematic	NOUN
ejpam-5875	397	18	and	and	CCONJ
ejpam-5875	397	19	informatics	informatic	NOUN
ejpam-5875	397	20	,	,	PUNCT
ejpam-5875	397	21	31(2):503–512	31(2):503–512	PROPN
ejpam-5875	397	22	,	,	PUNCT
ejpam-5875	397	23	2016	2016	NUM
ejpam-5875	397	24	.	.	PUNCT
ejpam-5875	398	1	[	[	X
ejpam-5875	398	2	33	33	NUM
ejpam-5875	398	3	]	]	X
ejpam-5875	398	4	o.	o.	PROPN
ejpam-5875	398	5	b.	b.	PROPN
ejpam-5875	398	6	kalkan	kalkan	PROPN
ejpam-5875	398	7	.	.	PUNCT
ejpam-5875	399	1	position	position	NOUN
ejpam-5875	399	2	vector	vector	NOUN
ejpam-5875	399	3	of	of	ADP
ejpam-5875	399	4	a	a	DET
ejpam-5875	399	5	w	w	NOUN
ejpam-5875	399	6	-	-	PUNCT
ejpam-5875	399	7	curve	curve	NOUN
ejpam-5875	399	8	in	in	ADP
ejpam-5875	399	9	the	the	DET
ejpam-5875	399	10	4d	4d	NUM
ejpam-5875	399	11	galilean	galilean	PROPN
ejpam-5875	399	12	space	space	NOUN
ejpam-5875	399	13	.	.	PUNCT
ejpam-5875	400	1	facta	facta	PROPN
ejpam-5875	400	2	universitatis	universitatis	PROPN
ejpam-5875	400	3	,	,	PUNCT
ejpam-5875	400	4	series	series	NOUN
ejpam-5875	400	5	:	:	PUNCT
ejpam-5875	400	6	mathematics	mathematic	NOUN
ejpam-5875	400	7	and	and	CCONJ
ejpam-5875	400	8	informatics	informatic	NOUN
ejpam-5875	400	9	,	,	PUNCT
ejpam-5875	400	10	31(2):485–492	31(2):485–492	PROPN
ejpam-5875	400	11	,	,	PUNCT
ejpam-5875	400	12	2016	2016	NUM
ejpam-5875	400	13	.	.	PUNCT
ejpam-5875	401	1	[	[	X
ejpam-5875	401	2	34	34	NUM
ejpam-5875	401	3	]	]	X
ejpam-5875	401	4	s.	s.	PROPN
ejpam-5875	401	5	yilmaz	yilmaz	PROPN
ejpam-5875	401	6	,	,	PUNCT
ejpam-5875	401	7	u.	u.	PROPN
ejpam-5875	401	8	z.	z.	PROPN
ejpam-5875	401	9	savci	savci	PROPN
ejpam-5875	401	10	,	,	PUNCT
ejpam-5875	401	11	and	and	CCONJ
ejpam-5875	401	12	a.	a.	NOUN
ejpam-5875	401	13	gden	gden	PROPN
ejpam-5875	401	14	.	.	PUNCT
ejpam-5875	402	1	position	position	NOUN
ejpam-5875	402	2	vector	vector	NOUN
ejpam-5875	402	3	of	of	ADP
ejpam-5875	402	4	some	some	DET
ejpam-5875	402	5	special	special	ADJ
ejpam-5875	402	6	curves	curve	NOUN
ejpam-5875	402	7	in	in	ADP
ejpam-5875	402	8	galilean	galilean	PROPN
ejpam-5875	402	9	3	3	NUM
ejpam-5875	402	10	-	-	PUNCT
ejpam-5875	402	11	space	space	NOUN
ejpam-5875	402	12	g3	g3	NOUN
ejpam-5875	402	13	.	.	PUNCT
ejpam-5875	403	1	global	global	ADJ
ejpam-5875	403	2	journal	journal	PROPN
ejpam-5875	403	3	of	of	ADP
ejpam-5875	403	4	advanced	advanced	ADJ
ejpam-5875	403	5	research	research	NOUN
ejpam-5875	403	6	on	on	ADP
ejpam-5875	403	7	classical	classical	ADJ
ejpam-5875	403	8	and	and	CCONJ
ejpam-5875	403	9	modern	modern	ADJ
ejpam-5875	403	10	geometries	geometry	NOUN
ejpam-5875	403	11	,	,	PUNCT
ejpam-5875	403	12	3:7–11	3:7–11	NUM
ejpam-5875	403	13	,	,	PUNCT
ejpam-5875	403	14	2014	2014	NUM
ejpam-5875	403	15	.	.	PUNCT
ejpam-5875	404	1	a.	a.	NOUN
ejpam-5875	404	2	elsharkawy	elsharkawy	PROPN
ejpam-5875	404	3	,	,	PUNCT
ejpam-5875	404	4	n.	n.	NOUN
ejpam-5875	404	5	elsharkawy	elsharkawy	PROPN
ejpam-5875	404	6	/	/	SYM
ejpam-5875	404	7	eur	eur	PROPN
ejpam-5875	404	8	.	.	PUNCT
ejpam-5875	405	1	j.	j.	PROPN
ejpam-5875	405	2	pure	pure	PROPN
ejpam-5875	405	3	appl	appl	PROPN
ejpam-5875	405	4	.	.	PROPN
ejpam-5875	405	5	math	math	PROPN
ejpam-5875	405	6	,	,	PUNCT
ejpam-5875	405	7	18	18	NUM
ejpam-5875	405	8	(	(	PUNCT
ejpam-5875	405	9	2	2	NUM
ejpam-5875	405	10	)	)	PUNCT
ejpam-5875	405	11	(	(	PUNCT
ejpam-5875	405	12	2025	2025	NUM
ejpam-5875	405	13	)	)	PUNCT
ejpam-5875	405	14	,	,	PUNCT
ejpam-5875	405	15	5875	5875	NUM
ejpam-5875	405	16	15	15	NUM
ejpam-5875	405	17	of	of	ADP
ejpam-5875	405	18	15	15	NUM
ejpam-5875	405	19	[	[	SYM
ejpam-5875	405	20	35	35	NUM
ejpam-5875	405	21	]	]	PUNCT
ejpam-5875	405	22	a.	a.	NOUN
ejpam-5875	405	23	elsharkawy	elsharkawy	PROPN
ejpam-5875	405	24	,	,	PUNCT
ejpam-5875	405	25	m.	m.	NOUN
ejpam-5875	405	26	turan	turan	PROPN
ejpam-5875	405	27	,	,	PUNCT
ejpam-5875	405	28	and	and	CCONJ
ejpam-5875	405	29	h.	h.	PROPN
ejpam-5875	405	30	bozok	bozok	PROPN
ejpam-5875	405	31	.	.	PUNCT
ejpam-5875	406	1	involute	involute	ADJ
ejpam-5875	406	2	-	-	PUNCT
ejpam-5875	406	3	evolute	evolute	NOUN
ejpam-5875	406	4	curves	curve	NOUN
ejpam-5875	406	5	with	with	ADP
ejpam-5875	406	6	modified	modify	VERB
ejpam-5875	406	7	orthogonal	orthogonal	ADJ
ejpam-5875	406	8	frame	frame	NOUN
ejpam-5875	406	9	in	in	ADP
ejpam-5875	406	10	galilean	galilean	PROPN
ejpam-5875	406	11	space	space	PROPN
ejpam-5875	406	12	g3	g3	PROPN
ejpam-5875	406	13	.	.	PUNCT
ejpam-5875	407	1	ukr	ukr	PROPN
ejpam-5875	407	2	.	.	PROPN
ejpam-5875	407	3	math	math	PROPN
ejpam-5875	407	4	.	.	PUNCT
ejpam-5875	408	1	j	j	PROPN
ejpam-5875	408	2	,	,	PUNCT
ejpam-5875	408	3	76(10):1444–1454	76(10):1444–1454	PROPN
ejpam-5875	408	4	,	,	PUNCT
ejpam-5875	408	5	2024	2024	NUM
ejpam-5875	408	6	.	.	PUNCT