id	sid	tid	token	lemma	pos
ma-245	1	1	2024	2024	NUM
ma-245	1	2	ada	ada	PROPN
ma-245	1	3	academica	academica	PROPN
ma-245	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-245	1	5	.	.	PUNCT
ma-245	2	1	j.	j.	PROPN
ma-245	2	2	math	math	PROPN
ma-245	2	3	.	.	PUNCT
ma-245	3	1	anal	anal	ADJ
ma-245	3	2	.	.	PUNCT
ma-245	4	1	4	4	NUM
ma-245	4	2	(	(	PUNCT
ma-245	4	3	2024	2024	NUM
ma-245	4	4	)	)	PUNCT
ma-245	4	5	18doi	18doi	NOUN
ma-245	4	6	:	:	PUNCT
ma-245	4	7	10.28924	10.28924	NUM
ma-245	4	8	/	/	SYM
ma-245	4	9	ada	ada	PROPN
ma-245	4	10	/	/	SYM
ma-245	4	11	ma.4.18	ma.4.18	VERB
ma-245	4	12	the	the	DET
ma-245	4	13	jacobi	jacobi	PROPN
ma-245	4	14	mate	mate	NOUN
ma-245	4	15	of	of	ADP
ma-245	4	16	an	an	DET
ma-245	4	17	oval	oval	NOUN
ma-245	4	18	mircea	mircea	PROPN
ma-245	4	19	crasmareanu	crasmareanu	VERB
ma-245	4	20	faculty	faculty	NOUN
ma-245	4	21	of	of	ADP
ma-245	4	22	mathematics	mathematic	NOUN
ma-245	4	23	,	,	PUNCT
ma-245	4	24	university	university	NOUN
ma-245	4	25	"	"	PUNCT
ma-245	4	26	al	al	PROPN
ma-245	4	27	.	.	PROPN
ma-245	4	28	i.	i.	PROPN
ma-245	4	29	cuza	cuza	PROPN
ma-245	4	30	"	"	PUNCT
ma-245	4	31	,	,	PUNCT
ma-245	4	32	iasi	iasi	NOUN
ma-245	4	33	,	,	PUNCT
ma-245	4	34	700506	700506	NUM
ma-245	4	35	,	,	PUNCT
ma-245	4	36	romania	romania	PROPN
ma-245	4	37	mcrasm@uaic.ro	mcrasm@uaic.ro	PROPN
ma-245	4	38	abstract	abstract	NOUN
ma-245	4	39	.	.	PUNCT
ma-245	5	1	we	we	PRON
ma-245	5	2	introduce	introduce	VERB
ma-245	5	3	and	and	CCONJ
ma-245	5	4	study	study	VERB
ma-245	5	5	the	the	DET
ma-245	5	6	jacobi	jacobi	PROPN
ma-245	5	7	mate	mate	NOUN
ma-245	5	8	cj	cj	NOUN
ma-245	5	9	of	of	ADP
ma-245	5	10	an	an	DET
ma-245	5	11	euclidean	euclidean	ADJ
ma-245	5	12	oval	oval	NOUN
ma-245	5	13	c.	c.	NOUN
ma-245	5	14	we	we	PRON
ma-245	5	15	focus	focus	VERB
ma-245	5	16	here	here	ADV
ma-245	5	17	on	on	ADP
ma-245	5	18	thecurvature	thecurvature	NOUN
ma-245	5	19	of	of	ADP
ma-245	5	20	cj	cj	NOUN
ma-245	5	21	and	and	CCONJ
ma-245	5	22	on	on	ADP
ma-245	5	23	some	some	DET
ma-245	5	24	examples	example	NOUN
ma-245	5	25	.	.	PUNCT
ma-245	6	1	1	1	X
ma-245	6	2	.	.	X
ma-245	6	3	introduction	introduction	NOUN
ma-245	6	4	the	the	DET
ma-245	6	5	enormous	enormous	ADJ
ma-245	6	6	influence	influence	NOUN
ma-245	6	7	of	of	ADP
ma-245	6	8	convexity	convexity	NOUN
ma-245	6	9	in	in	ADP
ma-245	6	10	practically	practically	ADV
ma-245	6	11	every	every	DET
ma-245	6	12	area	area	NOUN
ma-245	6	13	of	of	ADP
ma-245	6	14	mathematics	mathematic	NOUN
ma-245	6	15	is	be	AUX
ma-245	6	16	widely	widely	ADV
ma-245	6	17	known.we	known.we	X
ma-245	6	18	highlight	highlight	VERB
ma-245	6	19	the	the	DET
ma-245	6	20	idea	idea	NOUN
ma-245	6	21	of	of	ADP
ma-245	6	22	convex	convex	NOUN
ma-245	6	23	curve	curve	NOUN
ma-245	6	24	by	by	ADP
ma-245	6	25	limiting	limit	VERB
ma-245	6	26	the	the	DET
ma-245	6	27	discussion	discussion	NOUN
ma-245	6	28	to	to	ADP
ma-245	6	29	geometry	geometry	NOUN
ma-245	6	30	,	,	PUNCT
ma-245	6	31	namely	namely	ADV
ma-245	6	32	euclideanplane	euclideanplane	ADJ
ma-245	6	33	geometry	geometry	NOUN
ma-245	6	34	.	.	PUNCT
ma-245	7	1	the	the	DET
ma-245	7	2	recent	recent	ADJ
ma-245	7	3	book	book	NOUN
ma-245	7	4	[	[	X
ma-245	7	5	2	2	NUM
ma-245	7	6	]	]	PUNCT
ma-245	7	7	dedicates	dedicate	VERB
ma-245	7	8	an	an	DET
ma-245	7	9	entire	entire	ADJ
ma-245	7	10	chapter	chapter	NOUN
ma-245	7	11	,	,	PUNCT
ma-245	7	12	specifically	specifically	ADV
ma-245	7	13	chapter	chapter	NOUN
ma-245	7	14	6	6	NUM
ma-245	7	15	,	,	PUNCT
ma-245	7	16	to	to	SCONJ
ma-245	7	17	thistopic.this	thistopic.this	PRON
ma-245	7	18	brief	brief	ADJ
ma-245	7	19	note	note	NOUN
ma-245	7	20	aims	aim	VERB
ma-245	7	21	to	to	PART
ma-245	7	22	relate	relate	VERB
ma-245	7	23	,	,	PUNCT
ma-245	7	24	via	via	ADP
ma-245	7	25	the	the	DET
ma-245	7	26	first	first	ADJ
ma-245	7	27	two	two	NUM
ma-245	7	28	jacobi	jacobi	PROPN
ma-245	7	29	elliptic	elliptic	ADJ
ma-245	7	30	functions	function	NOUN
ma-245	7	31	,	,	PUNCT
ma-245	7	32	a	a	DET
ma-245	7	33	second	second	ADJ
ma-245	7	34	curve	curve	NOUN
ma-245	7	35	,	,	PUNCT
ma-245	7	36	cj	cj	X
ma-245	7	37	,	,	PUNCT
ma-245	7	38	toa	toa	PROPN
ma-245	7	39	given	give	VERB
ma-245	7	40	specific	specific	ADJ
ma-245	7	41	convex	convex	NOUN
ma-245	7	42	curve	curve	NOUN
ma-245	7	43	c	c	PROPN
ma-245	7	44	,	,	PUNCT
ma-245	7	45	called	call	VERB
ma-245	7	46	oval	oval	NOUN
ma-245	7	47	.	.	PUNCT
ma-245	8	1	given	give	VERB
ma-245	8	2	that	that	SCONJ
ma-245	8	3	these	these	DET
ma-245	8	4	elliptic	elliptic	ADJ
ma-245	8	5	functions	function	NOUN
ma-245	8	6	are	be	AUX
ma-245	8	7	1	1	NUM
ma-245	8	8	-	-	PUNCT
ma-245	8	9	parametricextensions	parametricextension	NOUN
ma-245	8	10	of	of	ADP
ma-245	8	11	the	the	DET
ma-245	8	12	standard	standard	ADJ
ma-245	8	13	cosinus	cosinus	NOUN
ma-245	8	14	and	and	CCONJ
ma-245	8	15	sinus	sinus	NOUN
ma-245	8	16	functions	function	NOUN
ma-245	8	17	,	,	PUNCT
ma-245	8	18	which	which	PRON
ma-245	8	19	determine	determine	VERB
ma-245	8	20	c	c	PROPN
ma-245	8	21	,	,	PUNCT
ma-245	8	22	this	this	DET
ma-245	8	23	link	link	NOUN
ma-245	8	24	makes	make	VERB
ma-245	8	25	sense.the	sense.the	DET
ma-245	8	26	support	support	NOUN
ma-245	8	27	function	function	NOUN
ma-245	8	28	defining	define	VERB
ma-245	8	29	c	c	PROPN
ma-245	8	30	serves	serve	VERB
ma-245	8	31	as	as	ADP
ma-245	8	32	the	the	DET
ma-245	8	33	foundation	foundation	NOUN
ma-245	8	34	for	for	ADP
ma-245	8	35	the	the	DET
ma-245	8	36	full	full	ADJ
ma-245	8	37	analysis	analysis	NOUN
ma-245	8	38	of	of	ADP
ma-245	8	39	this	this	DET
ma-245	8	40	pair	pair	NOUN
ma-245	8	41	of	of	ADP
ma-245	8	42	curves.more	curves.more	NOUN
ma-245	8	43	specifically	specifically	ADV
ma-245	8	44	,	,	PUNCT
ma-245	8	45	we	we	PRON
ma-245	8	46	concentrate	concentrate	VERB
ma-245	8	47	on	on	ADP
ma-245	8	48	the	the	DET
ma-245	8	49	curvature	curvature	NOUN
ma-245	8	50	,	,	PUNCT
ma-245	8	51	which	which	PRON
ma-245	8	52	is	be	AUX
ma-245	8	53	the	the	DET
ma-245	8	54	only	only	ADJ
ma-245	8	55	differential	differential	ADJ
ma-245	8	56	invariant	invariant	ADJ
ma-245	8	57	for	for	ADP
ma-245	8	58	aplane	aplane	NOUN
ma-245	8	59	curve	curve	NOUN
ma-245	8	60	.	.	PUNCT
ma-245	9	1	as	as	ADP
ma-245	9	2	possible	possible	ADJ
ma-245	9	3	area	area	NOUN
ma-245	9	4	of	of	ADP
ma-245	9	5	applications	application	NOUN
ma-245	9	6	for	for	ADP
ma-245	9	7	our	our	PRON
ma-245	9	8	results	result	NOUN
ma-245	9	9	we	we	PRON
ma-245	9	10	mention	mention	VERB
ma-245	9	11	the	the	DET
ma-245	9	12	very	very	ADV
ma-245	9	13	recent	recent	ADJ
ma-245	9	14	(	(	PUNCT
ma-245	9	15	computerbased	computerbased	ADJ
ma-245	9	16	)	)	PUNCT
ma-245	9	17	shape	shape	NOUN
ma-245	9	18	analysis	analysis	NOUN
ma-245	9	19	or	or	CCONJ
ma-245	9	20	topology	topology	NOUN
ma-245	9	21	optimization.the	optimization.the	DET
ma-245	9	22	following	follow	VERB
ma-245	9	23	is	be	AUX
ma-245	9	24	a	a	DET
ma-245	9	25	list	list	NOUN
ma-245	9	26	of	of	ADP
ma-245	9	27	the	the	DET
ma-245	9	28	contents	content	NOUN
ma-245	9	29	.	.	PUNCT
ma-245	10	1	the	the	DET
ma-245	10	2	differential	differential	NOUN
ma-245	10	3	(	(	PUNCT
ma-245	10	4	and	and	CCONJ
ma-245	10	5	integral	integral	ADJ
ma-245	10	6	)	)	PUNCT
ma-245	10	7	geometry	geometry	NOUN
ma-245	10	8	of	of	ADP
ma-245	10	9	the	the	DET
ma-245	10	10	ovals	oval	NOUN
ma-245	10	11	isreviewed	isreviewe	VERB
ma-245	10	12	in	in	ADP
ma-245	10	13	the	the	DET
ma-245	10	14	second	second	ADJ
ma-245	10	15	section	section	NOUN
ma-245	10	16	.	.	PUNCT
ma-245	11	1	our	our	PRON
ma-245	11	2	new	new	ADJ
ma-245	11	3	idea	idea	NOUN
ma-245	11	4	of	of	ADP
ma-245	11	5	jacobi	jacobi	PROPN
ma-245	11	6	mate	mate	NOUN
ma-245	11	7	of	of	ADP
ma-245	11	8	the	the	DET
ma-245	11	9	given	give	VERB
ma-245	11	10	oval	oval	NOUN
ma-245	11	11	c	c	NOUN
ma-245	11	12	is	be	AUX
ma-245	11	13	presented	present	VERB
ma-245	11	14	inthe	inthe	DET
ma-245	11	15	next	next	ADJ
ma-245	11	16	section	section	NOUN
ma-245	11	17	.	.	PUNCT
ma-245	12	1	it	it	PRON
ma-245	12	2	is	be	AUX
ma-245	12	3	important	important	ADJ
ma-245	12	4	to	to	PART
ma-245	12	5	note	note	VERB
ma-245	12	6	that	that	SCONJ
ma-245	12	7	,	,	PUNCT
ma-245	12	8	apart	apart	ADV
ma-245	12	9	from	from	ADP
ma-245	12	10	the	the	DET
ma-245	12	11	pair	pair	NOUN
ma-245	12	12	(	(	PUNCT
ma-245	12	13	c	c	X
ma-245	12	14	,	,	PUNCT
ma-245	12	15	cj	cj	NOUN
ma-245	12	16	)	)	PUNCT
ma-245	12	17	,	,	PUNCT
ma-245	12	18	there	there	PRON
ma-245	12	19	exists	exist	VERB
ma-245	12	20	another	another	DET
ma-245	12	21	curve	curve	NOUN
ma-245	12	22	p	p	NOUN
ma-245	12	23	that	that	PRON
ma-245	12	24	is	be	AUX
ma-245	12	25	naturally	naturally	ADV
ma-245	12	26	connected	connect	VERB
ma-245	12	27	to	to	ADP
ma-245	12	28	the	the	DET
ma-245	12	29	support	support	NOUN
ma-245	12	30	function	function	VERB
ma-245	12	31	p	p	NOUN
ma-245	12	32	of	of	ADP
ma-245	12	33	c	c	PROPN
ma-245	12	34	and	and	CCONJ
ma-245	12	35	hence	hence	ADV
ma-245	12	36	we	we	PRON
ma-245	12	37	will	will	AUX
ma-245	12	38	call	call	VERB
ma-245	12	39	the	the	DET
ma-245	12	40	support	support	NOUN
ma-245	12	41	curve	curve	NOUN
ma-245	12	42	.	.	PUNCT
ma-245	13	1	in	in	ADP
ma-245	13	2	fact	fact	NOUN
ma-245	13	3	,	,	PUNCT
ma-245	13	4	we	we	PRON
ma-245	13	5	study	study	VERB
ma-245	13	6	three	three	NUM
ma-245	13	7	curves	curve	NOUN
ma-245	13	8	.	.	PUNCT
ma-245	14	1	after	after	ADP
ma-245	14	2	the	the	DET
ma-245	14	3	computation	computation	NOUN
ma-245	14	4	of	of	ADP
ma-245	14	5	p	p	NOUN
ma-245	14	6	and	and	CCONJ
ma-245	14	7	cj	cj	NOUN
ma-245	14	8	curvatures	curvature	NOUN
ma-245	14	9	,	,	PUNCT
ma-245	14	10	we	we	PRON
ma-245	14	11	focus	focus	VERB
ma-245	14	12	on	on	ADP
ma-245	14	13	afew	afew	NOUN
ma-245	14	14	cases	case	NOUN
ma-245	14	15	.	.	PUNCT
ma-245	15	1	we	we	PRON
ma-245	15	2	point	point	VERB
ma-245	15	3	out	out	ADP
ma-245	15	4	that	that	SCONJ
ma-245	15	5	certain	certain	ADJ
ma-245	15	6	complicated	complicated	ADJ
ma-245	15	7	calculations	calculation	NOUN
ma-245	15	8	require	require	VERB
ma-245	15	9	software	software	NOUN
ma-245	15	10	and	and	CCONJ
ma-245	15	11	we	we	PRON
ma-245	15	12	make	make	VERB
ma-245	15	13	useof	useof	ADJ
ma-245	15	14	wolframalpha	wolframalpha	NOUN
ma-245	15	15	.	.	PUNCT
ma-245	16	1	2	2	X
ma-245	16	2	.	.	X
ma-245	16	3	the	the	DET
ma-245	16	4	differential	differential	ADJ
ma-245	16	5	geometry	geometry	NOUN
ma-245	16	6	of	of	ADP
ma-245	16	7	euclidean	euclidean	ADJ
ma-245	16	8	ovals	oval	NOUN
ma-245	16	9	a	a	DET
ma-245	16	10	brief	brief	ADJ
ma-245	16	11	overview	overview	NOUN
ma-245	16	12	of	of	ADP
ma-245	16	13	the	the	DET
ma-245	16	14	differential	differential	ADJ
ma-245	16	15	geometry	geometry	NOUN
ma-245	16	16	of	of	ADP
ma-245	16	17	ovals	oval	NOUN
ma-245	16	18	is	be	AUX
ma-245	16	19	given	give	VERB
ma-245	16	20	in	in	ADP
ma-245	16	21	this	this	DET
ma-245	16	22	first	first	ADJ
ma-245	16	23	part	part	NOUN
ma-245	16	24	.	.	PUNCT
ma-245	17	1	hence	hence	ADV
ma-245	17	2	,	,	PUNCT
ma-245	17	3	ourframework	ourframework	NOUN
ma-245	17	4	is	be	AUX
ma-245	17	5	the	the	DET
ma-245	17	6	euclidean	euclidean	ADJ
ma-245	17	7	linear	linear	PROPN
ma-245	17	8	space	space	NOUN
ma-245	17	9	e2	e2	NOUN
ma-245	17	10	:	:	PUNCT
ma-245	17	11	=	=	SYM
ma-245	17	12	(	(	PUNCT
ma-245	17	13	r2	r2	PROPN
ma-245	17	14	,	,	PUNCT
ma-245	17	15	〈	〈	PROPN
ma-245	17	16	·	·	SYM
ma-245	17	17	,	,	PUNCT
ma-245	17	18	·	·	SYM
ma-245	17	19	〉	〉	NUM
ma-245	17	20	)	)	PUNCT
ma-245	17	21	with	with	ADP
ma-245	17	22	to	to	ADP
ma-245	17	23	the	the	DET
ma-245	17	24	canonical	canonical	ADJ
ma-245	17	25	inner	inner	ADJ
ma-245	17	26	product	product	NOUN
ma-245	17	27	:	:	PUNCT
ma-245	17	28	received	receive	VERB
ma-245	17	29	:	:	PUNCT
ma-245	17	30	28	28	NUM
ma-245	17	31	may	may	PROPN
ma-245	17	32	2024	2024	NUM
ma-245	17	33	.	.	PUNCT
ma-245	18	1	key	key	ADJ
ma-245	18	2	words	word	NOUN
ma-245	18	3	and	and	CCONJ
ma-245	18	4	phrases	phrase	NOUN
ma-245	18	5	.	.	PUNCT
ma-245	19	1	jacobi	jacobi	PROPN
ma-245	19	2	elliptic	elliptic	ADJ
ma-245	19	3	functions	function	NOUN
ma-245	19	4	;	;	PUNCT
ma-245	19	5	oval	oval	NOUN
ma-245	19	6	;	;	PUNCT
ma-245	19	7	support	support	NOUN
ma-245	19	8	function	function	NOUN
ma-245	19	9	;	;	PUNCT
ma-245	19	10	curvature.1	curvature.1	PROPN
ma-245	19	11	https://adac.ee	https://adac.ee	PROPN
ma-245	19	12	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	19	13	https://orcid.org/0000-0002-5230-2751	https://orcid.org/0000-0002-5230-2751	PROPN
ma-245	19	14	eur	eur	PROPN
ma-245	19	15	.	.	PUNCT
ma-245	20	1	j.	j.	PROPN
ma-245	20	2	math	math	PROPN
ma-245	20	3	.	.	PUNCT
ma-245	21	1	anal	anal	PROPN
ma-245	21	2	.	.	PUNCT
ma-245	22	1	10.28924	10.28924	NUM
ma-245	22	2	/	/	SYM
ma-245	22	3	ada	ada	PROPN
ma-245	22	4	/	/	SYM
ma-245	22	5	ma.4.18	ma.4.18	VERB
ma-245	22	6	2	2	NUM
ma-245	22	7	〈	〈	PROPN
ma-245	22	8	u	u	NOUN
ma-245	22	9	,	,	PUNCT
ma-245	22	10	v	v	NOUN
ma-245	22	11	〉	〉	NOUN
ma-245	22	12	=	=	SYM
ma-245	22	13	x1y1	x1y1	PUNCT
ma-245	23	1	+	+	CCONJ
ma-245	23	2	x2y2	x2y2	NUM
ma-245	23	3	,	,	PUNCT
ma-245	23	4	u	u	NOUN
ma-245	23	5	=	=	SYM
ma-245	23	6	(	(	PUNCT
ma-245	23	7	x1	x1	PROPN
ma-245	23	8	,	,	PUNCT
ma-245	23	9	x2	x2	ADJ
ma-245	23	10	)	)	PUNCT
ma-245	23	11	∈	∈	PROPN
ma-245	23	12	r2	r2	NOUN
ma-245	23	13	,	,	PUNCT
ma-245	23	14	v	v	NOUN
ma-245	23	15	=	=	SYM
ma-245	23	16	(	(	PUNCT
ma-245	23	17	y1	y1	INTJ
ma-245	23	18	,	,	PUNCT
ma-245	23	19	y2	y2	NOUN
ma-245	23	20	)	)	PUNCT
ma-245	23	21	∈	∈	PROPN
ma-245	23	22	r2	r2	NOUN
ma-245	23	23	,	,	PUNCT
ma-245	23	24	0	0	NUM
ma-245	23	25	≤	≤	NUM
ma-245	23	26	‖u‖2	‖u‖2	PROPN
ma-245	23	27	=	=	SYM
ma-245	24	1	〈	〈	PROPN
ma-245	24	2	u	u	NOUN
ma-245	24	3	,	,	PUNCT
ma-245	24	4	u	u	NOUN
ma-245	24	5	〉	〉	NOUN
ma-245	24	6	.	.	PUNCT
ma-245	25	1	(	(	PUNCT
ma-245	25	2	2.1	2.1	NUM
ma-245	25	3	)	)	PUNCT
ma-245	25	4	fix	fix	VERB
ma-245	25	5	an	an	DET
ma-245	25	6	open	open	ADJ
ma-245	25	7	interval	interval	NOUN
ma-245	25	8	i	i	NOUN
ma-245	25	9	⊆	⊆	NUM
ma-245	25	10	r	r	NOUN
ma-245	25	11	and	and	CCONJ
ma-245	25	12	consider	consider	VERB
ma-245	25	13	c	c	PROPN
ma-245	25	14	⊂	⊂	PROPN
ma-245	25	15	e2	e2	VERB
ma-245	25	16	a	a	DET
ma-245	25	17	regular	regular	ADJ
ma-245	25	18	parametrized	parametrized	ADJ
ma-245	25	19	curve	curve	NOUN
ma-245	25	20	of	of	ADP
ma-245	25	21	equation	equation	NOUN
ma-245	25	22	:	:	PUNCT
ma-245	26	1	c	c	NOUN
ma-245	26	2	:	:	PUNCT
ma-245	26	3	r(t	r(t	NOUN
ma-245	26	4	)	)	PUNCT
ma-245	26	5	=	=	SYM
ma-245	26	6	(	(	PUNCT
ma-245	26	7	x(t	x(t	PROPN
ma-245	26	8	)	)	PUNCT
ma-245	26	9	,	,	PUNCT
ma-245	26	10	y(t	y(t	NOUN
ma-245	26	11	)	)	PUNCT
ma-245	26	12	)	)	PUNCT
ma-245	26	13	,	,	PUNCT
ma-245	26	14	r	r	NOUN
ma-245	26	15	∈	∈	PROPN
ma-245	26	16	c∞	c∞	PROPN
ma-245	26	17	,	,	PUNCT
ma-245	26	18	‖r	‖r	PROPN
ma-245	26	19	′(t)‖	′(t)‖	PROPN
ma-245	26	20	>	>	X
ma-245	26	21	0	0	PROPN
ma-245	26	22	,	,	PUNCT
ma-245	26	23	t	t	PROPN
ma-245	26	24	∈	∈	PROPN
ma-245	26	25	i.	i.	NOUN
ma-245	26	26	(	(	PUNCT
ma-245	26	27	2.2	2.2	NUM
ma-245	26	28	)	)	PUNCT
ma-245	26	29	suppose	suppose	VERB
ma-245	26	30	that	that	SCONJ
ma-245	26	31	c	c	PROPN
ma-245	26	32	is	be	AUX
ma-245	26	33	closed	closed	ADJ
ma-245	26	34	,	,	PUNCT
ma-245	26	35	simple	simple	ADJ
ma-245	26	36	and	and	CCONJ
ma-245	26	37	strictly	strictly	ADV
ma-245	26	38	convex	convex	ADJ
ma-245	26	39	;	;	PUNCT
ma-245	26	40	then	then	ADV
ma-245	26	41	will	will	AUX
ma-245	26	42	be	be	AUX
ma-245	26	43	called	call	VERB
ma-245	26	44	oval	oval	NOUN
ma-245	26	45	.	.	PUNCT
ma-245	27	1	all	all	DET
ma-245	27	2	its	its	PRON
ma-245	27	3	geometryis	geometryis	NOUN
ma-245	27	4	provided	provide	VERB
ma-245	27	5	by	by	ADP
ma-245	27	6	a	a	DET
ma-245	27	7	smooth	smooth	ADJ
ma-245	27	8	support	support	NOUN
ma-245	27	9	function	function	NOUN
ma-245	27	10	p	p	NOUN
ma-245	27	11	:	:	PUNCT
ma-245	27	12	i	i	PRON
ma-245	27	13	=	=	PUNCT
ma-245	28	1	[	[	X
ma-245	28	2	0	0	NUM
ma-245	28	3	,	,	PUNCT
ma-245	28	4	l	l	NOUN
ma-245	28	5	>	>	X
ma-245	28	6	0]→	0]→	NOUN
ma-245	28	7	(	(	PUNCT
ma-245	28	8	0,+∞	0,+∞	NUM
ma-245	28	9	)	)	PUNCT
ma-245	28	10	with	with	ADP
ma-245	28	11	:	:	PUNCT
ma-245	28	12	p(0	p(0	ADJ
ma-245	28	13	)	)	PUNCT
ma-245	28	14	=	=	SYM
ma-245	28	15	p(l	p(l	PROPN
ma-245	28	16	)	)	PUNCT
ma-245	28	17	,	,	PUNCT
ma-245	28	18	p(t	p(t	NOUN
ma-245	28	19	)	)	PUNCT
ma-245	28	20	+	+	NUM
ma-245	28	21	p′′(t	p′′(t	NOUN
ma-245	28	22	)	)	PUNCT
ma-245	28	23	>	>	X
ma-245	28	24	0	0	NUM
ma-245	28	25	,	,	PUNCT
ma-245	28	26	t	t	PROPN
ma-245	28	27	∈	∈	PROPN
ma-245	29	1	i	i	PRON
ma-245	29	2	(	(	PUNCT
ma-245	29	3	2.3	2.3	NUM
ma-245	29	4	)	)	PUNCT
ma-245	29	5	through	through	ADP
ma-245	29	6	the	the	DET
ma-245	29	7	relations	relation	NOUN
ma-245	29	8	:(	:(	PUNCT
ma-245	29	9	x(t	x(t	PROPN
ma-245	29	10	)	)	PUNCT
ma-245	29	11	y(t	y(t	NUM
ma-245	29	12	)	)	PUNCT
ma-245	29	13	)	)	PUNCT
ma-245	29	14	:	:	PUNCT
ma-245	29	15	=	=	SYM
ma-245	29	16	r(t	r(t	NOUN
ma-245	29	17	)	)	PUNCT
ma-245	29	18	·	·	PUNCT
ma-245	29	19	(	(	PUNCT
ma-245	29	20	p(t	p(t	NOUN
ma-245	29	21	)	)	PUNCT
ma-245	29	22	p′(t	p′(t	NOUN
ma-245	29	23	)	)	PUNCT
ma-245	29	24	)	)	PUNCT
ma-245	29	25	,	,	PUNCT
ma-245	29	26	r(t	r(t	NOUN
ma-245	29	27	)	)	PUNCT
ma-245	29	28	:	:	PUNCT
ma-245	29	29	=	=	SYM
ma-245	29	30	(	(	PUNCT
ma-245	29	31	cos	cos	ADP
ma-245	29	32	t	t	PROPN
ma-245	29	33	−	−	PROPN
ma-245	29	34	sin	sin	NOUN
ma-245	29	35	t	t	PROPN
ma-245	29	36	sin	sin	PROPN
ma-245	29	37	t	t	PROPN
ma-245	29	38	cos	cos	PROPN
ma-245	29	39	t	t	PROPN
ma-245	29	40	)	)	PUNCT
ma-245	29	41	∈	∈	PROPN
ma-245	29	42	so(2	so(2	NOUN
ma-245	29	43	)	)	PUNCT
ma-245	29	44	=	=	SYM
ma-245	29	45	s1	s1	NOUN
ma-245	29	46	,	,	PUNCT
ma-245	29	47	‖r(t)‖2	‖r(t)‖2	NOUN
ma-245	29	48	=	=	NOUN
ma-245	29	49	(	(	PUNCT
ma-245	29	50	p(t))2	p(t))2	NOUN
ma-245	29	51	+	+	CCONJ
ma-245	29	52	(	(	PUNCT
ma-245	29	53	p′(t))2	p′(t))2	NOUN
ma-245	29	54	.	.	PUNCT
ma-245	30	1	(	(	PUNCT
ma-245	30	2	2.4)we	2.4)we	NOUN
ma-245	30	3	point	point	VERB
ma-245	30	4	out	out	ADP
ma-245	30	5	that	that	SCONJ
ma-245	30	6	the	the	DET
ma-245	30	7	function	function	NOUN
ma-245	30	8	p	p	NOUN
ma-245	30	9	was	be	AUX
ma-245	30	10	firstly	firstly	ADV
ma-245	30	11	considered	consider	VERB
ma-245	30	12	by	by	ADP
ma-245	30	13	minkowski	minkowski	PROPN
ma-245	30	14	and	and	CCONJ
ma-245	30	15	the	the	DET
ma-245	30	16	function	function	NOUN
ma-245	30	17	t	t	PROPN
ma-245	30	18	→	→	SYM
ma-245	30	19	‖r(t)‖	‖r(t)‖	X
ma-245	30	20	>	>	X
ma-245	30	21	0	0	NUM
ma-245	30	22	is	be	AUX
ma-245	30	23	exactly	exactly	ADV
ma-245	30	24	the	the	DET
ma-245	30	25	first	first	ADJ
ma-245	30	26	legendre	legendre	PROPN
ma-245	30	27	transformation	transformation	NOUN
ma-245	30	28	of	of	ADP
ma-245	30	29	the	the	DET
ma-245	30	30	convex	convex	NOUN
ma-245	30	31	function	function	NOUN
ma-245	30	32	p.	p.	NOUN
ma-245	30	33	let	let	VERB
ma-245	30	34	f(c	f(c	PRON
ma-245	30	35	)	)	PUNCT
ma-245	31	1	=	=	PRON
ma-245	31	2	{	{	PUNCT
ma-245	31	3	t	t	PROPN
ma-245	31	4	,	,	PUNCT
ma-245	31	5	n	n	CCONJ
ma-245	31	6	}	}	PUNCT
ma-245	31	7	be	be	AUX
ma-245	31	8	thefrenet	thefrenet	NOUN
ma-245	31	9	frame	frame	NOUN
ma-245	31	10	of	of	ADP
ma-245	31	11	c	c	PROPN
ma-245	31	12	and	and	CCONJ
ma-245	31	13	k	k	NOUN
ma-245	31	14	:	:	PUNCT
ma-245	32	1	i	i	PRON
ma-245	32	2	=	=	PUNCT
ma-245	33	1	[	[	X
ma-245	33	2	0	0	NUM
ma-245	33	3	,	,	PUNCT
ma-245	33	4	l]→	l]→	NOUN
ma-245	33	5	r∗+	r∗+	PROPN
ma-245	33	6	=	=	SYM
ma-245	33	7	(	(	PUNCT
ma-245	33	8	0,+∞	0,+∞	NUM
ma-245	33	9	)	)	PUNCT
ma-245	33	10	its	its	PRON
ma-245	33	11	curvature	curvature	NOUN
ma-245	33	12	function	function	NOUN
ma-245	33	13	.	.	PUNCT
ma-245	34	1	then	then	ADV
ma-245	34	2	,	,	PUNCT
ma-245	34	3	it	it	PRON
ma-245	34	4	is	be	AUX
ma-245	34	5	well	well	ADV
ma-245	34	6	knownthat	knownthat	ADJ
ma-245	34	7	these	these	DET
ma-245	34	8	main	main	ADJ
ma-245	34	9	functions	function	NOUN
ma-245	34	10	are	be	AUX
ma-245	34	11	given	give	VERB
ma-245	34	12	by	by	ADP
ma-245	34	13	:	:	PUNCT
ma-245	34	14	p(t	p(t	NOUN
ma-245	34	15	)	)	PUNCT
ma-245	34	16	:	:	PUNCT
ma-245	35	1	=	=	SYM
ma-245	35	2	−〈r(t	−〈r(t	NUM
ma-245	35	3	)	)	PUNCT
ma-245	35	4	,	,	PUNCT
ma-245	35	5	n(t	n(t	PROPN
ma-245	35	6	)	)	PUNCT
ma-245	35	7	〉	〉	PROPN
ma-245	35	8	>	>	X
ma-245	35	9	0	0	NUM
ma-245	35	10	,	,	PUNCT
ma-245	35	11	k(t	k(t	PROPN
ma-245	35	12	)	)	PUNCT
ma-245	35	13	:	:	PUNCT
ma-245	35	14	=	=	NOUN
ma-245	35	15	1	1	NUM
ma-245	35	16	p(t	p(t	NOUN
ma-245	35	17	)	)	PUNCT
ma-245	35	18	+	+	SYM
ma-245	35	19	p′′(t	p′′(t	NOUN
ma-245	35	20	)	)	PUNCT
ma-245	35	21	=	=	SYM
ma-245	35	22	1	1	NUM
ma-245	35	23	‖r	‖r	PROPN
ma-245	35	24	′(t)‖	′(t)‖	PROPN
ma-245	35	25	>	>	X
ma-245	35	26	0	0	PUNCT
ma-245	35	27	(	(	PUNCT
ma-245	35	28	2.5	2.5	NUM
ma-245	35	29	)	)	PUNCT
ma-245	35	30	since	since	SCONJ
ma-245	35	31	:	:	PUNCT
ma-245	35	32	t	t	PROPN
ma-245	35	33	(	(	PUNCT
ma-245	35	34	t	t	PROPN
ma-245	35	35	)	)	PUNCT
ma-245	35	36	=	=	PUNCT
ma-245	35	37	(	(	PUNCT
ma-245	35	38	−	−	PROPN
ma-245	35	39	sin	sin	PROPN
ma-245	35	40	t	t	PROPN
ma-245	35	41	,	,	PUNCT
ma-245	35	42	cos	cos	PROPN
ma-245	35	43	t	t	PROPN
ma-245	35	44	)	)	PUNCT
ma-245	35	45	=	=	PUNCT
ma-245	36	1	ie	ie	X
ma-245	36	2	it	it	PRON
ma-245	36	3	,	,	PUNCT
ma-245	36	4	n(t	n(t	PROPN
ma-245	36	5	)	)	PUNCT
ma-245	36	6	=	=	VERB
ma-245	37	1	it	it	PRON
ma-245	37	2	(	(	PUNCT
ma-245	37	3	t	t	NOUN
ma-245	37	4	)	)	PUNCT
ma-245	37	5	=	=	PUNCT
ma-245	37	6	−e	−e	NOUN
ma-245	37	7	it	it	PRON
ma-245	37	8	=	=	PUNCT
ma-245	37	9	(	(	PUNCT
ma-245	37	10	−	−	PROPN
ma-245	37	11	cos	cos	ADP
ma-245	37	12	t,−	t,−	ADJ
ma-245	37	13	sin	sin	NOUN
ma-245	37	14	t	t	PROPN
ma-245	37	15	)	)	PUNCT
ma-245	37	16	(	(	PUNCT
ma-245	37	17	2.6)which	2.6)which	PROPN
ma-245	37	18	means	mean	VERB
ma-245	37	19	that	that	SCONJ
ma-245	37	20	the	the	DET
ma-245	37	21	frenet	frenet	ADJ
ma-245	37	22	frame	frame	NOUN
ma-245	37	23	is	be	AUX
ma-245	37	24	universal	universal	ADJ
ma-245	37	25	for	for	ADP
ma-245	37	26	the	the	DET
ma-245	37	27	set	set	NOUN
ma-245	37	28	of	of	ADP
ma-245	37	29	ovals	oval	NOUN
ma-245	37	30	defined	define	VERB
ma-245	37	31	on	on	ADP
ma-245	37	32	the	the	DET
ma-245	37	33	same	same	ADJ
ma-245	37	34	interval	interval	NOUN
ma-245	37	35	i	i	PRON
ma-245	37	36	.the	.the	PUNCT
ma-245	37	37	geometry	geometry	NOUN
ma-245	37	38	of	of	ADP
ma-245	37	39	the	the	DET
ma-245	37	40	ovals	oval	NOUN
ma-245	37	41	has	have	VERB
ma-245	37	42	two	two	NUM
ma-245	37	43	well	well	ADV
ma-245	37	44	-	-	PUNCT
ma-245	37	45	known	know	VERB
ma-245	37	46	integral	integral	ADJ
ma-245	37	47	relations	relation	NOUN
ma-245	37	48	:	:	PUNCT
ma-245	37	49	i	i	X
ma-245	37	50	)	)	PUNCT
ma-245	37	51	the	the	DET
ma-245	37	52	cauchy	cauchy	ADJ
ma-245	37	53	formula	formula	NOUN
ma-245	37	54	:	:	PUNCT
ma-245	38	1	l	l	NOUN
ma-245	38	2	=	=	SYM
ma-245	39	1	∫	∫	PROPN
ma-245	39	2	2π	2π	PROPN
ma-245	39	3	0	0	PUNCT
ma-245	39	4	p(t)dt	p(t)dt	PROPN
ma-245	39	5	.	.	PUNCT
ma-245	40	1	(	(	PUNCT
ma-245	40	2	2.7	2.7	NUM
ma-245	40	3	)	)	PUNCT
ma-245	40	4	ii	ii	PROPN
ma-245	40	5	)	)	PUNCT
ma-245	40	6	the	the	DET
ma-245	40	7	blaschke	blaschke	ADJ
ma-245	40	8	formula	formula	NOUN
ma-245	40	9	for	for	ADP
ma-245	40	10	the	the	DET
ma-245	40	11	area	area	NOUN
ma-245	40	12	a(c	a(c	NOUN
ma-245	40	13	)	)	PUNCT
ma-245	40	14	enclosed	enclose	VERB
ma-245	40	15	by	by	ADP
ma-245	40	16	c	c	NOUN
ma-245	40	17	:	:	PUNCT
ma-245	40	18	a(c	a(c	ADJ
ma-245	40	19	)	)	PUNCT
ma-245	41	1	=	=	SYM
ma-245	41	2	1	1	NUM
ma-245	41	3	2	2	NUM
ma-245	41	4	∫	∫	NOUN
ma-245	41	5	2π	2π	NOUN
ma-245	41	6	0	0	PUNCT
ma-245	42	1	[	[	X
ma-245	42	2	(	(	PUNCT
ma-245	42	3	p(t))2	p(t))2	PROPN
ma-245	42	4	−	−	PROPN
ma-245	42	5	(	(	PUNCT
ma-245	42	6	p′(t))2]dt	p′(t))2]dt	NOUN
ma-245	42	7	≤	≤	NUM
ma-245	42	8	1	1	NUM
ma-245	42	9	2	2	NUM
ma-245	42	10	∫	∫	PROPN
ma-245	42	11	2π	2π	PROPN
ma-245	42	12	0	0	NUM
ma-245	42	13	‖r(t)‖2dt	‖r(t)‖2dt	PRON
ma-245	42	14	,	,	PUNCT
ma-245	42	15	4πa(c	4πa(c	NUM
ma-245	42	16	)	)	PUNCT
ma-245	42	17	≤	≤	NOUN
ma-245	42	18	l2	l2	NOUN
ma-245	42	19	(	(	PUNCT
ma-245	42	20	2.8	2.8	NUM
ma-245	42	21	)	)	PUNCT
ma-245	42	22	with	with	ADP
ma-245	42	23	equality	equality	NOUN
ma-245	42	24	in	in	ADP
ma-245	42	25	the	the	DET
ma-245	42	26	isoperimetric	isoperimetric	ADJ
ma-245	42	27	inequality	inequality	NOUN
ma-245	42	28	(	(	PUNCT
ma-245	42	29	2.8	2.8	NUM
ma-245	42	30	)	)	PUNCT
ma-245	42	31	provided	provide	VERB
ma-245	42	32	by	by	ADP
ma-245	42	33	the	the	DET
ma-245	42	34	circle	circle	NOUN
ma-245	42	35	;	;	PUNCT
ma-245	42	36	we	we	PRON
ma-245	42	37	will	will	AUX
ma-245	42	38	treat	treat	VERB
ma-245	42	39	the	the	DET
ma-245	42	40	circleas	circleas	ADJ
ma-245	42	41	oval	oval	NOUN
ma-245	42	42	in	in	ADP
ma-245	42	43	the	the	DET
ma-245	42	44	example	example	NOUN
ma-245	42	45	3.4	3.4	NUM
ma-245	42	46	.	.	PUNCT
ma-245	43	1	remarks	remark	VERB
ma-245	43	2	2.1	2.1	NUM
ma-245	43	3	i	i	NOUN
ma-245	43	4	)	)	PUNCT
ma-245	43	5	the	the	DET
ma-245	43	6	decomposition	decomposition	NOUN
ma-245	43	7	of	of	ADP
ma-245	43	8	the	the	DET
ma-245	43	9	position	position	NOUN
ma-245	43	10	vector	vector	NOUN
ma-245	43	11	field	field	NOUN
ma-245	43	12	r	r	NOUN
ma-245	43	13	in	in	ADP
ma-245	43	14	the	the	DET
ma-245	43	15	frenet	frenet	ADJ
ma-245	43	16	basis	basis	NOUN
ma-245	43	17	is	be	AUX
ma-245	43	18	:	:	PUNCT
ma-245	43	19	r(t	r(t	NOUN
ma-245	43	20	)	)	PUNCT
ma-245	44	1	=	=	SYM
ma-245	44	2	p′(t)t	p′(t)t	ADP
ma-245	44	3	(	(	PUNCT
ma-245	44	4	t)−	t)−	PROPN
ma-245	44	5	p(t)n(t	p(t)n(t	NOUN
ma-245	44	6	)	)	PUNCT
ma-245	44	7	.	.	PUNCT
ma-245	45	1	(	(	PUNCT
ma-245	45	2	2.9	2.9	NUM
ma-245	45	3	)	)	PUNCT
ma-245	45	4	a	a	DET
ma-245	45	5	plane	plane	NOUN
ma-245	45	6	curve	curve	NOUN
ma-245	45	7	satisfying	satisfy	VERB
ma-245	45	8	k(t	k(t	NOUN
ma-245	45	9	)	)	PUNCT
ma-245	45	10	=	=	SYM
ma-245	45	11	1	1	NUM
ma-245	45	12	‖r	‖r	NOUN
ma-245	45	13	′(t)‖	′(t)‖	NOUN
ma-245	45	14	for	for	ADP
ma-245	45	15	all	all	DET
ma-245	45	16	t	t	PROPN
ma-245	45	17	is	be	AUX
ma-245	45	18	called	call	VERB
ma-245	45	19	flat	flat	ADJ
ma-245	45	20	-	-	PUNCT
ma-245	45	21	flow	flow	NOUN
ma-245	45	22	curve	curve	NOUN
ma-245	45	23	in	in	ADP
ma-245	45	24	[	[	X
ma-245	45	25	6	6	NUM
ma-245	45	26	]	]	PUNCT
ma-245	45	27	.	.	PUNCT
ma-245	46	1	hence	hence	ADV
ma-245	46	2	,	,	PUNCT
ma-245	46	3	any	any	DET
ma-245	46	4	oval	oval	NOUN
ma-245	46	5	issuch	issuch	NOUN
ma-245	46	6	a	a	DET
ma-245	46	7	curve	curve	NOUN
ma-245	46	8	,	,	PUNCT
ma-245	46	9	a	a	DET
ma-245	46	10	fact	fact	NOUN
ma-245	46	11	that	that	PRON
ma-245	46	12	explains	explain	VERB
ma-245	46	13	the	the	DET
ma-245	46	14	equality	equality	NOUN
ma-245	46	15	with	with	ADP
ma-245	46	16	2π	2π	NOUN
ma-245	46	17	of	of	ADP
ma-245	46	18	its	its	PRON
ma-245	46	19	total	total	ADJ
ma-245	46	20	curvature.ii	curvature.ii	NOUN
ma-245	46	21	)	)	PUNCT
ma-245	46	22	an	an	DET
ma-245	46	23	important	important	ADJ
ma-245	46	24	tool	tool	NOUN
ma-245	46	25	in	in	ADP
ma-245	46	26	one	one	NUM
ma-245	46	27	-	-	PUNCT
ma-245	46	28	dimensional	dimensional	ADJ
ma-245	46	29	dynamics	dynamic	NOUN
ma-245	46	30	is	be	AUX
ma-245	46	31	the	the	DET
ma-245	46	32	fermi	fermi	PROPN
ma-245	46	33	-	-	PUNCT
ma-245	46	34	walker	walker	PROPN
ma-245	46	35	derivative	derivative	NOUN
ma-245	46	36	.	.	PUNCT
ma-245	47	1	let	let	VERB
ma-245	47	2	x(c	x(c	PROPN
ma-245	47	3	)	)	PUNCT
ma-245	47	4	be	be	VERB
ma-245	47	5	the	the	DET
ma-245	47	6	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	47	7	eur	eur	PROPN
ma-245	47	8	.	.	PUNCT
ma-245	48	1	j.	j.	PROPN
ma-245	48	2	math	math	PROPN
ma-245	48	3	.	.	PUNCT
ma-245	49	1	anal	anal	PROPN
ma-245	49	2	.	.	PUNCT
ma-245	50	1	10.28924	10.28924	NUM
ma-245	50	2	/	/	SYM
ma-245	50	3	ada	ada	PROPN
ma-245	50	4	/	/	SYM
ma-245	50	5	ma.4.18	ma.4.18	PROPN
ma-245	50	6	3set	3set	NOUN
ma-245	50	7	of	of	ADP
ma-245	50	8	vector	vector	NOUN
ma-245	50	9	fields	field	NOUN
ma-245	50	10	along	along	ADP
ma-245	50	11	the	the	DET
ma-245	50	12	curve	curve	NOUN
ma-245	50	13	c.	c.	NOUN
ma-245	50	14	then	then	ADV
ma-245	50	15	the	the	DET
ma-245	50	16	fermi	fermi	PROPN
ma-245	50	17	-	-	PUNCT
ma-245	50	18	walker	walker	PROPN
ma-245	50	19	derivative	derivative	NOUN
ma-245	50	20	is	be	AUX
ma-245	50	21	the	the	DET
ma-245	50	22	map	map	NOUN
ma-245	50	23	(	(	PUNCT
ma-245	50	24	[	[	X
ma-245	50	25	6	6	NUM
ma-245	50	26	,	,	PUNCT
ma-245	50	27	p.	p.	NOUN
ma-245	50	28	420	420	NUM
ma-245	50	29	]	]	PUNCT
ma-245	50	30	)	)	PUNCT
ma-245	51	1	∇fw	∇fw	NOUN
ma-245	51	2	:	:	PUNCT
ma-245	51	3	x(c)→	x(c)→	PROPN
ma-245	51	4	x(c	x(c	PROPN
ma-245	51	5	):	):	PUNCT
ma-245	51	6	∇fw	∇fw	ADJ
ma-245	51	7	(	(	PUNCT
ma-245	51	8	x	x	NOUN
ma-245	51	9	)	)	PUNCT
ma-245	51	10	:	:	PUNCT
ma-245	52	1	=	=	SYM
ma-245	52	2	d	d	X
ma-245	52	3	dt	dt	X
ma-245	52	4	x	x	PUNCT
ma-245	52	5	+	+	CCONJ
ma-245	52	6	‖r	‖r	NOUN
ma-245	52	7	′(·)‖k	′(·)‖k	VERB
ma-245	52	8	[	[	X
ma-245	52	9	〈	〈	PROPN
ma-245	52	10	x	x	X
ma-245	52	11	,	,	PUNCT
ma-245	52	12	n〉t	n〉t	VERB
ma-245	52	13	−	−	PROPN
ma-245	52	14	〈	〈	PROPN
ma-245	52	15	x	x	X
ma-245	52	16	,	,	PUNCT
ma-245	52	17	t	t	PROPN
ma-245	52	18	〉	〉	PROPN
ma-245	52	19	n	n	CCONJ
ma-245	52	20	]	]	PUNCT
ma-245	52	21	.	.	PUNCT
ma-245	53	1	(	(	PUNCT
ma-245	53	2	2.10	2.10	NUM
ma-245	53	3	)	)	PUNCT
ma-245	53	4	the	the	DET
ma-245	53	5	frenet	frenet	ADJ
ma-245	53	6	frame	frame	NOUN
ma-245	53	7	is	be	AUX
ma-245	53	8	fermi	fermi	NOUN
ma-245	53	9	-	-	PUNCT
ma-245	53	10	walker	walker	NOUN
ma-245	53	11	conserved	conserve	VERB
ma-245	53	12	:	:	PUNCT
ma-245	54	1	∇fw	∇fw	INTJ
ma-245	54	2	(	(	PUNCT
ma-245	54	3	t	t	NOUN
ma-245	54	4	)	)	PUNCT
ma-245	55	1	=	=	PUNCT
ma-245	55	2	∇fw	∇fw	NOUN
ma-245	55	3	(	(	PUNCT
ma-245	55	4	n	n	CCONJ
ma-245	55	5	)	)	PUNCT
ma-245	55	6	=	=	SYM
ma-245	55	7	0	0	X
ma-245	55	8	.	.	PUNCT
ma-245	56	1	for	for	ADP
ma-245	56	2	our	our	PRON
ma-245	56	3	oval	oval	NOUN
ma-245	56	4	c	c	NOUN
ma-245	56	5	we	we	PRON
ma-245	56	6	derive	derive	VERB
ma-245	56	7	:	:	PUNCT
ma-245	56	8	∇fw	∇fw	NOUN
ma-245	56	9	(	(	PUNCT
ma-245	56	10	r)(t	r)(t	ADJ
ma-245	56	11	)	)	PUNCT
ma-245	56	12	=	=	SYM
ma-245	56	13	r	r	NOUN
ma-245	56	14	′(t)−	′(t)−	PROPN
ma-245	56	15	‖r	‖r	NOUN
ma-245	56	16	′(t)‖k(t)[p(t)t	′(t)‖k(t)[p(t)t	X
ma-245	56	17	(	(	PUNCT
ma-245	56	18	t	t	PROPN
ma-245	56	19	)	)	PUNCT
ma-245	56	20	+	+	NOUN
ma-245	56	21	p′(t)n(t	p′(t)n(t	NOUN
ma-245	56	22	)	)	PUNCT
ma-245	56	23	]	]	PUNCT
ma-245	57	1	=	=	PRON
ma-245	57	2	p′′(t)t	p′′(t)t	NOUN
ma-245	57	3	(	(	PUNCT
ma-245	57	4	t)−	t)−	PROPN
ma-245	57	5	p′(t)n(t	p′(t)n(t	PROPN
ma-245	57	6	)	)	PUNCT
ma-245	57	7	.	.	PUNCT
ma-245	58	1	(	(	PUNCT
ma-245	58	2	2.11	2.11	NUM
ma-245	58	3	)	)	PUNCT
ma-245	58	4	hence	hence	ADV
ma-245	58	5	if	if	SCONJ
ma-245	58	6	we	we	PRON
ma-245	58	7	denote	denote	VERB
ma-245	58	8	r	r	NOUN
ma-245	58	9	=	=	SYM
ma-245	58	10	rotation(p	rotation(p	PROPN
ma-245	58	11	)	)	PUNCT
ma-245	58	12	then	then	ADV
ma-245	58	13	the	the	DET
ma-245	58	14	curve	curve	NOUN
ma-245	58	15	t	t	PROPN
ma-245	58	16	→	→	PUNCT
ma-245	58	17	∇fw	∇fw	ADJ
ma-245	58	18	(	(	PUNCT
ma-245	58	19	r)(t	r)(t	X
ma-245	58	20	)	)	PUNCT
ma-245	58	21	is	be	AUX
ma-245	58	22	exactly	exactly	ADV
ma-245	58	23	the	the	DET
ma-245	58	24	curve	curve	NOUN
ma-245	58	25	rotation(p′).iii	rotation(p′).iii	NOUN
ma-245	58	26	)	)	PUNCT
ma-245	58	27	associated	associate	VERB
ma-245	58	28	to	to	ADP
ma-245	58	29	the	the	DET
ma-245	58	30	support	support	NOUN
ma-245	58	31	function	function	NOUN
ma-245	58	32	p	p	NOUN
ma-245	58	33	there	there	PRON
ma-245	58	34	exists	exist	VERB
ma-245	58	35	the	the	DET
ma-245	58	36	width	width	ADJ
ma-245	58	37	function	function	NOUN
ma-245	58	38	w	w	NOUN
ma-245	58	39	:	:	PUNCT
ma-245	59	1	[	[	X
ma-245	59	2	0	0	NUM
ma-245	59	3	,	,	PUNCT
ma-245	59	4	l/2	l/2	NUM
ma-245	59	5	]	]	PUNCT
ma-245	59	6	→	→	X
ma-245	59	7	(	(	PUNCT
ma-245	59	8	0,+∞	0,+∞	NUM
ma-245	59	9	)	)	PUNCT
ma-245	59	10	,	,	PUNCT
ma-245	59	11	w	w	PROPN
ma-245	59	12	(	(	PUNCT
ma-245	59	13	t	t	PROPN
ma-245	59	14	)	)	PUNCT
ma-245	59	15	:	:	PUNCT
ma-245	59	16	=	=	PUNCT
ma-245	59	17	p(t	p(t	NOUN
ma-245	59	18	)	)	PUNCT
ma-245	60	1	+	+	CCONJ
ma-245	60	2	p	p	X
ma-245	60	3	(	(	PUNCT
ma-245	60	4	t	t	PROPN
ma-245	60	5	+	+	CCONJ
ma-245	60	6	l	l	NOUN
ma-245	60	7	2	2	NUM
ma-245	60	8	)	)	PUNCT
ma-245	60	9	.	.	PUNCT
ma-245	61	1	hence	hence	ADV
ma-245	61	2	,	,	PUNCT
ma-245	61	3	its	its	PRON
ma-245	61	4	period	period	NOUN
ma-245	61	5	is	be	AUX
ma-245	61	6	l	l	NOUN
ma-245	61	7	2	2	NUM
ma-245	61	8	.iv	.iv	PUNCT
ma-245	61	9	)	)	PUNCT
ma-245	61	10	concerning	concern	VERB
ma-245	61	11	the	the	DET
ma-245	61	12	possible	possible	ADJ
ma-245	61	13	relationship	relationship	NOUN
ma-245	61	14	between	between	ADP
ma-245	61	15	the	the	DET
ma-245	61	16	periodicity	periodicity	NOUN
ma-245	61	17	and	and	CCONJ
ma-245	61	18	the	the	DET
ma-245	61	19	curvature	curvature	NOUN
ma-245	61	20	of	of	ADP
ma-245	61	21	a	a	DET
ma-245	61	22	plane	plane	NOUN
ma-245	61	23	curvea	curvea	NOUN
ma-245	61	24	very	very	ADV
ma-245	61	25	interesting	interesting	ADJ
ma-245	61	26	problem	problem	NOUN
ma-245	61	27	is	be	AUX
ma-245	61	28	solved	solve	VERB
ma-245	61	29	in	in	ADP
ma-245	61	30	the	the	DET
ma-245	61	31	paper	paper	NOUN
ma-245	62	1	[	[	X
ma-245	62	2	1	1	NUM
ma-245	62	3	]	]	X
ma-245	62	4	:	:	PUNCT
ma-245	62	5	when	when	SCONJ
ma-245	62	6	is	be	AUX
ma-245	62	7	a	a	DET
ma-245	62	8	periodic	periodic	ADJ
ma-245	62	9	function	function	NOUN
ma-245	62	10	the	the	DET
ma-245	62	11	curvature	curvature	NOUN
ma-245	62	12	ofa	ofa	PROPN
ma-245	62	13	closed	close	VERB
ma-245	62	14	plane	plane	NOUN
ma-245	62	15	curve	curve	NOUN
ma-245	62	16	?	?	PUNCT
ma-245	63	1	2	2	NUM
ma-245	63	2	3	3	NUM
ma-245	63	3	.	.	PUNCT
ma-245	64	1	the	the	DET
ma-245	64	2	jacobi	jacobi	PROPN
ma-245	64	3	mate	mate	NOUN
ma-245	64	4	of	of	ADP
ma-245	64	5	an	an	DET
ma-245	64	6	oval	oval	NOUN
ma-245	64	7	fix	fix	VERB
ma-245	64	8	the	the	DET
ma-245	64	9	real	real	ADJ
ma-245	64	10	number	number	NOUN
ma-245	64	11	ρ	ρ	PROPN
ma-245	64	12	∈	∈	PROPN
ma-245	64	13	(	(	PUNCT
ma-245	64	14	−1	−1	NOUN
ma-245	64	15	,	,	PUNCT
ma-245	64	16	1	1	NUM
ma-245	64	17	)	)	PUNCT
ma-245	64	18	as	as	ADP
ma-245	64	19	the	the	DET
ma-245	64	20	modulus	modulus	NOUN
ma-245	64	21	for	for	ADP
ma-245	64	22	the	the	DET
ma-245	64	23	differential	differential	ADJ
ma-245	64	24	system	system	NOUN
ma-245	64	25	(	(	PUNCT
ma-245	64	26	[	[	X
ma-245	64	27	7	7	NUM
ma-245	64	28	,	,	PUNCT
ma-245	64	29	p.	p.	NOUN
ma-245	64	30	130]):	130]):	NUM
ma-245	65	1	du	du	PROPN
ma-245	65	2	dt	dt	PROPN
ma-245	65	3	=	=	SYM
ma-245	65	4	−wv	−wv	PROPN
ma-245	65	5	,	,	PUNCT
ma-245	65	6	u(0	u(0	NOUN
ma-245	65	7	)	)	PUNCT
ma-245	65	8	=	=	SYM
ma-245	65	9	1	1	NUM
ma-245	65	10	,	,	PUNCT
ma-245	65	11	dv	dv	PROPN
ma-245	65	12	dt	dt	PROPN
ma-245	65	13	=	=	SYM
ma-245	65	14	wu	wu	PROPN
ma-245	65	15	,	,	PUNCT
ma-245	65	16	v(0	v(0	PROPN
ma-245	65	17	)	)	PUNCT
ma-245	65	18	=	=	SYM
ma-245	65	19	0	0	NUM
ma-245	65	20	,	,	PUNCT
ma-245	65	21	dw	dw	NOUN
ma-245	65	22	dt	dt	NOUN
ma-245	66	1	=	=	PROPN
ma-245	66	2	−ρ	−ρ	NOUN
ma-245	66	3	2uv	2uv	ADJ
ma-245	66	4	,	,	PUNCT
ma-245	66	5	w(0	w(0	PROPN
ma-245	66	6	)	)	PUNCT
ma-245	66	7	=	=	SYM
ma-245	67	1	1	1	X
ma-245	67	2	.	.	PUNCT
ma-245	67	3	(	(	PUNCT
ma-245	67	4	3.1	3.1	NUM
ma-245	67	5	)	)	PUNCT
ma-245	67	6	recall	recall	NOUN
ma-245	67	7	that	that	SCONJ
ma-245	67	8	its	its	PRON
ma-245	67	9	solutions	solution	NOUN
ma-245	67	10	are	be	AUX
ma-245	67	11	called	call	VERB
ma-245	67	12	jacobi	jacobi	PROPN
ma-245	67	13	elliptic	elliptic	ADJ
ma-245	67	14	functions	function	NOUN
ma-245	67	15	and	and	CCONJ
ma-245	67	16	there	there	PRON
ma-245	67	17	are	be	VERB
ma-245	67	18	usually	usually	ADV
ma-245	67	19	denoted	denote	VERB
ma-245	67	20	cn	cn	PROPN
ma-245	67	21	(	(	PUNCT
ma-245	67	22	·	·	PUNCT
ma-245	67	23	,	,	PUNCT
ma-245	67	24	ρ	ρ	PROPN
ma-245	67	25	)	)	PUNCT
ma-245	67	26	,	,	PUNCT
ma-245	67	27	sn	sn	PROPN
ma-245	67	28	(	(	PUNCT
ma-245	67	29	·	·	PUNCT
ma-245	67	30	,	,	PUNCT
ma-245	67	31	ρ	ρ	NOUN
ma-245	67	32	)	)	PUNCT
ma-245	67	33	respectively	respectively	ADV
ma-245	67	34	dn	dn	PROPN
ma-245	67	35	(	(	PUNCT
ma-245	67	36	·	·	PUNCT
ma-245	67	37	,	,	PUNCT
ma-245	67	38	ρ	ρ	PROPN
ma-245	67	39	)	)	PUNCT
ma-245	67	40	;	;	PUNCT
ma-245	67	41	we	we	PRON
ma-245	67	42	prefer	prefer	VERB
ma-245	67	43	the	the	DET
ma-245	67	44	simple	simple	ADJ
ma-245	67	45	notation	notation	NOUN
ma-245	67	46	used	use	VERB
ma-245	67	47	above	above	ADV
ma-245	67	48	.	.	PUNCT
ma-245	68	1	as	as	SCONJ
ma-245	68	2	solutions	solution	NOUN
ma-245	68	3	of	of	ADP
ma-245	68	4	the	the	DET
ma-245	68	5	odesystem	odesystem	NOUN
ma-245	68	6	(	(	PUNCT
ma-245	68	7	3.1	3.1	NUM
ma-245	68	8	)	)	PUNCT
ma-245	68	9	these	these	DET
ma-245	68	10	functions	function	NOUN
ma-245	68	11	satisfy	satisfy	VERB
ma-245	68	12	two	two	NUM
ma-245	68	13	remarkable	remarkable	ADJ
ma-245	68	14	identities	identity	NOUN
ma-245	68	15	:	:	PUNCT
ma-245	68	16	u2	u2	NOUN
ma-245	68	17	+	+	CCONJ
ma-245	68	18	v2	v2	PROPN
ma-245	68	19	=	=	SYM
ma-245	68	20	1	1	NUM
ma-245	68	21	,	,	PUNCT
ma-245	68	22	ρ2v2	ρ2v2	X
ma-245	68	23	+	+	NUM
ma-245	68	24	w2	w2	NOUN
ma-245	68	25	=	=	SYM
ma-245	68	26	1	1	X
ma-245	68	27	.	.	PUNCT
ma-245	69	1	(	(	PUNCT
ma-245	69	2	3.2	3.2	NUM
ma-245	69	3	)	)	PUNCT
ma-245	69	4	also	also	ADV
ma-245	69	5	,	,	PUNCT
ma-245	69	6	both	both	PRON
ma-245	69	7	functions	function	VERB
ma-245	69	8	u	u	NOUN
ma-245	69	9	(	(	PUNCT
ma-245	69	10	·	·	PUNCT
ma-245	69	11	)	)	PUNCT
ma-245	69	12	and	and	CCONJ
ma-245	69	13	v	v	NOUN
ma-245	69	14	(	(	PUNCT
ma-245	69	15	·	·	PUNCT
ma-245	69	16	)	)	PUNCT
ma-245	69	17	are	be	AUX
ma-245	69	18	periodic	periodic	ADJ
ma-245	69	19	with	with	ADP
ma-245	69	20	l	l	NOUN
ma-245	69	21	=	=	SYM
ma-245	69	22	4l̃	4l̃	PROPN
ma-245	69	23	for	for	ADP
ma-245	69	24	(	(	PUNCT
ma-245	69	25	[	[	X
ma-245	69	26	7	7	NUM
ma-245	69	27	,	,	PUNCT
ma-245	69	28	p.	p.	NOUN
ma-245	69	29	131	131	NUM
ma-245	69	30	]	]	PUNCT
ma-245	69	31	):	):	PUNCT
ma-245	70	1	l̃	l̃	PROPN
ma-245	70	2	=	=	SYM
ma-245	70	3	l̃(ρ	l̃(ρ	NOUN
ma-245	70	4	)	)	PUNCT
ma-245	70	5	:	:	PUNCT
ma-245	71	1	=	=	SYM
ma-245	71	2	∫	∫	PROPN
ma-245	71	3	1	1	NUM
ma-245	71	4	0	0	NUM
ma-245	71	5	ds√	ds√	PUNCT
ma-245	71	6	(	(	PUNCT
ma-245	71	7	1−	1−	NUM
ma-245	71	8	s2)(1−	s2)(1−	PROPN
ma-245	71	9	ρ2s2	ρ2s2	NOUN
ma-245	71	10	)	)	PUNCT
ma-245	71	11	(	(	PUNCT
ma-245	71	12	3.3	3.3	NUM
ma-245	71	13	)	)	PUNCT
ma-245	71	14	while	while	SCONJ
ma-245	71	15	w	w	NOUN
ma-245	71	16	is	be	AUX
ma-245	71	17	periodic	periodic	ADJ
ma-245	71	18	of	of	ADP
ma-245	71	19	period	period	NOUN
ma-245	71	20	2l̃.	2l̃.	NUM
ma-245	71	21	in	in	ADP
ma-245	71	22	particular	particular	ADJ
ma-245	71	23	,	,	PUNCT
ma-245	71	24	l̃(0	l̃(0	X
ma-245	71	25	)	)	PUNCT
ma-245	71	26	=	=	SYM
ma-245	71	27	arcsin	arcsin	PROPN
ma-245	71	28	s|10	s|10	PROPN
ma-245	71	29	=	=	PROPN
ma-245	72	1	π	π	PROPN
ma-245	72	2	2	2	NUM
ma-245	72	3	for	for	ADP
ma-245	72	4	the	the	DET
ma-245	72	5	usual	usual	ADJ
ma-245	72	6	trigonometricalfunctions	trigonometricalfunction	NOUN
ma-245	72	7	cn	cn	PROPN
ma-245	72	8	(	(	PUNCT
ma-245	72	9	·	·	PUNCT
ma-245	72	10	,	,	PUNCT
ma-245	72	11	0	0	NUM
ma-245	72	12	)	)	PUNCT
ma-245	72	13	=	=	SYM
ma-245	72	14	cos	cos	PROPN
ma-245	72	15	(	(	PUNCT
ma-245	72	16	·	·	PUNCT
ma-245	72	17	)	)	PUNCT
ma-245	72	18	and	and	CCONJ
ma-245	72	19	sn	sn	PROPN
ma-245	72	20	(	(	PUNCT
ma-245	72	21	·	·	PUNCT
ma-245	72	22	,	,	PUNCT
ma-245	72	23	0	0	NUM
ma-245	72	24	)	)	PUNCT
ma-245	72	25	=	=	VERB
ma-245	72	26	sin	sin	NOUN
ma-245	72	27	(	(	PUNCT
ma-245	72	28	·	·	PUNCT
ma-245	72	29	)	)	PUNCT
ma-245	72	30	.	.	PUNCT
ma-245	73	1	the	the	DET
ma-245	73	2	complementary	complementary	ADJ
ma-245	73	3	modulus	modulus	NOUN
ma-245	73	4	is	be	AUX
ma-245	73	5	ρ′	ρ′	PUNCT
ma-245	73	6	:	:	PUNCT
ma-245	73	7	=	=	PUNCT
ma-245	74	1	√1−	√1−	X
ma-245	74	2	ρ2	ρ2	NOUN
ma-245	74	3	∈	∈	PROPN
ma-245	74	4	(	(	PUNCT
ma-245	74	5	0	0	NUM
ma-245	74	6	,	,	PUNCT
ma-245	74	7	1	1	NUM
ma-245	74	8	]	]	PUNCT
ma-245	74	9	and	and	CCONJ
ma-245	74	10	the	the	DET
ma-245	74	11	third	third	ADJ
ma-245	74	12	jacobi	jacobi	PROPN
ma-245	74	13	function	function	NOUN
ma-245	74	14	is	be	AUX
ma-245	74	15	bounded	bound	VERB
ma-245	74	16	by	by	ADP
ma-245	74	17	:	:	PUNCT
ma-245	74	18	0	0	NUM
ma-245	74	19	<	<	X
ma-245	74	20	ρ′	ρ′	PRON
ma-245	74	21	≤	≤	X
ma-245	74	22	w(t	w(t	PROPN
ma-245	74	23	)	)	PUNCT
ma-245	74	24	≤	≤	NOUN
ma-245	74	25	1	1	NUM
ma-245	74	26	.	.	PUNCT
ma-245	75	1	(	(	PUNCT
ma-245	75	2	3.4	3.4	NUM
ma-245	75	3	)	)	PUNCT
ma-245	75	4	the	the	DET
ma-245	75	5	self	self	NOUN
ma-245	75	6	-	-	PUNCT
ma-245	75	7	complementary	complementary	ADJ
ma-245	75	8	case	case	NOUN
ma-245	75	9	ρ′	ρ′	PUNCT
ma-245	75	10	=	=	SYM
ma-245	75	11	ρ	ρ	PROPN
ma-245	75	12	is	be	AUX
ma-245	75	13	provided	provide	VERB
ma-245	75	14	by	by	ADP
ma-245	75	15	ρ	ρ	PROPN
ma-245	75	16	=	=	SYM
ma-245	75	17	1√	1√	PROPN
ma-245	75	18	2	2	NUM
ma-245	75	19	and	and	CCONJ
ma-245	75	20	being	be	AUX
ma-245	75	21	in	in	ADP
ma-245	75	22	the	the	DET
ma-245	75	23	interval	interval	NOUN
ma-245	75	24	(	(	PUNCT
ma-245	75	25	0	0	NUM
ma-245	75	26	,	,	PUNCT
ma-245	75	27	1	1	NUM
ma-245	75	28	)	)	PUNCT
ma-245	75	29	is	be	AUX
ma-245	75	30	theeccentricity	theeccentricity	NOUN
ma-245	75	31	of	of	ADP
ma-245	75	32	an	an	DET
ma-245	75	33	ellipse	ellipse	NOUN
ma-245	75	34	,	,	PUNCT
ma-245	75	35	called	call	VERB
ma-245	75	36	self	self	NOUN
ma-245	75	37	-	-	PUNCT
ma-245	75	38	complementary	complementary	ADJ
ma-245	75	39	and	and	CCONJ
ma-245	75	40	studied	study	VERB
ma-245	75	41	in	in	ADP
ma-245	75	42	[	[	X
ma-245	75	43	5	5	NUM
ma-245	75	44	]	]	PUNCT
ma-245	75	45	.	.	PUNCT
ma-245	76	1	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	76	2	eur	eur	PROPN
ma-245	76	3	.	.	PUNCT
ma-245	77	1	j.	j.	PROPN
ma-245	77	2	math	math	PROPN
ma-245	77	3	.	.	PUNCT
ma-245	78	1	anal	anal	PROPN
ma-245	78	2	.	.	PUNCT
ma-245	79	1	10.28924	10.28924	NUM
ma-245	79	2	/	/	SYM
ma-245	79	3	ada	ada	PROPN
ma-245	79	4	/	/	SYM
ma-245	79	5	ma.4.18	ma.4.18	PROPN
ma-245	79	6	4due	4due	PROPN
ma-245	79	7	to	to	ADP
ma-245	79	8	the	the	DET
ma-245	79	9	increasing	increase	VERB
ma-245	79	10	interest	interest	NOUN
ma-245	79	11	in	in	ADP
ma-245	79	12	the	the	DET
ma-245	79	13	geometry	geometry	NOUN
ma-245	79	14	of	of	ADP
ma-245	79	15	ovals	oval	NOUN
ma-245	79	16	this	this	DET
ma-245	79	17	short	short	ADJ
ma-245	79	18	note	note	NOUN
ma-245	79	19	defines	define	VERB
ma-245	79	20	the	the	DET
ma-245	79	21	jacobi	jacobi	NOUN
ma-245	79	22	matefor	matefor	ADP
ma-245	79	23	the	the	DET
ma-245	79	24	given	give	VERB
ma-245	79	25	oval	oval	NOUN
ma-245	79	26	c.	c.	NOUN
ma-245	79	27	as	as	ADP
ma-245	79	28	basic	basic	ADJ
ma-245	79	29	tool	tool	NOUN
ma-245	79	30	we	we	PRON
ma-245	79	31	use	use	VERB
ma-245	79	32	the	the	DET
ma-245	79	33	new	new	ADJ
ma-245	79	34	rotation	rotation	NOUN
ma-245	79	35	matrix	matrix	NOUN
ma-245	79	36	:	:	PUNCT
ma-245	80	1	jacobi(t	jacobi(t	X
ma-245	80	2	,	,	PUNCT
ma-245	80	3	ρ	ρ	PROPN
ma-245	80	4	)	)	PUNCT
ma-245	80	5	:	:	PUNCT
ma-245	80	6	=	=	SYM
ma-245	80	7	(	(	PUNCT
ma-245	80	8	u(t	u(t	NOUN
ma-245	80	9	)	)	PUNCT
ma-245	80	10	−v(t	−v(t	NOUN
ma-245	80	11	)	)	PUNCT
ma-245	80	12	v(t	v(t	NOUN
ma-245	80	13	)	)	PUNCT
ma-245	80	14	u(t	u(t	NOUN
ma-245	80	15	)	)	PUNCT
ma-245	80	16	)	)	PUNCT
ma-245	81	1	∈	∈	PROPN
ma-245	81	2	so(2	so(2	NOUN
ma-245	81	3	)	)	PUNCT
ma-245	81	4	=	=	SYM
ma-245	81	5	s1	s1	PROPN
ma-245	81	6	.	.	PUNCT
ma-245	82	1	(	(	PUNCT
ma-245	82	2	3.5	3.5	NUM
ma-245	82	3	)	)	PUNCT
ma-245	82	4	definition	definition	NOUN
ma-245	82	5	3.1	3.1	NUM
ma-245	82	6	the	the	DET
ma-245	82	7	curve	curve	NOUN
ma-245	82	8	cj	cj	PROPN
ma-245	82	9	is	be	AUX
ma-245	82	10	the	the	DET
ma-245	82	11	ρ	ρ	PROPN
ma-245	82	12	-	-	PUNCT
ma-245	82	13	jacobi	jacobi	PROPN
ma-245	82	14	mate	mate	NOUN
ma-245	82	15	of	of	ADP
ma-245	82	16	c	c	PROPN
ma-245	82	17	if	if	SCONJ
ma-245	82	18	its	its	PRON
ma-245	82	19	parametrization	parametrization	NOUN
ma-245	82	20	is	be	AUX
ma-245	82	21	:	:	PUNCT
ma-245	82	22	rj(t	rj(t	X
ma-245	82	23	)	)	PUNCT
ma-245	83	1	=	=	SYM
ma-245	83	2	(	(	PUNCT
ma-245	83	3	xj	xj	PROPN
ma-245	83	4	yj	yj	PROPN
ma-245	83	5	)	)	PUNCT
ma-245	83	6	(	(	PUNCT
ma-245	83	7	t	t	NOUN
ma-245	83	8	)	)	PUNCT
ma-245	83	9	:	:	PUNCT
ma-245	84	1	=	=	PUNCT
ma-245	84	2	jacobi(t	jacobi(t	PROPN
ma-245	84	3	,	,	PUNCT
ma-245	84	4	ρ	ρ	PROPN
ma-245	84	5	)	)	PUNCT
ma-245	84	6	(	(	PUNCT
ma-245	84	7	p	p	NOUN
ma-245	84	8	p′	p′	NOUN
ma-245	84	9	)	)	PUNCT
ma-245	84	10	(	(	PUNCT
ma-245	84	11	t	t	NOUN
ma-245	84	12	)	)	PUNCT
ma-245	84	13	=	=	PUNCT
ma-245	84	14	(	(	PUNCT
ma-245	84	15	p(t)u(t)−	p(t)u(t)−	PROPN
ma-245	84	16	p′(t)v(t	p′(t)v(t	PROPN
ma-245	84	17	)	)	PUNCT
ma-245	84	18	p′(t)u(t	p′(t)u(t	NOUN
ma-245	84	19	)	)	PUNCT
ma-245	84	20	+	+	NUM
ma-245	84	21	p(t)v(t	p(t)v(t	NOUN
ma-245	84	22	)	)	PUNCT
ma-245	84	23	)	)	PUNCT
ma-245	84	24	,	,	PUNCT
ma-245	84	25	t	t	PROPN
ma-245	84	26	∈	∈	PROPN
ma-245	85	1	i	i	PRON
ma-245	85	2	=	=	PUNCT
ma-245	86	1	[	[	X
ma-245	86	2	0	0	NUM
ma-245	86	3	,	,	PUNCT
ma-245	86	4	l	l	NOUN
ma-245	86	5	]	]	X
ma-245	86	6	.	.	PUNCT
ma-245	87	1	(	(	PUNCT
ma-245	87	2	3.6)since	3.6)since	NUM
ma-245	87	3	the	the	DET
ma-245	87	4	derivative	derivative	NOUN
ma-245	87	5	of	of	ADP
ma-245	87	6	rj	rj	PROPN
ma-245	87	7	is	be	AUX
ma-245	87	8	:	:	PUNCT
ma-245	87	9	r	r	NOUN
ma-245	87	10	′j	′j	NOUN
ma-245	87	11	(	(	PUNCT
ma-245	87	12	t	t	PROPN
ma-245	87	13	)	)	PUNCT
ma-245	87	14	=	=	PUNCT
ma-245	87	15	(	(	PUNCT
ma-245	87	16	p	p	X
ma-245	87	17	′(t)u(t)(1−	′(t)u(t)(1−	PROPN
ma-245	87	18	w(t))−	w(t))−	NOUN
ma-245	87	19	v(t)(p(t)w(t	v(t)(p(t)w(t	NOUN
ma-245	87	20	)	)	PUNCT
ma-245	87	21	+	+	NUM
ma-245	87	22	p′′(t	p′′(t	NOUN
ma-245	87	23	)	)	PUNCT
ma-245	87	24	)	)	PUNCT
ma-245	87	25	,	,	PUNCT
ma-245	87	26	p′(t)v(t)(1−	p′(t)v(t)(1−	X
ma-245	87	27	w(t	w(t	PROPN
ma-245	87	28	)	)	PUNCT
ma-245	87	29	)	)	PUNCT
ma-245	88	1	+	+	CCONJ
ma-245	89	1	u(t)(p(t)w(t	u(t)(p(t)w(t	X
ma-245	89	2	)	)	PUNCT
ma-245	89	3	+	+	NUM
ma-245	89	4	p′′(t)))(3.7)it	p′′(t)))(3.7)it	NOUN
ma-245	89	5	results	result	NOUN
ma-245	89	6	:	:	PUNCT
ma-245	89	7	‖r	‖r	NOUN
ma-245	89	8	′j	′j	NOUN
ma-245	89	9	(	(	PUNCT
ma-245	89	10	t)‖2	t)‖2	ADJ
ma-245	89	11	=	=	SYM
ma-245	89	12	(	(	PUNCT
ma-245	89	13	p′(t))2[1−w(t)]2+[p(t)w(t)+p′′(t)]2	p′(t))2[1−w(t)]2+[p(t)w(t)+p′′(t)]2	PROPN
ma-245	89	14	∈	∈	PROPN
ma-245	89	15	(	(	PUNCT
ma-245	89	16	(	(	PUNCT
ma-245	89	17	ρ′p(t)+p′′(t))2	ρ′p(t)+p′′(t))2	X
ma-245	89	18	,	,	PUNCT
ma-245	89	19	(	(	PUNCT
ma-245	89	20	p′(t))2+[p(t)+p′′(t)]2)(3.8)and	p′(t))2+[p(t)+p′′(t)]2)(3.8)and	NOUN
ma-245	89	21	then	then	ADV
ma-245	89	22	cj	cj	PROPN
ma-245	89	23	is	be	AUX
ma-245	89	24	a	a	DET
ma-245	89	25	regular	regular	ADJ
ma-245	89	26	curve	curve	NOUN
ma-245	89	27	.	.	PUNCT
ma-245	90	1	it	it	PRON
ma-245	90	2	results	result	VERB
ma-245	90	3	also	also	ADV
ma-245	90	4	immediately	immediately	ADV
ma-245	90	5	:	:	PUNCT
ma-245	90	6	{	{	PUNCT
ma-245	90	7	x	x	SYM
ma-245	90	8	′′j	′′j	NOUN
ma-245	90	9	=	=	SYM
ma-245	90	10	p	p	PRON
ma-245	90	11	′′u(1−	′′u(1−	PROPN
ma-245	90	12	2w	2w	NUM
ma-245	90	13	)	)	PUNCT
ma-245	91	1	+	+	NUM
ma-245	91	2	p′v(ρ2u2	p′v(ρ2u2	NOUN
ma-245	91	3	+	+	CCONJ
ma-245	91	4	w2	w2	NOUN
ma-245	91	5	−	−	PROPN
ma-245	91	6	2w	2w	NUM
ma-245	91	7	)	)	PUNCT
ma-245	92	1	+	+	NUM
ma-245	92	2	v(pρ2uv	v(pρ2uv	AUX
ma-245	92	3	−	−	NOUN
ma-245	92	4	p′′′)−	p′′′)−	NOUN
ma-245	92	5	pw2u	pw2u	NOUN
ma-245	92	6	y	y	PROPN
ma-245	92	7	′′j	′′j	NOUN
ma-245	92	8	=	=	SYM
ma-245	92	9	p	p	X
ma-245	92	10	′′v(1−	′′v(1−	PROPN
ma-245	92	11	2w	2w	NUM
ma-245	92	12	)	)	PUNCT
ma-245	93	1	+	+	NUM
ma-245	93	2	p′u(ρ2v2	p′u(ρ2v2	ADJ
ma-245	93	3	−	−	PROPN
ma-245	93	4	w2	w2	NOUN
ma-245	93	5	+	+	CCONJ
ma-245	93	6	2w)−	2w)−	NUM
ma-245	93	7	u(pρ2uv	u(pρ2uv	ADJ
ma-245	93	8	−	−	NOUN
ma-245	93	9	p′′′)−	p′′′)−	NOUN
ma-245	93	10	pw2v	pw2v	NOUN
ma-245	93	11	(	(	PUNCT
ma-245	93	12	3.9	3.9	NUM
ma-245	93	13	)	)	PUNCT
ma-245	93	14	and	and	CCONJ
ma-245	93	15	then	then	ADV
ma-245	93	16	,	,	PUNCT
ma-245	93	17	considering	consider	VERB
ma-245	93	18	the	the	DET
ma-245	93	19	map	map	NOUN
ma-245	93	20	(	(	PUNCT
ma-245	93	21	·	·	PUNCT
ma-245	93	22	,	,	PUNCT
ma-245	93	23	ρ)→	ρ)→	PROPN
ma-245	93	24	rj	rj	PROPN
ma-245	93	25	(	(	PUNCT
ma-245	93	26	·	·	PUNCT
ma-245	93	27	)	)	PUNCT
ma-245	93	28	as	as	ADP
ma-245	93	29	a	a	DET
ma-245	93	30	flow	flow	NOUN
ma-245	93	31	of	of	ADP
ma-245	93	32	curves	curve	NOUN
ma-245	93	33	,	,	PUNCT
ma-245	93	34	we	we	PRON
ma-245	93	35	compute	compute	VERB
ma-245	93	36	its	its	PRON
ma-245	93	37	first	first	ADJ
ma-245	93	38	derivative	derivative	ADJ
ma-245	93	39	witha	witha	NOUN
ma-245	93	40	possible	possible	ADJ
ma-245	93	41	application	application	NOUN
ma-245	93	42	to	to	ADP
ma-245	93	43	a	a	DET
ma-245	93	44	parabolic	parabolic	ADJ
ma-245	93	45	flow	flow	NOUN
ma-245	93	46	(	(	PUNCT
ma-245	93	47	for	for	ADP
ma-245	93	48	example	example	NOUN
ma-245	93	49	,	,	PUNCT
ma-245	93	50	of	of	ADP
ma-245	93	51	curve	curve	NOUN
ma-245	93	52	shortening	shortening	NOUN
ma-245	93	53	type	type	NOUN
ma-245	93	54	,	,	PUNCT
ma-245	93	55	see	see	VERB
ma-245	93	56	the	the	DET
ma-245	93	57	chapter	chapter	NOUN
ma-245	93	58	2	2	NUM
ma-245	93	59	in	in	ADP
ma-245	93	60	[	[	PUNCT
ma-245	93	61	3	3	NUM
ma-245	93	62	]	]	PUNCT
ma-245	93	63	):	):	PUNCT
ma-245	93	64	{	{	PUNCT
ma-245	93	65	∂	∂	NUM
ma-245	93	66	∂ρ	∂ρ	NOUN
ma-245	93	67	r	r	PROPN
ma-245	93	68	′′	′′	PROPN
ma-245	93	69	j	j	PROPN
ma-245	93	70	(	(	PUNCT
ma-245	93	71	t	t	PROPN
ma-245	93	72	)	)	PUNCT
ma-245	93	73	=	=	PUNCT
ma-245	94	1	2ρu(t)v(t)[p	2ρu(t)v(t)[p	NUM
ma-245	94	2	′(t)(u(t	′(t)(u(t	PROPN
ma-245	94	3	)	)	PUNCT
ma-245	94	4	,	,	PUNCT
ma-245	94	5	v(t	v(t	NOUN
ma-245	94	6	)	)	PUNCT
ma-245	94	7	)	)	PUNCT
ma-245	95	1	+	+	CCONJ
ma-245	95	2	p(t)(v(t),−u(t	p(t)(v(t),−u(t	PROPN
ma-245	95	3	)	)	PUNCT
ma-245	95	4	)	)	PUNCT
ma-245	95	5	]	]	PUNCT
ma-245	96	1	=	=	PUNCT
ma-245	96	2	2ρu(t)v(t)[−i	2ρu(t)v(t)[−i	NUM
ma-245	96	3	rj(t	rj(t	NUM
ma-245	96	4	)	)	PUNCT
ma-245	96	5	]	]	PUNCT
ma-245	97	1	=	=	SYM
ma-245	97	2	2w	2w	NUM
ma-245	97	3	′(t)[i	′(t)[i	NOUN
ma-245	97	4	rj(t	rj(t	NUM
ma-245	97	5	)	)	PUNCT
ma-245	97	6	]	]	PUNCT
ma-245	97	7	,	,	PUNCT
ma-245	97	8	‖	‖	PROPN
ma-245	97	9	∂∂ρ	∂∂ρ	ADV
ma-245	98	1	r	r	NOUN
ma-245	98	2	′′	′′	PROPN
ma-245	98	3	j	j	PROPN
ma-245	98	4	(	(	PUNCT
ma-245	98	5	t)‖	t)‖	NOUN
ma-245	98	6	=	=	SYM
ma-245	98	7	2|ρ||u(t)||v(t)|‖r(t)‖.	2|ρ||u(t)||v(t)|‖r(t)‖.	NUM
ma-245	98	8	(	(	PUNCT
ma-245	98	9	3.10)therefore	3.10)therefore	NUM
ma-245	98	10	,	,	PUNCT
ma-245	98	11	∂	∂	NOUN
ma-245	98	12	∂ρ	∂ρ	NOUN
ma-245	98	13	r	r	PROPN
ma-245	98	14	′′	′′	PROPN
ma-245	98	15	j	j	PROPN
ma-245	98	16	(	(	PUNCT
ma-245	98	17	t	t	PROPN
ma-245	98	18	)	)	PUNCT
ma-245	98	19	is	be	AUX
ma-245	98	20	orthogonal	orthogonal	ADJ
ma-245	98	21	to	to	ADP
ma-245	98	22	rj(t	rj(t	NUM
ma-245	98	23	)	)	PUNCT
ma-245	98	24	,	,	PUNCT
ma-245	98	25	for	for	ADP
ma-245	98	26	all	all	DET
ma-245	98	27	t	t	NOUN
ma-245	98	28	∈	∈	PROPN
ma-245	99	1	[	[	X
ma-245	99	2	0	0	NUM
ma-245	99	3	,	,	PUNCT
ma-245	99	4	l	l	NOUN
ma-245	99	5	]	]	PUNCT
ma-245	99	6	.	.	PUNCT
ma-245	100	1	remark	remark	PROPN
ma-245	100	2	3.2	3.2	NUM
ma-245	100	3	we	we	PRON
ma-245	100	4	point	point	VERB
ma-245	100	5	out	out	ADP
ma-245	100	6	that	that	SCONJ
ma-245	100	7	following	follow	VERB
ma-245	100	8	the	the	DET
ma-245	100	9	approach	approach	NOUN
ma-245	100	10	of	of	ADP
ma-245	100	11	[	[	X
ma-245	100	12	8	8	NUM
ma-245	100	13	]	]	PUNCT
ma-245	100	14	we	we	PRON
ma-245	100	15	can	can	AUX
ma-245	100	16	think	think	VERB
ma-245	100	17	c	c	NOUN
ma-245	100	18	and	and	CCONJ
ma-245	100	19	cj	cj	NOUN
ma-245	100	20	as	as	ADP
ma-245	100	21	theeuclidean	theeuclidean	ADJ
ma-245	100	22	and	and	CCONJ
ma-245	100	23	jacobi	jacobi	NOUN
ma-245	100	24	deformations	deformation	NOUN
ma-245	100	25	of	of	ADP
ma-245	100	26	the	the	DET
ma-245	100	27	support	support	NOUN
ma-245	100	28	curve	curve	NOUN
ma-245	100	29	t	t	PROPN
ma-245	100	30	→	→	SYM
ma-245	100	31	p	p	X
ma-245	100	32	(	(	PUNCT
ma-245	100	33	t	t	PROPN
ma-245	100	34	)	)	PUNCT
ma-245	100	35	:	:	PUNCT
ma-245	100	36	=	=	SYM
ma-245	100	37	(	(	PUNCT
ma-245	100	38	p(t	p(t	NOUN
ma-245	100	39	)	)	PUNCT
ma-245	100	40	,	,	PUNCT
ma-245	100	41	p′(t	p′(t	PROPN
ma-245	100	42	)	)	PUNCT
ma-245	100	43	)	)	PUNCT
ma-245	100	44	.	.	PUNCT
ma-245	101	1	we	we	PRON
ma-245	101	2	have	have	VERB
ma-245	101	3	‖r(t)‖	‖r(t)‖	NOUN
ma-245	101	4	=	=	SYM
ma-245	101	5	‖p	‖p	PROPN
ma-245	101	6	(	(	PUNCT
ma-245	101	7	t)‖	t)‖	NOUN
ma-245	101	8	=	=	SYM
ma-245	101	9	‖rj(t)‖	‖rj(t)‖	PROPN
ma-245	101	10	,	,	PUNCT
ma-245	101	11	for	for	ADP
ma-245	101	12	all	all	DET
ma-245	101	13	t	t	NOUN
ma-245	101	14	.	.	PUNCT
ma-245	102	1	the	the	DET
ma-245	102	2	expression	expression	NOUN
ma-245	102	3	of	of	ADP
ma-245	102	4	p	p	PROPN
ma-245	102	5	recalls	recall	VERB
ma-245	102	6	the	the	DET
ma-245	102	7	well	well	ADV
ma-245	102	8	-	-	PUNCT
ma-245	102	9	known	know	VERB
ma-245	102	10	weierstrassparametrization	weierstrassparametrization	NOUN
ma-245	102	11	(	(	PUNCT
ma-245	102	12	℘(u	℘(u	NOUN
ma-245	102	13	)	)	PUNCT
ma-245	102	14	,	,	PUNCT
ma-245	102	15	℘′(u	℘′(u	NOUN
ma-245	102	16	)	)	PUNCT
ma-245	102	17	)	)	PUNCT
ma-245	102	18	of	of	ADP
ma-245	102	19	the	the	DET
ma-245	102	20	elliptic	elliptic	ADJ
ma-245	102	21	curve	curve	NOUN
ma-245	102	22	e(g2	e(g2	X
ma-245	102	23	,	,	PUNCT
ma-245	102	24	g3	g3	PROPN
ma-245	102	25	)	)	PUNCT
ma-245	102	26	:	:	PUNCT
ma-245	103	1	y2	y2	NOUN
ma-245	103	2	=	=	SYM
ma-245	103	3	4x3	4x3	NUM
ma-245	103	4	−	−	PROPN
ma-245	103	5	g2x	g2x	PROPN
ma-245	103	6	−	−	PROPN
ma-245	103	7	g3	g3	PROPN
ma-245	103	8	;	;	PUNCT
ma-245	103	9	see	see	VERB
ma-245	103	10	[	[	X
ma-245	103	11	9	9	NUM
ma-245	103	12	,	,	PUNCT
ma-245	103	13	p.	p.	NOUN
ma-245	103	14	77	77	NUM
ma-245	103	15	]	]	PUNCT
ma-245	103	16	.	.	PUNCT
ma-245	104	1	2	2	NUM
ma-245	104	2	our	our	PRON
ma-245	104	3	main	main	ADJ
ma-245	104	4	theoretical	theoretical	ADJ
ma-245	104	5	result	result	NOUN
ma-245	104	6	computes	compute	VERB
ma-245	104	7	the	the	DET
ma-245	104	8	curvature	curvature	NOUN
ma-245	104	9	of	of	ADP
ma-245	104	10	the	the	DET
ma-245	104	11	mate	mate	NOUN
ma-245	104	12	cj	cj	NOUN
ma-245	104	13	through	through	ADP
ma-245	104	14	a	a	DET
ma-245	104	15	long	long	ADJ
ma-245	104	16	but	but	CCONJ
ma-245	104	17	straight	straight	ADJ
ma-245	104	18	-	-	PUNCT
ma-245	104	19	forward	forward	NOUN
ma-245	104	20	computation	computation	NOUN
ma-245	104	21	:	:	PUNCT
ma-245	104	22	theorem	theorem	VERB
ma-245	104	23	3.3	3.3	NUM
ma-245	104	24	i	i	NOUN
ma-245	104	25	)	)	PUNCT
ma-245	104	26	if	if	SCONJ
ma-245	104	27	p	p	NOUN
ma-245	104	28	is	be	AUX
ma-245	104	29	not	not	PART
ma-245	104	30	a	a	DET
ma-245	104	31	constant	constant	ADJ
ma-245	104	32	then	then	ADV
ma-245	104	33	the	the	DET
ma-245	104	34	support	support	NOUN
ma-245	104	35	curve	curve	NOUN
ma-245	104	36	p	p	NOUN
ma-245	104	37	is	be	AUX
ma-245	104	38	a	a	DET
ma-245	104	39	regular	regular	ADJ
ma-245	104	40	one	one	NOUN
ma-245	104	41	having	have	VERB
ma-245	104	42	the	the	DET
ma-245	104	43	euclidean	euclidean	ADJ
ma-245	104	44	curvature	curvature	NOUN
ma-245	104	45	:	:	PUNCT
ma-245	104	46	kp	kp	PROPN
ma-245	104	47	(	(	PUNCT
ma-245	104	48	t	t	PROPN
ma-245	104	49	)	)	PUNCT
ma-245	105	1	=	=	SYM
ma-245	105	2	p′(t)p′′′(t)−	p′(t)p′′′(t)−	PROPN
ma-245	105	3	(	(	PUNCT
ma-245	105	4	p′′(t))2	p′′(t))2	NUM
ma-245	106	1	[	[	X
ma-245	106	2	(	(	PUNCT
ma-245	106	3	p′(t))2	p′(t))2	NOUN
ma-245	106	4	+	+	CCONJ
ma-245	106	5	(	(	PUNCT
ma-245	106	6	p′′(t))2	p′′(t))2	NUM
ma-245	106	7	]	]	X
ma-245	106	8	3	3	NUM
ma-245	106	9	2	2	NUM
ma-245	106	10	.	.	PUNCT
ma-245	107	1	(	(	PUNCT
ma-245	107	2	3.11	3.11	NUM
ma-245	107	3	)	)	PUNCT
ma-245	107	4	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	107	5	eur	eur	PROPN
ma-245	107	6	.	.	PUNCT
ma-245	108	1	j.	j.	PROPN
ma-245	108	2	math	math	PROPN
ma-245	108	3	.	.	PUNCT
ma-245	109	1	anal	anal	PROPN
ma-245	109	2	.	.	PUNCT
ma-245	110	1	10.28924	10.28924	NUM
ma-245	110	2	/	/	SYM
ma-245	110	3	ada	ada	PROPN
ma-245	110	4	/	/	SYM
ma-245	110	5	ma.4.18	ma.4.18	PROPN
ma-245	110	6	5	5	NUM
ma-245	110	7	let	let	VERB
ma-245	110	8	r(t0	r(t0	NOUN
ma-245	110	9	)	)	PUNCT
ma-245	110	10	be	be	AUX
ma-245	110	11	a	a	DET
ma-245	110	12	vertex	vertex	NOUN
ma-245	110	13	of	of	ADP
ma-245	110	14	the	the	DET
ma-245	110	15	oval	oval	NOUN
ma-245	110	16	c	c	X
ma-245	110	17	i.e.	i.e.	X
ma-245	110	18	p′′′(t0	p′′′(t0	NOUN
ma-245	110	19	)	)	PUNCT
ma-245	111	1	=	=	SYM
ma-245	111	2	−p′(t0	−p′(t0	PROPN
ma-245	111	3	)	)	PUNCT
ma-245	111	4	.	.	PUNCT
ma-245	112	1	then	then	ADV
ma-245	112	2	the	the	DET
ma-245	112	3	curvature	curvature	NOUN
ma-245	112	4	of	of	ADP
ma-245	112	5	p	p	NOUN
ma-245	112	6	in	in	ADP
ma-245	112	7	t0	t0	PROPN
ma-245	112	8	is	be	AUX
ma-245	112	9	:	:	PUNCT
ma-245	112	10	kp	kp	PROPN
ma-245	112	11	(	(	PUNCT
ma-245	112	12	t0	t0	NOUN
ma-245	112	13	)	)	PUNCT
ma-245	112	14	=	=	PUNCT
ma-245	113	1	−1	−1	NOUN
ma-245	114	1	[	[	X
ma-245	114	2	(	(	PUNCT
ma-245	114	3	p′(t0))2	p′(t0))2	X
ma-245	114	4	+	+	CCONJ
ma-245	114	5	(	(	PUNCT
ma-245	114	6	p′′(t0))2	p′′(t0))2	X
ma-245	114	7	]	]	X
ma-245	114	8	1	1	NUM
ma-245	114	9	2	2	NUM
ma-245	114	10	<	<	X
ma-245	114	11	0	0	NUM
ma-245	114	12	.	.	PUNCT
ma-245	114	13	ii	ii	PROPN
ma-245	114	14	)	)	PUNCT
ma-245	114	15	the	the	DET
ma-245	114	16	curvature	curvature	NOUN
ma-245	114	17	of	of	ADP
ma-245	114	18	the	the	DET
ma-245	114	19	ρ	ρ	PROPN
ma-245	114	20	-	-	PUNCT
ma-245	114	21	jacobi	jacobi	PROPN
ma-245	114	22	mate	mate	NOUN
ma-245	114	23	cj	cj	NOUN
ma-245	114	24	of	of	ADP
ma-245	114	25	the	the	DET
ma-245	114	26	oval	oval	NOUN
ma-245	114	27	c	c	NOUN
ma-245	114	28	is	be	AUX
ma-245	114	29	a	a	DET
ma-245	114	30	quadratic	quadratic	ADJ
ma-245	114	31	function	function	NOUN
ma-245	114	32	in	in	ADP
ma-245	114	33	ρ	ρ	PROPN
ma-245	114	34	:	:	PUNCT
ma-245	114	35	kj	kj	PROPN
ma-245	114	36	=	=	PUNCT
ma-245	114	37	(	(	PUNCT
ma-245	114	38	p′)2(1−	p′)2(1−	PROPN
ma-245	114	39	w)(2w	w)(2w	NUM
ma-245	114	40	−	−	PROPN
ma-245	114	41	w2	w2	NOUN
ma-245	114	42	)	)	PUNCT
ma-245	114	43	+	+	SYM
ma-245	114	44	p′[(1−	p′[(1−	X
ma-245	114	45	w)p′′′	w)p′′′	PROPN
ma-245	114	46	−	−	PROPN
ma-245	114	47	ρ2uv(p	ρ2uv(p	PROPN
ma-245	114	48	+	+	CCONJ
ma-245	114	49	p′′	p′′	NOUN
ma-245	114	50	)	)	PUNCT
ma-245	114	51	]	]	PUNCT
ma-245	115	1	+	+	CCONJ
ma-245	116	1	[	[	X
ma-245	116	2	pw2	pw2	X
ma-245	116	3	+	+	PUNCT
ma-245	117	1	p′′(2w	p′′(2w	ADV
ma-245	117	2	−	−	NUM
ma-245	117	3	1)](pw	1)](pw	NUM
ma-245	117	4	+	+	NUM
ma-245	117	5	p′′	p′′	ADJ
ma-245	117	6	)	)	PUNCT
ma-245	118	1	[	[	X
ma-245	118	2	(	(	PUNCT
ma-245	118	3	p′)2(1−	p′)2(1−	PROPN
ma-245	118	4	w)2	w)2	VERB
ma-245	118	5	+	+	CCONJ
ma-245	118	6	(	(	PUNCT
ma-245	118	7	pw	pw	X
ma-245	119	1	+	+	PUNCT
ma-245	119	2	p′′)2	p′′)2	VERB
ma-245	119	3	]	]	X
ma-245	119	4	3	3	NUM
ma-245	119	5	2	2	NUM
ma-245	119	6	.(3.12	.(3.12	NOUN
ma-245	119	7	)	)	PUNCT
ma-245	119	8	if	if	SCONJ
ma-245	119	9	the	the	DET
ma-245	119	10	modulus	modulus	ADJ
ma-245	119	11	ρ	ρ	NOUN
ma-245	119	12	is	be	AUX
ma-245	119	13	zero	zero	NUM
ma-245	119	14	then	then	ADV
ma-245	119	15	w	w	PROPN
ma-245	119	16	≡	≡	PROPN
ma-245	119	17	1	1	NUM
ma-245	119	18	and	and	CCONJ
ma-245	119	19	kj	kj	PROPN
ma-245	119	20	reduces	reduce	VERB
ma-245	119	21	to	to	ADP
ma-245	119	22	the	the	DET
ma-245	119	23	usual	usual	ADJ
ma-245	119	24	curvature	curvature	NOUN
ma-245	119	25	k	k	X
ma-245	119	26	from	from	ADP
ma-245	119	27	(	(	PUNCT
ma-245	119	28	2.5	2.5	NUM
ma-245	119	29	)	)	PUNCT
ma-245	119	30	.	.	PUNCT
ma-245	120	1	moreover	moreover	ADV
ma-245	120	2	,	,	PUNCT
ma-245	120	3	for	for	ADP
ma-245	120	4	the	the	DET
ma-245	120	5	flow	flow	NOUN
ma-245	120	6	interpretation	interpretation	NOUN
ma-245	120	7	before	before	ADP
ma-245	120	8	the	the	DET
ma-245	120	9	remark	remark	NOUN
ma-245	120	10	3.2	3.2	NUM
ma-245	120	11	we	we	PRON
ma-245	120	12	have	have	VERB
ma-245	120	13	:	:	PUNCT
ma-245	120	14	∂2kj(t	∂2kj(t	X
ma-245	120	15	)	)	PUNCT
ma-245	120	16	∂ρ2	∂ρ2	NOUN
ma-245	120	17	|ρ=0	|ρ=0	ADP
ma-245	120	18	=	=	SYM
ma-245	120	19	−2u(t)v(t)p′(t)[k(t)]2	−2u(t)v(t)p′(t)[k(t)]2	X
ma-245	120	20	.	.	PUNCT
ma-245	121	1	(	(	PUNCT
ma-245	121	2	3.13	3.13	NUM
ma-245	121	3	)	)	PUNCT
ma-245	121	4	we	we	PRON
ma-245	121	5	focus	focus	VERB
ma-245	121	6	now	now	ADV
ma-245	121	7	on	on	ADP
ma-245	121	8	some	some	DET
ma-245	121	9	concrete	concrete	ADJ
ma-245	121	10	examples	example	NOUN
ma-245	121	11	.	.	PUNCT
ma-245	122	1	example	example	NOUN
ma-245	122	2	3.4	3.4	NUM
ma-245	122	3	the	the	DET
ma-245	122	4	circle	circle	NOUN
ma-245	122	5	c(o	c(o	NOUN
ma-245	122	6	,	,	PUNCT
ma-245	122	7	r	r	NOUN
ma-245	122	8	>	>	X
ma-245	122	9	0	0	NUM
ma-245	122	10	)	)	PUNCT
ma-245	122	11	of	of	ADP
ma-245	122	12	the	the	DET
ma-245	122	13	euclidean	euclidean	ADJ
ma-245	122	14	plane	plane	NOUN
ma-245	122	15	geometry	geometry	NOUN
ma-245	122	16	is	be	AUX
ma-245	122	17	the	the	DET
ma-245	122	18	oval	oval	NOUN
ma-245	122	19	provided	provide	VERB
ma-245	122	20	bythe	bythe	DET
ma-245	122	21	constant	constant	ADJ
ma-245	122	22	support	support	NOUN
ma-245	122	23	function	function	NOUN
ma-245	122	24	p	p	PROPN
ma-245	122	25	≡	≡	PROPN
ma-245	122	26	r	r	NOUN
ma-245	122	27	and	and	CCONJ
ma-245	122	28	hence	hence	ADV
ma-245	122	29	w	w	PROPN
ma-245	122	30	≡	≡	PROPN
ma-245	122	31	2r	2r	NUM
ma-245	122	32	;	;	PUNCT
ma-245	122	33	the	the	DET
ma-245	122	34	curve	curve	NOUN
ma-245	122	35	p	p	NOUN
ma-245	122	36	consists	consist	VERB
ma-245	122	37	in	in	ADP
ma-245	122	38	the	the	DET
ma-245	122	39	unique	unique	ADJ
ma-245	122	40	point	point	NOUN
ma-245	122	41	(	(	PUNCT
ma-245	122	42	r	r	NOUN
ma-245	122	43	,	,	PUNCT
ma-245	122	44	0	0	NUM
ma-245	122	45	)	)	PUNCT
ma-245	122	46	.	.	PUNCT
ma-245	123	1	its	its	PRON
ma-245	123	2	ρ	ρ	PROPN
ma-245	123	3	-	-	PUNCT
ma-245	123	4	jacobi	jacobi	PROPN
ma-245	123	5	mate	mate	NOUN
ma-245	123	6	coincides	coincide	VERB
ma-245	123	7	cu	cu	PROPN
ma-245	123	8	c(o	c(o	NOUN
ma-245	123	9	,	,	PUNCT
ma-245	123	10	r	r	NOUN
ma-245	123	11	)	)	PUNCT
ma-245	123	12	,	,	PUNCT
ma-245	123	13	hence	hence	ADV
ma-245	123	14	kj	kj	PROPN
ma-245	123	15	=	=	PUNCT
ma-245	123	16	k	k	PROPN
ma-245	123	17	≡	≡	PROPN
ma-245	123	18	1	1	NUM
ma-245	123	19	r	r	NOUN
ma-245	123	20	,	,	PUNCT
ma-245	123	21	but	but	CCONJ
ma-245	123	22	now	now	ADV
ma-245	123	23	with	with	ADP
ma-245	123	24	the	the	DET
ma-245	123	25	parametrization	parametrization	NOUN
ma-245	123	26	:	:	PUNCT
ma-245	123	27	circlej(t	circlej(t	NOUN
ma-245	123	28	)	)	PUNCT
ma-245	123	29	=	=	SYM
ma-245	123	30	r(u(t	r(u(t	PROPN
ma-245	123	31	)	)	PUNCT
ma-245	123	32	,	,	PUNCT
ma-245	123	33	v(t	v(t	NOUN
ma-245	123	34	)	)	PUNCT
ma-245	123	35	)	)	PUNCT
ma-245	123	36	,	,	PUNCT
ma-245	123	37	ci	ci	PROPN
ma-245	123	38	rcle	rcle	NOUN
ma-245	123	39	′	′	PROPN
ma-245	123	40	j	j	PROPN
ma-245	123	41	(	(	PUNCT
ma-245	123	42	t	t	PROPN
ma-245	123	43	)	)	PUNCT
ma-245	123	44	=	=	SYM
ma-245	123	45	rw(t)(−v(t	rw(t)(−v(t	PROPN
ma-245	123	46	)	)	PUNCT
ma-245	123	47	,	,	PUNCT
ma-245	123	48	u(t	u(t	PROPN
ma-245	123	49	)	)	PUNCT
ma-245	123	50	)	)	PUNCT
ma-245	123	51	,	,	PUNCT
ma-245	123	52	‖circle	‖circle	X
ma-245	124	1	′j	′j	NOUN
ma-245	124	2	(	(	PUNCT
ma-245	124	3	t)‖	t)‖	NOUN
ma-245	124	4	=	=	SYM
ma-245	124	5	rw(t	rw(t	PUNCT
ma-245	124	6	)	)	PUNCT
ma-245	124	7	∈	∈	PROPN
ma-245	125	1	[	[	X
ma-245	125	2	rρ′	rρ′	NUM
ma-245	125	3	,	,	PUNCT
ma-245	125	4	r].(3.14)hence	r].(3.14)hence	NOUN
ma-245	125	5	,	,	PUNCT
ma-245	125	6	the	the	DET
ma-245	125	7	frenet	frenet	ADJ
ma-245	125	8	frame	frame	NOUN
ma-245	125	9	is	be	AUX
ma-245	125	10	:	:	PUNCT
ma-245	125	11	t	t	PROPN
ma-245	125	12	(	(	PUNCT
ma-245	125	13	t	t	PROPN
ma-245	125	14	)	)	PUNCT
ma-245	125	15	=	=	PUNCT
ma-245	125	16	(	(	PUNCT
ma-245	125	17	−v(t	−v(t	NOUN
ma-245	125	18	)	)	PUNCT
ma-245	125	19	,	,	PUNCT
ma-245	125	20	u(t	u(t	NOUN
ma-245	125	21	)	)	PUNCT
ma-245	125	22	)	)	PUNCT
ma-245	125	23	,	,	PUNCT
ma-245	125	24	n(t	n(t	PROPN
ma-245	125	25	)	)	PUNCT
ma-245	125	26	=	=	VERB
ma-245	126	1	it	it	PRON
ma-245	126	2	(	(	PUNCT
ma-245	126	3	t	t	NOUN
ma-245	126	4	)	)	PUNCT
ma-245	126	5	=	=	PUNCT
ma-245	126	6	(	(	PUNCT
ma-245	126	7	−u(t),−v(t	−u(t),−v(t	NOUN
ma-245	126	8	)	)	PUNCT
ma-245	126	9	)	)	PUNCT
ma-245	126	10	(	(	PUNCT
ma-245	126	11	3.15	3.15	NUM
ma-245	126	12	)	)	PUNCT
ma-245	126	13	as	as	ADP
ma-245	126	14	natural	natural	ADJ
ma-245	126	15	generalization	generalization	NOUN
ma-245	126	16	of	of	ADP
ma-245	126	17	(	(	PUNCT
ma-245	126	18	2.6	2.6	NUM
ma-245	126	19	)	)	PUNCT
ma-245	126	20	.	.	PUNCT
ma-245	127	1	by	by	ADP
ma-245	127	2	defining	define	VERB
ma-245	127	3	a	a	DET
ma-245	127	4	new	new	ADJ
ma-245	127	5	function	function	NOUN
ma-245	127	6	:	:	PUNCT
ma-245	128	1	w	w	X
ma-245	128	2	(	(	PUNCT
ma-245	128	3	t	t	PROPN
ma-245	128	4	)	)	PUNCT
ma-245	128	5	=	=	SYM
ma-245	129	1	∫	∫	PROPN
ma-245	129	2	t	t	PROPN
ma-245	129	3	0	0	NUM
ma-245	130	1	w(λ)dλ	w(λ)dλ	PROPN
ma-245	130	2	(	(	PUNCT
ma-245	130	3	3.16	3.16	NUM
ma-245	130	4	)	)	PUNCT
ma-245	130	5	we	we	PRON
ma-245	130	6	can	can	AUX
ma-245	130	7	write	write	VERB
ma-245	130	8	the	the	DET
ma-245	130	9	parametrization	parametrization	NOUN
ma-245	130	10	by	by	ADP
ma-245	130	11	arc	arc	NOUN
ma-245	130	12	-	-	PUNCT
ma-245	130	13	length	length	NOUN
ma-245	130	14	:	:	PUNCT
ma-245	130	15	circlej(s	circlej(s	NOUN
ma-245	130	16	)	)	PUNCT
ma-245	131	1	=	=	SYM
ma-245	131	2	r	r	NOUN
ma-245	131	3	(	(	PUNCT
ma-245	131	4	u	u	NOUN
ma-245	131	5	◦	◦	NOUN
ma-245	131	6	w−1	w−1	PROPN
ma-245	131	7	(	(	PUNCT
ma-245	131	8	s	s	NOUN
ma-245	131	9	r	r	NOUN
ma-245	131	10	)	)	PUNCT
ma-245	131	11	,	,	PUNCT
ma-245	131	12	v	v	X
ma-245	131	13	◦	◦	NOUN
ma-245	131	14	w−1	w−1	PROPN
ma-245	131	15	(	(	PUNCT
ma-245	131	16	s	s	NOUN
ma-245	131	17	r	r	NOUN
ma-245	131	18	)	)	PUNCT
ma-245	131	19	)	)	PUNCT
ma-245	131	20	,	,	PUNCT
ma-245	131	21	s	s	VERB
ma-245	131	22	∈	∈	PROPN
ma-245	132	1	[	[	X
ma-245	132	2	0	0	NUM
ma-245	132	3	,	,	PUNCT
ma-245	132	4	2πr	2πr	NOUN
ma-245	132	5	]	]	X
ma-245	132	6	.	.	PUNCT
ma-245	133	1	(	(	PUNCT
ma-245	133	2	3.17	3.17	NUM
ma-245	133	3	)	)	PUNCT
ma-245	133	4	the	the	DET
ma-245	133	5	value	value	NOUN
ma-245	133	6	of	of	ADP
ma-245	133	7	l̃	l̃	PROPN
ma-245	133	8	from	from	ADP
ma-245	133	9	(	(	PUNCT
ma-245	133	10	3.3	3.3	NUM
ma-245	133	11	)	)	PUNCT
ma-245	133	12	in	in	ADP
ma-245	133	13	the	the	DET
ma-245	133	14	self	self	NOUN
ma-245	133	15	-	-	PUNCT
ma-245	133	16	complementary	complementary	ADJ
ma-245	133	17	case	case	NOUN
ma-245	133	18	is	be	AUX
ma-245	133	19	:	:	PUNCT
ma-245	133	20	l̃	l̃	PROPN
ma-245	133	21	(	(	PUNCT
ma-245	133	22	1√	1√	PROPN
ma-245	133	23	2	2	NUM
ma-245	133	24	)	)	PUNCT
ma-245	133	25	'	'	PART
ma-245	133	26	1.85	1.85	NUM
ma-245	133	27	>	>	X
ma-245	133	28	π	π	X
ma-245	133	29	2	2	X
ma-245	133	30	'	'	NUM
ma-245	133	31	1.57	1.57	NUM
ma-245	133	32	(	(	PUNCT
ma-245	133	33	3.18	3.18	NUM
ma-245	133	34	)	)	PUNCT
ma-245	133	35	while	while	SCONJ
ma-245	133	36	the	the	DET
ma-245	133	37	second	second	ADJ
ma-245	133	38	identity	identity	NOUN
ma-245	133	39	from	from	ADP
ma-245	133	40	(	(	PUNCT
ma-245	133	41	3.2	3.2	NUM
ma-245	133	42	)	)	PUNCT
ma-245	133	43	provides	provide	VERB
ma-245	133	44	,	,	PUNCT
ma-245	133	45	in	in	ADP
ma-245	133	46	the	the	DET
ma-245	133	47	case	case	NOUN
ma-245	133	48	ρ	ρ	PROPN
ma-245	133	49	6=	6=	PROPN
ma-245	133	50	0	0	NUM
ma-245	133	51	,	,	PUNCT
ma-245	133	52	a	a	DET
ma-245	133	53	second	second	ADJ
ma-245	133	54	jacobi	jacobi	NOUN
ma-245	133	55	parametrizationof	parametrizationof	PROPN
ma-245	133	56	the	the	DET
ma-245	133	57	circle	circle	NOUN
ma-245	133	58	:	:	PUNCT
ma-245	133	59	scirclej(t	scirclej(t	X
ma-245	133	60	)	)	PUNCT
ma-245	133	61	=	=	PUNCT
ma-245	133	62	r(ρv(t	r(ρv(t	NOUN
ma-245	133	63	)	)	PUNCT
ma-245	133	64	,	,	PUNCT
ma-245	133	65	w(t	w(t	PROPN
ma-245	133	66	)	)	PUNCT
ma-245	133	67	)	)	PUNCT
ma-245	133	68	,	,	PUNCT
ma-245	133	69	scircle	scircle	NOUN
ma-245	133	70	′j	′j	NOUN
ma-245	133	71	(	(	PUNCT
ma-245	133	72	t	t	PROPN
ma-245	133	73	)	)	PUNCT
ma-245	133	74	=	=	SYM
ma-245	133	75	rρu(t)(w(t),−ρv(t	rρu(t)(w(t),−ρv(t	NOUN
ma-245	133	76	)	)	PUNCT
ma-245	133	77	)	)	PUNCT
ma-245	133	78	.	.	PUNCT
ma-245	134	1	(	(	PUNCT
ma-245	134	2	3.19	3.19	NUM
ma-245	134	3	)	)	PUNCT
ma-245	134	4	now	now	ADV
ma-245	134	5	,	,	PUNCT
ma-245	134	6	this	this	DET
ma-245	134	7	second	second	ADJ
ma-245	134	8	parametrization	parametrization	NOUN
ma-245	134	9	has	have	VERB
ma-245	134	10	singularities	singularity	NOUN
ma-245	134	11	,	,	PUNCT
ma-245	134	12	namely	namely	ADV
ma-245	134	13	the	the	DET
ma-245	134	14	zeros	zero	NOUN
ma-245	134	15	l̃	l̃	PROPN
ma-245	134	16	,	,	PUNCT
ma-245	134	17	3l̃	3l̃	NUM
ma-245	134	18	of	of	ADP
ma-245	134	19	the	the	DET
ma-245	134	20	function	function	NOUN
ma-245	134	21	u.	u.	NOUN
ma-245	134	22	2	2	NUM
ma-245	134	23	example	example	NOUN
ma-245	134	24	3.5	3.5	NUM
ma-245	134	25	fix	fix	NOUN
ma-245	134	26	the	the	DET
ma-245	134	27	smooth	smooth	ADJ
ma-245	134	28	real	real	ADJ
ma-245	134	29	function	function	NOUN
ma-245	134	30	p(t	p(t	NOUN
ma-245	134	31	)	)	PUNCT
ma-245	134	32	:	:	PUNCT
ma-245	135	1	=	=	SYM
ma-245	135	2	r	r	NOUN
ma-245	135	3	−	−	PROPN
ma-245	135	4	cos	cos	PROPN
ma-245	135	5	3	3	NUM
ma-245	135	6	t	t	NOUN
ma-245	135	7	;	;	PUNCT
ma-245	135	8	hence	hence	ADV
ma-245	135	9	p(t	p(t	NOUN
ma-245	135	10	)	)	PUNCT
ma-245	136	1	=	=	SYM
ma-245	136	2	p(t	p(t	NOUN
ma-245	136	3	+	+	NUM
ma-245	136	4	2π	2π	NOUN
ma-245	136	5	)	)	PUNCT
ma-245	137	1	andagain	andagain	VERB
ma-245	137	2	the	the	DET
ma-245	137	3	width	width	NOUN
ma-245	137	4	is	be	AUX
ma-245	137	5	constant	constant	ADJ
ma-245	137	6	w	w	PROPN
ma-245	137	7	≡	≡	PROPN
ma-245	137	8	2r	2r	NUM
ma-245	137	9	.	.	PUNCT
ma-245	138	1	in	in	ADP
ma-245	138	2	[	[	X
ma-245	138	3	4	4	NUM
ma-245	138	4	,	,	PUNCT
ma-245	138	5	p.	p.	NOUN
ma-245	138	6	23	23	NUM
ma-245	138	7	]	]	PUNCT
ma-245	138	8	it	it	PRON
ma-245	138	9	is	be	AUX
ma-245	138	10	proved	prove	VERB
ma-245	138	11	that	that	SCONJ
ma-245	138	12	if	if	SCONJ
ma-245	138	13	r	r	NOUN
ma-245	138	14	>	>	X
ma-245	138	15	8	8	NUM
ma-245	138	16	then	then	ADV
ma-245	138	17	p	p	NOUN
ma-245	138	18	is	be	AUX
ma-245	138	19	the	the	DET
ma-245	138	20	supportfunction	supportfunction	NOUN
ma-245	138	21	of	of	ADP
ma-245	138	22	an	an	DET
ma-245	138	23	oval	oval	NOUN
ma-245	138	24	c.	c.	NOUN
ma-245	138	25	if	if	SCONJ
ma-245	138	26	r	r	NOUN
ma-245	138	27	>	>	X
ma-245	138	28	8	8	NUM
ma-245	138	29	is	be	AUX
ma-245	138	30	a	a	DET
ma-245	138	31	positive	positive	ADJ
ma-245	138	32	integer	integer	NOUN
ma-245	138	33	then	then	ADV
ma-245	138	34	the	the	DET
ma-245	138	35	oval	oval	NOUN
ma-245	139	1	c	c	NOUN
ma-245	139	2	contains	contain	VERB
ma-245	139	3	the	the	DET
ma-245	139	4	integral	integral	ADJ
ma-245	139	5	point	point	NOUN
ma-245	139	6	(	(	PUNCT
ma-245	139	7	−(r	−(r	NOUN
ma-245	139	8	+	+	NOUN
ma-245	139	9	1	1	NUM
ma-245	139	10	)	)	PUNCT
ma-245	139	11	,	,	PUNCT
ma-245	139	12	0	0	X
ma-245	139	13	)	)	PUNCT
ma-245	139	14	corresponding	correspond	VERB
ma-245	139	15	to	to	ADP
ma-245	139	16	t	t	NOUN
ma-245	139	17	=	=	PUNCT
ma-245	139	18	π	π	NOUN
ma-245	139	19	while	while	SCONJ
ma-245	139	20	the	the	DET
ma-245	139	21	curve	curve	NOUN
ma-245	139	22	p	p	NOUN
ma-245	139	23	contains	contain	VERB
ma-245	139	24	the	the	DET
ma-245	139	25	4	4	NUM
ma-245	139	26	integral	integral	ADJ
ma-245	139	27	points	point	NOUN
ma-245	139	28	(	(	PUNCT
ma-245	139	29	r	r	NOUN
ma-245	139	30	−	−	PROPN
ma-245	139	31	1	1	NUM
ma-245	139	32	,	,	PUNCT
ma-245	139	33	0	0	NUM
ma-245	139	34	)	)	PUNCT
ma-245	139	35	,	,	PUNCT
ma-245	139	36	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	139	37	eur	eur	PROPN
ma-245	139	38	.	.	PUNCT
ma-245	140	1	j.	j.	PROPN
ma-245	140	2	math	math	PROPN
ma-245	140	3	.	.	PUNCT
ma-245	141	1	anal	anal	PROPN
ma-245	141	2	.	.	PUNCT
ma-245	142	1	10.28924	10.28924	NUM
ma-245	142	2	/	/	SYM
ma-245	142	3	ada	ada	PROPN
ma-245	142	4	/	/	PROPN
ma-245	142	5	ma.4.18	ma.4.18	X
ma-245	142	6	6	6	NUM
ma-245	142	7	(	(	PUNCT
ma-245	142	8	r	r	NOUN
ma-245	142	9	,	,	PUNCT
ma-245	142	10	3	3	NUM
ma-245	142	11	)	)	PUNCT
ma-245	142	12	,	,	PUNCT
ma-245	142	13	(	(	PUNCT
ma-245	142	14	r	r	NOUN
ma-245	142	15	+	+	NOUN
ma-245	142	16	1	1	NUM
ma-245	142	17	,	,	PUNCT
ma-245	142	18	0	0	NUM
ma-245	142	19	)	)	PUNCT
ma-245	142	20	,	,	PUNCT
ma-245	142	21	(	(	PUNCT
ma-245	142	22	r,−3	r,−3	PROPN
ma-245	142	23	)	)	PUNCT
ma-245	142	24	corresponding	correspond	VERB
ma-245	142	25	respectively	respectively	ADV
ma-245	142	26	to	to	ADP
ma-245	142	27	t	t	PROPN
ma-245	142	28	=	=	SYM
ma-245	142	29	0	0	NUM
ma-245	142	30	,	,	PUNCT
ma-245	142	31	t	t	NOUN
ma-245	143	1	=	=	PUNCT
ma-245	143	2	π	π	PROPN
ma-245	143	3	2	2	NUM
ma-245	143	4	,	,	PUNCT
ma-245	143	5	t	t	NOUN
ma-245	143	6	=	=	SYM
ma-245	143	7	π	π	PROPN
ma-245	143	8	and	and	CCONJ
ma-245	143	9	t	t	NOUN
ma-245	143	10	=	=	SYM
ma-245	143	11	3π	3π	NOUN
ma-245	143	12	2	2	NUM
ma-245	143	13	.	.	PUNCT
ma-245	144	1	withthe	withthe	ADJ
ma-245	144	2	derivatives	derivative	NOUN
ma-245	144	3	:	:	PUNCT
ma-245	144	4	p′(t	p′(t	X
ma-245	144	5	)	)	PUNCT
ma-245	144	6	=	=	SYM
ma-245	144	7	3	3	NUM
ma-245	144	8	sin	sin	NOUN
ma-245	144	9	3	3	NUM
ma-245	144	10	t	t	NOUN
ma-245	144	11	,	,	PUNCT
ma-245	144	12	p′′(t	p′′(t	NOUN
ma-245	144	13	)	)	PUNCT
ma-245	144	14	=	=	SYM
ma-245	144	15	9	9	NUM
ma-245	144	16	cos	cos	PROPN
ma-245	144	17	3	3	NUM
ma-245	144	18	t	t	PROPN
ma-245	144	19	,	,	PUNCT
ma-245	144	20	p′′′(t	p′′′(t	NOUN
ma-245	144	21	)	)	PUNCT
ma-245	144	22	=	=	PUNCT
ma-245	144	23	−27	−27	PRON
ma-245	144	24	sin	sin	VERB
ma-245	144	25	3	3	NUM
ma-245	144	26	t	t	NOUN
ma-245	144	27	(	(	PUNCT
ma-245	144	28	3.20	3.20	NUM
ma-245	144	29	)	)	PUNCT
ma-245	144	30	it	it	PRON
ma-245	144	31	results	result	VERB
ma-245	144	32	the	the	DET
ma-245	144	33	curvatures	curvature	NOUN
ma-245	144	34	:	:	PUNCT
ma-245	145	1	kp	kp	PROPN
ma-245	145	2	(	(	PUNCT
ma-245	145	3	t	t	PROPN
ma-245	145	4	)	)	PUNCT
ma-245	145	5	=	=	PUNCT
ma-245	146	1	−3	−3	PROPN
ma-245	147	1	[	[	X
ma-245	147	2	(	(	PUNCT
ma-245	147	3	sin	sin	NOUN
ma-245	147	4	3t)2	3t)2	NUM
ma-245	147	5	+	+	SYM
ma-245	147	6	9(cos	9(cos	NUM
ma-245	147	7	3t)2	3t)2	NUM
ma-245	147	8	]	]	SYM
ma-245	147	9	3	3	NUM
ma-245	147	10	2	2	NUM
ma-245	147	11	<	<	X
ma-245	147	12	0	0	NUM
ma-245	147	13	,	,	PUNCT
ma-245	147	14	k(t	k(t	NOUN
ma-245	147	15	)	)	PUNCT
ma-245	147	16	=	=	SYM
ma-245	147	17	1	1	NUM
ma-245	147	18	r	r	NOUN
ma-245	147	19	+	+	NOUN
ma-245	147	20	8cos	8cos	NUM
ma-245	147	21	3	3	NUM
ma-245	147	22	t	t	NOUN
ma-245	147	23	∈	∈	NOUN
ma-245	147	24	[	[	PUNCT
ma-245	147	25	1	1	NUM
ma-245	147	26	r	r	NOUN
ma-245	147	27	+	+	NUM
ma-245	147	28	8	8	NUM
ma-245	147	29	,	,	PUNCT
ma-245	147	30	1	1	NUM
ma-245	147	31	r	r	NOUN
ma-245	147	32	−	−	NOUN
ma-245	147	33	8	8	NUM
ma-245	147	34	]	]	PUNCT
ma-245	147	35	.	.	PUNCT
ma-245	148	1	(	(	PUNCT
ma-245	148	2	3.21	3.21	NUM
ma-245	148	3	)	)	PUNCT
ma-245	148	4	we	we	PRON
ma-245	148	5	note	note	VERB
ma-245	148	6	that	that	SCONJ
ma-245	148	7	kp	kp	PROPN
ma-245	148	8	does	do	AUX
ma-245	148	9	not	not	PART
ma-245	148	10	depend	depend	VERB
ma-245	148	11	on	on	ADP
ma-245	148	12	r	r	NOUN
ma-245	148	13	while	while	SCONJ
ma-245	148	14	the	the	DET
ma-245	148	15	curvature	curvature	NOUN
ma-245	148	16	k	k	PROPN
ma-245	148	17	solves	solve	VERB
ma-245	148	18	the	the	DET
ma-245	148	19	differential	differential	ADJ
ma-245	148	20	equation	equation	NOUN
ma-245	148	21	∂k	∂k	X
ma-245	148	22	∂r	∂r	VERB
ma-245	148	23	=	=	PUNCT
ma-245	148	24	−k	−k	PROPN
ma-245	148	25	2	2	NUM
ma-245	148	26	.	.	PUNCT
ma-245	149	1	the	the	DET
ma-245	149	2	cauchy	cauchy	NOUN
ma-245	149	3	and	and	CCONJ
ma-245	149	4	the	the	DET
ma-245	149	5	blaschke	blaschke	ADJ
ma-245	149	6	formulae	formulae	NOUN
ma-245	149	7	give	give	NOUN
ma-245	149	8	:	:	PUNCT
ma-245	149	9	l(c	l(c	NOUN
ma-245	149	10	)	)	PUNCT
ma-245	150	1	=	=	SYM
ma-245	150	2	2πr	2πr	NOUN
ma-245	150	3	,	,	PUNCT
ma-245	150	4	a(c	a(c	PROPN
ma-245	150	5	)	)	PUNCT
ma-245	150	6	=	=	NOUN
ma-245	150	7	π(r2	π(r2	NOUN
ma-245	151	1	−	−	NOUN
ma-245	151	2	4	4	NUM
ma-245	151	3	)	)	PUNCT
ma-245	151	4	>	>	X
ma-245	151	5	60π	60π	NOUN
ma-245	151	6	.	.	PUNCT
ma-245	152	1	(	(	PUNCT
ma-245	152	2	3.22	3.22	NUM
ma-245	152	3	)	)	PUNCT
ma-245	152	4	the	the	DET
ma-245	152	5	length	length	NOUN
ma-245	152	6	of	of	ADP
ma-245	152	7	the	the	DET
ma-245	152	8	curve	curve	NOUN
ma-245	152	9	p	p	NOUN
ma-245	152	10	is	be	AUX
ma-245	152	11	:	:	PUNCT
ma-245	152	12	l(p	l(p	NOUN
ma-245	152	13	)	)	PUNCT
ma-245	153	1	=	=	SYM
ma-245	153	2	3	3	NUM
ma-245	153	3	∫	∫	PROPN
ma-245	153	4	2π	2π	PROPN
ma-245	153	5	0	0	NUM
ma-245	154	1	√	√	NUM
ma-245	154	2	(	(	PUNCT
ma-245	154	3	sin	sin	NOUN
ma-245	154	4	3t)2	3t)2	NUM
ma-245	154	5	+	+	SYM
ma-245	154	6	9(cos	9(cos	NOUN
ma-245	154	7	3t)2dt	3t)2dt	NUM
ma-245	154	8	=	=	SYM
ma-245	154	9	3	3	NUM
ma-245	154	10	∫	∫	PROPN
ma-245	154	11	2π	2π	PROPN
ma-245	154	12	0	0	NUM
ma-245	155	1	√	√	NUM
ma-245	155	2	5	5	NUM
ma-245	155	3	+	+	SYM
ma-245	155	4	4	4	NUM
ma-245	155	5	cos	cos	ADP
ma-245	155	6	6tdt	6tdt	NUM
ma-245	155	7	'	'	NUM
ma-245	155	8	40.09	40.09	NUM
ma-245	155	9	(	(	PUNCT
ma-245	155	10	3.23	3.23	NUM
ma-245	155	11	)	)	PUNCT
ma-245	156	1	and	and	CCONJ
ma-245	156	2	we	we	PRON
ma-245	156	3	point	point	VERB
ma-245	156	4	out	out	ADP
ma-245	156	5	that	that	SCONJ
ma-245	156	6	the	the	DET
ma-245	156	7	argument	argument	NOUN
ma-245	156	8	3	3	NUM
ma-245	156	9	t	t	NOUN
ma-245	156	10	involved	involve	VERB
ma-245	156	11	in	in	ADP
ma-245	156	12	its	its	PRON
ma-245	156	13	components	component	NOUN
ma-245	156	14	recalls	recall	VERB
ma-245	156	15	the	the	DET
ma-245	156	16	cayley	cayley	ADJ
ma-245	156	17	sextic	sextic	ADJ
ma-245	156	18	,	,	PUNCT
ma-245	156	19	whichis	whichi	VERB
ma-245	156	20	not	not	PART
ma-245	156	21	an	an	DET
ma-245	156	22	oval	oval	NOUN
ma-245	156	23	but	but	CCONJ
ma-245	156	24	a	a	DET
ma-245	156	25	closed	closed	ADJ
ma-245	156	26	curve	curve	NOUN
ma-245	156	27	:	:	PUNCT
ma-245	156	28	cay	cay	NOUN
ma-245	156	29	ley(t	ley(t	PROPN
ma-245	156	30	)	)	PUNCT
ma-245	157	1	:	:	PUNCT
ma-245	157	2	=	=	PUNCT
ma-245	157	3	cos3	cos3	NOUN
ma-245	157	4	t(cos	t(cos	ADP
ma-245	157	5	3	3	NUM
ma-245	157	6	t	t	NOUN
ma-245	157	7	,	,	PUNCT
ma-245	157	8	sin	sin	NOUN
ma-245	157	9	3	3	NUM
ma-245	157	10	t	t	NOUN
ma-245	157	11	)	)	PUNCT
ma-245	157	12	,	,	PUNCT
ma-245	157	13	t	t	PROPN
ma-245	157	14	∈	∈	PROPN
ma-245	158	1	[	[	X
ma-245	158	2	0	0	NUM
ma-245	158	3	,	,	PUNCT
ma-245	158	4	2π	2π	NOUN
ma-245	158	5	]	]	PUNCT
ma-245	158	6	,	,	PUNCT
ma-245	158	7	l(cay	l(cay	X
ma-245	158	8	ley	ley	PROPN
ma-245	158	9	)	)	PUNCT
ma-245	158	10	=	=	SYM
ma-245	158	11	3π	3π	NUM
ma-245	158	12	.	.	PUNCT
ma-245	159	1	(	(	PUNCT
ma-245	159	2	3.24	3.24	NUM
ma-245	159	3	)	)	PUNCT
ma-245	159	4	for	for	ADP
ma-245	159	5	the	the	DET
ma-245	159	6	ρ	ρ	PROPN
ma-245	159	7	-	-	PUNCT
ma-245	159	8	jacobi	jacobi	PROPN
ma-245	159	9	mate	mate	NOUN
ma-245	159	10	we	we	PRON
ma-245	159	11	compute	compute	VERB
ma-245	159	12	only	only	ADV
ma-245	159	13	the	the	DET
ma-245	159	14	velocity	velocity	NOUN
ma-245	159	15	since	since	SCONJ
ma-245	159	16	its	its	PRON
ma-245	159	17	curvature	curvature	NOUN
ma-245	159	18	has	have	VERB
ma-245	159	19	a	a	DET
ma-245	159	20	complicated	complicated	ADJ
ma-245	159	21	expression	expression	NOUN
ma-245	159	22	:	:	PUNCT
ma-245	159	23	{	{	PUNCT
ma-245	159	24	‖r	‖r	NOUN
ma-245	159	25	′j	′j	NOUN
ma-245	159	26	(	(	PUNCT
ma-245	159	27	t)‖2	t)‖2	ADJ
ma-245	159	28	=	=	SYM
ma-245	159	29	9(sin	9(sin	PROPN
ma-245	160	1	3t)2[1−	3t)2[1−	NUM
ma-245	160	2	w(t)]2	w(t)]2	NOUN
ma-245	160	3	+	+	X
ma-245	161	1	[	[	X
ma-245	161	2	9	9	NUM
ma-245	161	3	cos	co	NOUN
ma-245	161	4	3	3	NUM
ma-245	161	5	t	t	NOUN
ma-245	161	6	+	+	NOUN
ma-245	161	7	w(t)(r	w(t)(r	NUM
ma-245	161	8	−	−	PUNCT
ma-245	161	9	cos	cos	ADP
ma-245	161	10	3t)]2	3t)]2	NUM
ma-245	161	11	,	,	PUNCT
ma-245	161	12	0	0	PUNCT
ma-245	161	13	<	<	X
ma-245	161	14	(	(	PUNCT
ma-245	161	15	ρ′r	ρ′r	NOUN
ma-245	161	16	+	+	CCONJ
ma-245	161	17	(	(	PUNCT
ma-245	161	18	9−	9−	NUM
ma-245	161	19	ρ′	ρ′	NUM
ma-245	161	20	)	)	PUNCT
ma-245	161	21	cos	cos	ADP
ma-245	161	22	3t)2	3t)2	NUM
ma-245	161	23	<	<	X
ma-245	161	24	‖r	‖r	NOUN
ma-245	161	25	′j	′j	NOUN
ma-245	161	26	(	(	PUNCT
ma-245	161	27	t)‖2	t)‖2	PROPN
ma-245	161	28	<	<	X
ma-245	161	29	81(sin	81(sin	NOUN
ma-245	161	30	3t)2	3t)2	NUM
ma-245	161	31	+	+	CCONJ
ma-245	161	32	(	(	PUNCT
ma-245	161	33	r	r	NOUN
ma-245	161	34	+	+	NUM
ma-245	161	35	8cos	8cos	NUM
ma-245	161	36	3t)2	3t)2	NUM
ma-245	161	37	.	.	PUNCT
ma-245	162	1	(	(	PUNCT
ma-245	162	2	3.25	3.25	NUM
ma-245	162	3	)	)	PUNCT
ma-245	162	4	2	2	NUM
ma-245	162	5	example	example	NOUN
ma-245	162	6	3.6	3.6	NUM
ma-245	162	7	for	for	ADP
ma-245	162	8	α	α	PRON
ma-245	162	9	∈	∈	PROPN
ma-245	163	1	[	[	X
ma-245	163	2	1,+∞	1,+∞	NUM
ma-245	163	3	)	)	PUNCT
ma-245	163	4	the	the	DET
ma-245	163	5	2π	2π	NOUN
ma-245	163	6	-	-	ADJ
ma-245	163	7	periodic	periodic	ADJ
ma-245	163	8	function	function	NOUN
ma-245	163	9	pα	pα	INTJ
ma-245	163	10	:	:	PUNCT
ma-245	164	1	[	[	X
ma-245	164	2	0	0	NUM
ma-245	164	3	,	,	PUNCT
ma-245	164	4	2π]→	2π]→	NUM
ma-245	164	5	r∗+	r∗+	PROPN
ma-245	164	6	,	,	PUNCT
ma-245	164	7	pα(t	pα(t	NOUN
ma-245	164	8	)	)	PUNCT
ma-245	164	9	:	:	PUNCT
ma-245	164	10	=	=	SYM
ma-245	164	11	1	1	NUM
ma-245	164	12	α	α	NOUN
ma-245	164	13	√	√	NOUN
ma-245	164	14	α4	α4	NOUN
ma-245	164	15	cos2	cos2	PROPN
ma-245	164	16	t	t	PROPN
ma-245	164	17	+	+	CCONJ
ma-245	164	18	sin2	sin2	PROPN
ma-245	164	19	t	t	PROPN
ma-245	164	20	is	be	AUX
ma-245	164	21	the	the	DET
ma-245	164	22	support	support	NOUN
ma-245	164	23	function	function	NOUN
ma-245	164	24	of	of	ADP
ma-245	164	25	an	an	DET
ma-245	164	26	ellipse	ellipse	NOUN
ma-245	164	27	since	since	SCONJ
ma-245	164	28	:	:	PUNCT
ma-245	164	29	pα(t	pα(t	VERB
ma-245	164	30	)	)	PUNCT
ma-245	165	1	+	+	CCONJ
ma-245	165	2	p	p	X
ma-245	165	3	′′	′′	PROPN
ma-245	165	4	α(t	α(t	X
ma-245	165	5	)	)	PUNCT
ma-245	165	6	=	=	SYM
ma-245	165	7	α3	α3	NOUN
ma-245	165	8	(	(	PUNCT
ma-245	165	9	α4	α4	NOUN
ma-245	165	10	cos2	cos2	PROPN
ma-245	165	11	t	t	PROPN
ma-245	165	12	+	+	CCONJ
ma-245	165	13	sin2	sin2	PROPN
ma-245	165	14	t	t	PROPN
ma-245	165	15	)	)	PUNCT
ma-245	165	16	3	3	NUM
ma-245	165	17	2	2	NUM
ma-245	165	18	>	>	X
ma-245	165	19	0	0	NUM
ma-245	165	20	.	.	PUNCT
ma-245	166	1	(	(	PUNCT
ma-245	166	2	3.26	3.26	NUM
ma-245	166	3	)	)	PUNCT
ma-245	166	4	now	now	ADV
ma-245	166	5	,	,	PUNCT
ma-245	166	6	the	the	DET
ma-245	166	7	width	width	ADJ
ma-245	166	8	function	function	NOUN
ma-245	166	9	is	be	AUX
ma-245	166	10	non	non	ADJ
ma-245	166	11	-	-	ADJ
ma-245	166	12	constant	constant	ADJ
ma-245	166	13	being	being	NOUN
ma-245	166	14	2pα	2pα	NOUN
ma-245	166	15	.	.	PUNCT
ma-245	167	1	as	as	ADP
ma-245	167	2	example	example	NOUN
ma-245	167	3	,	,	PUNCT
ma-245	167	4	with	with	ADP
ma-245	167	5	wolframalpha	wolframalpha	NOUN
ma-245	167	6	we	we	PRON
ma-245	167	7	obtain	obtain	VERB
ma-245	167	8	thelength	thelength	NOUN
ma-245	167	9	l	l	NOUN
ma-245	167	10	(	(	PUNCT
ma-245	167	11	pα=2	pα=2	ADJ
ma-245	167	12	)	)	PUNCT
ma-245	167	13	'	'	PART
ma-245	167	14	45.51	45.51	NUM
ma-245	167	15	.	.	PUNCT
ma-245	168	1	the	the	DET
ma-245	168	2	rhs	rhs	PROPN
ma-245	168	3	of	of	ADP
ma-245	168	4	the	the	DET
ma-245	168	5	inequality	inequality	NOUN
ma-245	168	6	(	(	PUNCT
ma-245	168	7	3.8	3.8	NUM
ma-245	168	8	)	)	PUNCT
ma-245	168	9	reads	read	VERB
ma-245	168	10	:	:	PUNCT
ma-245	168	11	‖r	‖r	NOUN
ma-245	168	12	′j	′j	NOUN
ma-245	168	13	(	(	PUNCT
ma-245	168	14	t)‖2	t)‖2	ADJ
ma-245	168	15	<	<	X
ma-245	168	16	(	(	PUNCT
ma-245	168	17	α4	α4	NOUN
ma-245	168	18	−	−	PROPN
ma-245	168	19	1)2	1)2	NUM
ma-245	168	20	2α2(α4	2α2(α4	NUM
ma-245	168	21	cos2	cos2	NOUN
ma-245	168	22	t	t	PROPN
ma-245	168	23	+	+	NUM
ma-245	168	24	sin2	sin2	PROPN
ma-245	168	25	t	t	PROPN
ma-245	168	26	)	)	PUNCT
ma-245	169	1	+	+	NUM
ma-245	169	2	α6	α6	NOUN
ma-245	169	3	(	(	PUNCT
ma-245	169	4	α4	α4	NOUN
ma-245	169	5	cos2	cos2	PROPN
ma-245	169	6	t	t	PROPN
ma-245	169	7	+	+	CCONJ
ma-245	169	8	sin2	sin2	PROPN
ma-245	169	9	t)3	t)3	PROPN
ma-245	169	10	.	.	PUNCT
ma-245	170	1	(	(	PUNCT
ma-245	170	2	3.27	3.27	NUM
ma-245	170	3	)	)	PUNCT
ma-245	170	4	with	with	ADP
ma-245	170	5	the	the	DET
ma-245	170	6	same	same	ADJ
ma-245	170	7	possible	possible	ADJ
ma-245	170	8	flow	flow	NOUN
ma-245	170	9	interpretation	interpretation	NOUN
ma-245	170	10	in	in	ADP
ma-245	170	11	mind	mind	NOUN
ma-245	170	12	we	we	PRON
ma-245	170	13	compute	compute	VERB
ma-245	170	14	the	the	DET
ma-245	170	15	first	first	ADJ
ma-245	170	16	derivative	derivative	NOUN
ma-245	170	17	of	of	ADP
ma-245	170	18	the	the	DET
ma-245	170	19	supportfunction	supportfunction	NOUN
ma-245	170	20	:	:	PUNCT
ma-245	171	1	∂pα	∂pα	PROPN
ma-245	171	2	∂α	∂α	PROPN
ma-245	171	3	(	(	PUNCT
ma-245	171	4	t	t	NOUN
ma-245	171	5	)	)	PUNCT
ma-245	171	6	=	=	SYM
ma-245	171	7	α4	α4	NOUN
ma-245	171	8	cos2	cos2	PROPN
ma-245	171	9	t	t	PROPN
ma-245	171	10	−	−	PROPN
ma-245	171	11	sin2	sin2	PROPN
ma-245	171	12	t	t	PROPN
ma-245	171	13	α2	α2	VERB
ma-245	171	14	√	√	PROPN
ma-245	171	15	α4	α4	PROPN
ma-245	171	16	cos2	cos2	PROPN
ma-245	171	17	t	t	PROPN
ma-245	171	18	+	+	NUM
ma-245	171	19	sin2	sin2	PROPN
ma-245	171	20	t	t	PROPN
ma-245	171	21	.	.	PUNCT
ma-245	172	1	(	(	PUNCT
ma-245	172	2	3.28	3.28	NUM
ma-245	172	3	)	)	PUNCT
ma-245	172	4	we	we	PRON
ma-245	172	5	note	note	VERB
ma-245	172	6	also	also	ADV
ma-245	172	7	that	that	PRON
ma-245	172	8	for	for	ADP
ma-245	172	9	α	α	PROPN
ma-245	172	10	>	>	X
ma-245	172	11	1	1	NUM
ma-245	172	12	the	the	DET
ma-245	172	13	given	give	VERB
ma-245	172	14	support	support	NOUN
ma-245	172	15	function	function	NOUN
ma-245	172	16	does	do	AUX
ma-245	172	17	not	not	PART
ma-245	172	18	has	have	VERB
ma-245	172	19	an	an	DET
ma-245	172	20	indicatrix	indicatrix	NOUN
ma-245	172	21	i.e.	i.e.	X
ma-245	172	22	the	the	DET
ma-245	172	23	planecurve	planecurve	NOUN
ma-245	172	24	defined	define	VERB
ma-245	172	25	implicitly	implicitly	ADV
ma-245	172	26	by	by	ADP
ma-245	172	27	{	{	PUNCT
ma-245	172	28	(	(	PUNCT
ma-245	172	29	α	α	PROPN
ma-245	172	30	,	,	PUNCT
ma-245	172	31	t	t	NOUN
ma-245	172	32	)	)	PUNCT
ma-245	172	33	∈	∈	PROPN
ma-245	172	34	r2	r2	NOUN
ma-245	172	35	;	;	PUNCT
ma-245	172	36	pα(t	pα(t	X
ma-245	172	37	)	)	PUNCT
ma-245	172	38	=	=	SYM
ma-245	172	39	1	1	X
ma-245	172	40	}	}	PUNCT
ma-245	172	41	is	be	AUX
ma-245	172	42	empty	empty	ADJ
ma-245	172	43	.	.	PUNCT
ma-245	173	1	the	the	DET
ma-245	173	2	same	same	ADJ
ma-245	173	3	fact	fact	NOUN
ma-245	173	4	holds	hold	VERB
ma-245	173	5	for	for	ADP
ma-245	173	6	the	the	DET
ma-245	173	7	supportfunction	supportfunction	NOUN
ma-245	173	8	of	of	ADP
ma-245	173	9	the	the	DET
ma-245	173	10	previous	previous	ADJ
ma-245	173	11	example	example	NOUN
ma-245	173	12	when	when	SCONJ
ma-245	173	13	r	r	NOUN
ma-245	173	14	>	>	X
ma-245	173	15	8	8	NUM
ma-245	173	16	.	.	SYM
ma-245	173	17	2	2	NUM
ma-245	173	18	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PROPN
ma-245	173	19	eur	eur	PROPN
ma-245	173	20	.	.	PUNCT
ma-245	174	1	j.	j.	PROPN
ma-245	174	2	math	math	PROPN
ma-245	174	3	.	.	PUNCT
ma-245	175	1	anal	anal	PROPN
ma-245	175	2	.	.	PUNCT
ma-245	176	1	10.28924	10.28924	NUM
ma-245	176	2	/	/	SYM
ma-245	176	3	ada	ada	PROPN
ma-245	176	4	/	/	SYM
ma-245	176	5	ma.4.18	ma.4.18	VERB
ma-245	176	6	7references	7reference	NOUN
ma-245	176	7	[	[	X
ma-245	176	8	1	1	X
ma-245	176	9	]	]	PUNCT
ma-245	176	10	j.	j.	PROPN
ma-245	176	11	arroyo	arroyo	PROPN
ma-245	176	12	,	,	PUNCT
ma-245	176	13	o.	o.	PROPN
ma-245	176	14	j.	j.	PROPN
ma-245	176	15	garay	garay	PROPN
ma-245	176	16	,	,	PUNCT
ma-245	176	17	j.	j.	PROPN
ma-245	176	18	j.	j.	PROPN
ma-245	176	19	mencia	mencia	PROPN
ma-245	176	20	,	,	PUNCT
ma-245	176	21	when	when	SCONJ
ma-245	176	22	is	be	AUX
ma-245	176	23	a	a	DET
ma-245	176	24	periodic	periodic	ADJ
ma-245	176	25	function	function	NOUN
ma-245	176	26	the	the	DET
ma-245	176	27	curvature	curvature	NOUN
ma-245	176	28	of	of	ADP
ma-245	176	29	a	a	DET
ma-245	176	30	closed	closed	ADJ
ma-245	176	31	plane	plane	NOUN
ma-245	176	32	curve	curve	NOUN
ma-245	176	33	,	,	PUNCT
ma-245	176	34	am	be	AUX
ma-245	176	35	.	.	PUNCT
ma-245	177	1	math.mon	math.mon	X
ma-245	177	2	.	.	PROPN
ma-245	177	3	115	115	NUM
ma-245	177	4	(	(	PUNCT
ma-245	177	5	2005	2005	NUM
ma-245	177	6	)	)	PUNCT
ma-245	177	7	,	,	PUNCT
ma-245	177	8	405	405	NUM
ma-245	177	9	-	-	SYM
ma-245	177	10	414	414	NUM
ma-245	177	11	.	.	PUNCT
ma-245	177	12	https://doi.org/10.1080/00029890.2008.11920543[2	https://doi.org/10.1080/00029890.2008.11920543[2	PROPN
ma-245	177	13	]	]	X
ma-245	177	14	h.	h.	PROPN
ma-245	177	15	alencar	alencar	PROPN
ma-245	177	16	,	,	PUNCT
ma-245	177	17	w.	w.	PROPN
ma-245	177	18	santos	santos	PROPN
ma-245	177	19	,	,	PUNCT
ma-245	177	20	g.	g.	PROPN
ma-245	177	21	silva	silva	PROPN
ma-245	177	22	neto	neto	PROPN
ma-245	177	23	,	,	PUNCT
ma-245	177	24	differential	differential	ADJ
ma-245	177	25	geometry	geometry	NOUN
ma-245	177	26	of	of	ADP
ma-245	177	27	plane	plane	NOUN
ma-245	177	28	curves	curve	NOUN
ma-245	177	29	,	,	PUNCT
ma-245	177	30	american	american	PROPN
ma-245	177	31	mathematical	mathematical	ADJ
ma-245	177	32	society	society	NOUN
ma-245	177	33	,	,	PUNCT
ma-245	177	34	providence	providence	NOUN
ma-245	177	35	,	,	PUNCT
ma-245	177	36	rhode	rhode	NOUN
ma-245	177	37	island	island	NOUN
ma-245	177	38	,	,	PUNCT
ma-245	177	39	2022	2022	NUM
ma-245	177	40	.	.	PUNCT
ma-245	178	1	https://doi.org/10.1090/stml/096.[3	https://doi.org/10.1090/stml/096.[3	PROPN
ma-245	178	2	]	]	X
ma-245	178	3	b.	b.	PROPN
ma-245	178	4	andrews	andrews	PROPN
ma-245	178	5	,	,	PUNCT
ma-245	178	6	b.	b.	PROPN
ma-245	178	7	chow	chow	PROPN
ma-245	178	8	,	,	PUNCT
ma-245	178	9	c.	c.	PROPN
ma-245	178	10	guenther	guenther	PROPN
ma-245	178	11	,	,	PUNCT
ma-245	178	12	m.	m.	NOUN
ma-245	178	13	langford	langford	PROPN
ma-245	178	14	,	,	PUNCT
ma-245	178	15	extrinsic	extrinsic	ADJ
ma-245	178	16	geometric	geometric	ADJ
ma-245	178	17	flows	flow	NOUN
ma-245	178	18	,	,	PUNCT
ma-245	178	19	american	american	PROPN
ma-245	178	20	mathematical	mathematical	PROPN
ma-245	178	21	society	society	NOUN
ma-245	178	22	,	,	PUNCT
ma-245	178	23	provi	provi	PROPN
ma-245	178	24	-	-	PUNCT
ma-245	178	25	dence	dence	NOUN
ma-245	178	26	,	,	PUNCT
ma-245	178	27	rhode	rhode	NOUN
ma-245	178	28	island	island	NOUN
ma-245	178	29	,	,	PUNCT
ma-245	178	30	2020	2020	NUM
ma-245	178	31	.	.	PUNCT
ma-245	179	1	https://doi.org/10.1090/gsm/206.[4	https://doi.org/10.1090/gsm/206.[4	PROPN
ma-245	179	2	]	]	X
ma-245	179	3	w.	w.	PROPN
ma-245	179	4	cieślak	cieślak	PROPN
ma-245	179	5	,	,	PUNCT
ma-245	179	6	w.	w.	PROPN
ma-245	179	7	mozgawa	mozgawa	PROPN
ma-245	179	8	,	,	PUNCT
ma-245	179	9	p.	p.	NOUN
ma-245	179	10	wlaź	wlaź	PROPN
ma-245	179	11	,	,	PUNCT
ma-245	179	12	on	on	ADP
ma-245	179	13	the	the	DET
ma-245	179	14	closest	close	ADJ
ma-245	179	15	distance	distance	NOUN
ma-245	179	16	between	between	ADP
ma-245	179	17	a	a	DET
ma-245	179	18	point	point	NOUN
ma-245	179	19	and	and	CCONJ
ma-245	179	20	a	a	DET
ma-245	179	21	convex	convex	NOUN
ma-245	179	22	body	body	NOUN
ma-245	179	23	,	,	PUNCT
ma-245	179	24	bull	bull	NOUN
ma-245	179	25	.	.	PUNCT
ma-245	180	1	soc	soc	PROPN
ma-245	180	2	.	.	PUNCT
ma-245	181	1	sci	sci	PROPN
ma-245	181	2	.	.	PUNCT
ma-245	182	1	lettr.łódź	lettr.łódź	PROPN
ma-245	182	2	,	,	PUNCT
ma-245	182	3	sér	sér	NOUN
ma-245	182	4	.	.	PUNCT
ma-245	182	5	:	:	PUNCT
ma-245	183	1	rech	rech	NOUN
ma-245	183	2	.	.	PUNCT
ma-245	184	1	déform	déform	NOUN
ma-245	184	2	.	.	PUNCT
ma-245	185	1	67	67	NUM
ma-245	185	2	(	(	PUNCT
ma-245	185	3	2017	2017	NUM
ma-245	185	4	)	)	PUNCT
ma-245	185	5	,	,	PUNCT
ma-245	185	6	21	21	NUM
ma-245	185	7	-	-	SYM
ma-245	185	8	30	30	NUM
ma-245	185	9	.	.	PUNCT
ma-245	186	1	https://doi.org/10.26485/0459-6854/2017/67.2/2.[5	https://doi.org/10.26485/0459-6854/2017/67.2/2.[5	NOUN
ma-245	186	2	]	]	X
ma-245	186	3	m.	m.	NOUN
ma-245	186	4	crasmareanu	crasmareanu	NOUN
ma-245	186	5	,	,	PUNCT
ma-245	186	6	magic	magic	ADJ
ma-245	186	7	conics	conic	NOUN
ma-245	186	8	,	,	PUNCT
ma-245	186	9	their	their	PRON
ma-245	186	10	integer	integer	NOUN
ma-245	186	11	points	point	NOUN
ma-245	186	12	and	and	CCONJ
ma-245	186	13	complementary	complementary	ADJ
ma-245	186	14	ellipses	ellipsis	NOUN
ma-245	186	15	,	,	PUNCT
ma-245	186	16	an	an	PRON
ma-245	186	17	.	.	NOUN
ma-245	186	18	ştiinţ	ştiinţ	PROPN
ma-245	186	19	.	.	PUNCT
ma-245	187	1	univ	univ	PROPN
ma-245	187	2	.	.	PUNCT
ma-245	188	1	al	al	PROPN
ma-245	188	2	.	.	PROPN
ma-245	188	3	i.	i.	PROPN
ma-245	188	4	cuza	cuza	PROPN
ma-245	188	5	iaşimat	iaşimat	PROPN
ma-245	188	6	.	.	PUNCT
ma-245	189	1	67	67	NUM
ma-245	189	2	(	(	PUNCT
ma-245	189	3	2021	2021	NUM
ma-245	189	4	)	)	PUNCT
ma-245	189	5	,	,	PUNCT
ma-245	189	6	129	129	NUM
ma-245	189	7	-	-	SYM
ma-245	189	8	148.[6	148.[6	NUM
ma-245	189	9	]	]	PUNCT
ma-245	189	10	m.	m.	NOUN
ma-245	189	11	crasmareanu	crasmareanu	NOUN
ma-245	189	12	,	,	PUNCT
ma-245	189	13	the	the	DET
ma-245	189	14	flow	flow	NOUN
ma-245	189	15	-	-	PUNCT
ma-245	189	16	curvature	curvature	NOUN
ma-245	189	17	of	of	ADP
ma-245	189	18	plane	plane	NOUN
ma-245	189	19	parametrized	parametrized	ADJ
ma-245	189	20	curves	curve	NOUN
ma-245	189	21	,	,	PUNCT
ma-245	189	22	commun	commun	PROPN
ma-245	189	23	.	.	PUNCT
ma-245	190	1	fac	fac	PROPN
ma-245	190	2	.	.	PUNCT
ma-245	190	3	sci	sci	PROPN
ma-245	190	4	.	.	PROPN
ma-245	190	5	univ	univ	PROPN
ma-245	190	6	.	.	PUNCT
ma-245	191	1	ankara	ankara	PROPN
ma-245	191	2	ser	ser	PROPN
ma-245	191	3	.	.	PUNCT
ma-245	192	1	a1	a1	PROPN
ma-245	192	2	math.stat	math.stat	PROPN
ma-245	192	3	.	.	PROPN
ma-245	192	4	72	72	NUM
ma-245	192	5	(	(	PUNCT
ma-245	192	6	2023	2023	NUM
ma-245	192	7	)	)	PUNCT
ma-245	192	8	,	,	PUNCT
ma-245	192	9	417	417	NUM
ma-245	192	10	-	-	SYM
ma-245	192	11	428	428	NUM
ma-245	192	12	.	.	PUNCT
ma-245	193	1	https://doi.org/10.31801/cfsuasmas.1165123.[7	https://doi.org/10.31801/cfsuasmas.1165123.[7	NOUN
ma-245	193	2	]	]	X
ma-245	193	3	r.h	r.h	PROPN
ma-245	193	4	.	.	PROPN
ma-245	193	5	cushman	cushman	PROPN
ma-245	193	6	,	,	PUNCT
ma-245	193	7	l.m	l.m	PROPN
ma-245	193	8	.	.	PROPN
ma-245	193	9	bates	bate	NOUN
ma-245	193	10	,	,	PUNCT
ma-245	193	11	global	global	ADJ
ma-245	193	12	aspects	aspect	NOUN
ma-245	193	13	of	of	ADP
ma-245	193	14	classical	classical	ADJ
ma-245	193	15	integrable	integrable	ADJ
ma-245	193	16	systems	system	NOUN
ma-245	193	17	,	,	PUNCT
ma-245	193	18	springer	springer	NOUN
ma-245	193	19	basel	basel	PROPN
ma-245	193	20	,	,	PUNCT
ma-245	193	21	basel	basel	PROPN
ma-245	193	22	,	,	PUNCT
ma-245	193	23	2015	2015	NUM
ma-245	193	24	.	.	PUNCT
ma-245	194	1	https	https	NOUN
ma-245	194	2	:	:	PUNCT
ma-245	194	3	//doi.org/10.1007/978	//doi.org/10.1007/978	NUM
ma-245	194	4	-	-	PUNCT
ma-245	194	5	3	3	NUM
ma-245	194	6	-	-	PUNCT
ma-245	194	7	0348	0348	NUM
ma-245	194	8	-	-	PUNCT
ma-245	194	9	0918	0918	NUM
ma-245	194	10	-	-	SYM
ma-245	194	11	4.[8	4.[8	PROPN
ma-245	194	12	]	]	PUNCT
ma-245	194	13	b.	b.	PROPN
ma-245	195	1	mazur	mazur	PROPN
ma-245	195	2	,	,	PUNCT
ma-245	195	3	perturbations	perturbation	NOUN
ma-245	195	4	,	,	PUNCT
ma-245	195	5	deformations	deformation	NOUN
ma-245	195	6	,	,	PUNCT
ma-245	195	7	and	and	CCONJ
ma-245	195	8	variations	variation	NOUN
ma-245	195	9	(	(	PUNCT
ma-245	195	10	and	and	CCONJ
ma-245	195	11	“	"	PUNCT
ma-245	195	12	near	near	ADJ
ma-245	195	13	-	-	PUNCT
ma-245	195	14	misses	miss	NOUN
ma-245	195	15	"	"	PUNCT
ma-245	195	16	)	)	PUNCT
ma-245	195	17	in	in	ADP
ma-245	195	18	geometry	geometry	NOUN
ma-245	195	19	,	,	PUNCT
ma-245	195	20	physics	physics	NOUN
ma-245	195	21	,	,	PUNCT
ma-245	195	22	and	and	CCONJ
ma-245	195	23	number	number	NOUN
ma-245	195	24	theory	theory	NOUN
ma-245	195	25	,	,	PUNCT
ma-245	195	26	bull	bull	PROPN
ma-245	195	27	.	.	PUNCT
ma-245	196	1	amer	amer	PROPN
ma-245	196	2	.	.	PUNCT
ma-245	196	3	math	math	PROPN
ma-245	196	4	.	.	PUNCT
ma-245	197	1	soc	soc	PROPN
ma-245	197	2	.	.	PUNCT
ma-245	198	1	41	41	NUM
ma-245	198	2	(	(	PUNCT
ma-245	198	3	2004	2004	NUM
ma-245	198	4	)	)	PUNCT
ma-245	198	5	,	,	PUNCT
ma-245	198	6	307–336	307–336	NUM
ma-245	198	7	.	.	PUNCT
ma-245	199	1	https://doi.org/10.1090/s0273-0979-04-01024-9.[9	https://doi.org/10.1090/s0273-0979-04-01024-9.[9	PROPN
ma-245	199	2	]	]	X
ma-245	199	3	r.	r.	PROPN
ma-245	199	4	takloo	takloo	NOUN
ma-245	199	5	-	-	PUNCT
ma-245	199	6	bighash	bighash	NOUN
ma-245	199	7	,	,	PUNCT
ma-245	199	8	a	a	DET
ma-245	199	9	pythagorean	pythagorean	ADJ
ma-245	199	10	introduction	introduction	NOUN
ma-245	199	11	to	to	ADP
ma-245	199	12	number	number	NOUN
ma-245	199	13	theory	theory	NOUN
ma-245	199	14	:	:	PUNCT
ma-245	199	15	right	right	ADJ
ma-245	199	16	triangles	triangle	NOUN
ma-245	199	17	,	,	PUNCT
ma-245	199	18	sums	sum	NOUN
ma-245	199	19	of	of	ADP
ma-245	199	20	squares	square	NOUN
ma-245	199	21	,	,	PUNCT
ma-245	199	22	and	and	CCONJ
ma-245	199	23	arithmetic	arithmetic	ADJ
ma-245	199	24	,	,	PUNCT
ma-245	199	25	springer	springer	NOUN
ma-245	199	26	,	,	PUNCT
ma-245	199	27	cham	cham	NOUN
ma-245	199	28	,	,	PUNCT
ma-245	199	29	2018	2018	NUM
ma-245	199	30	.	.	PUNCT
ma-245	200	1	https://doi.org/10.1007/978-3-030-02604-2	https://doi.org/10.1007/978-3-030-02604-2	PROPN
ma-245	200	2	.	.	PUNCT
ma-245	201	1	https://doi.org/10.28924/ada/ma.4.18	https://doi.org/10.28924/ada/ma.4.18	PRON
ma-245	201	2	https://doi.org/10.1080/00029890.2008.11920543	https://doi.org/10.1080/00029890.2008.11920543	PROPN
ma-245	201	3	https://doi.org/10.1090/stml/096	https://doi.org/10.1090/stml/096	PROPN
ma-245	201	4	https://doi.org/10.1090/gsm/206	https://doi.org/10.1090/gsm/206	NOUN
ma-245	201	5	https://doi.org/10.26485/0459-6854/2017/67.2/2	https://doi.org/10.26485/0459-6854/2017/67.2/2	VERB
ma-245	201	6	https://doi.org/10.31801/cfsuasmas.1165123	https://doi.org/10.31801/cfsuasmas.1165123	VERB
ma-245	201	7	https://doi.org/10.1007/978-3-0348-0918-4	https://doi.org/10.1007/978-3-0348-0918-4	PROPN
ma-245	201	8	https://doi.org/10.1007/978-3-0348-0918-4	https://doi.org/10.1007/978-3-0348-0918-4	PROPN
ma-245	201	9	https://doi.org/10.1090/s0273-0979-04-01024-9	https://doi.org/10.1090/s0273-0979-04-01024-9	NOUN
ma-245	201	10	https://doi.org/10.1007/978-3-030-02604-2	https://doi.org/10.1007/978-3-030-02604-2	PROPN
ma-245	201	11	1	1	NUM
ma-245	201	12	.	.	PUNCT
ma-245	202	1	introduction	introduction	NOUN
ma-245	202	2	2	2	NUM
ma-245	202	3	.	.	PUNCT
ma-245	203	1	the	the	DET
ma-245	203	2	differential	differential	ADJ
ma-245	203	3	geometry	geometry	NOUN
ma-245	203	4	of	of	ADP
ma-245	203	5	euclidean	euclidean	ADJ
ma-245	203	6	ovals	oval	NOUN
ma-245	203	7	3	3	NUM
ma-245	203	8	.	.	PUNCT
ma-245	204	1	the	the	DET
ma-245	204	2	jacobi	jacobi	PROPN
ma-245	204	3	mate	mate	NOUN
ma-245	204	4	of	of	ADP
ma-245	204	5	an	an	DET
ma-245	204	6	oval	oval	NOUN
ma-245	204	7	references	reference	NOUN