id	sid	tid	token	lemma	pos
ajrhss-2194	1	1	american	american	PROPN
ajrhss-2194	1	2	journal	journal	PROPN
ajrhss-2194	1	3	of	of	ADP
ajrhss-2194	1	4	research	research	NOUN
ajrhss-2194	1	5	in	in	ADP
ajrhss-2194	1	6	humanities	humanity	NOUN
ajrhss-2194	1	7	and	and	CCONJ
ajrhss-2194	1	8	social	social	ADJ
ajrhss-2194	1	9	sciences	science	NOUN
ajrhss-2194	1	10	issn	issn	PROPN
ajrhss-2194	1	11	(	(	PUNCT
ajrhss-2194	1	12	e	e	NOUN
ajrhss-2194	1	13	):	):	PUNCT
ajrhss-2194	1	14	2832	2832	NUM
ajrhss-2194	1	15	-	-	SYM
ajrhss-2194	1	16	8019	8019	NUM
ajrhss-2194	1	17	volume	volume	NOUN
ajrhss-2194	1	18	25	25	NUM
ajrhss-2194	1	19	,	,	PUNCT
ajrhss-2194	1	20	|	|	ADV
ajrhss-2194	1	21	june	june	PROPN
ajrhss-2194	1	22	2024	2024	NUM
ajrhss-2194	1	23	p	p	NOUN
ajrhss-2194	1	24	a	a	DET
ajrhss-2194	1	25	g	g	NOUN
ajrhss-2194	1	26	e	e	NOUN
ajrhss-2194	1	27	|	|	ADV
ajrhss-2194	1	28	41	41	NUM
ajrhss-2194	1	29	www.americanjournal.org	www.americanjournal.org	PROPN
ajrhss-2194	1	30	geometric	geometric	ADJ
ajrhss-2194	1	31	function	function	NOUN
ajrhss-2194	1	32	masimova	masimova	PROPN
ajrhss-2194	1	33	gulmira	gulmira	PROPN
ajrhss-2194	1	34	maxsutovna	maxsutovna	PROPN
ajrhss-2194	1	35	raxmatova	raxmatova	PROPN
ajrhss-2194	1	36	gulrux	gulrux	PROPN
ajrhss-2194	1	37	axatovna	axatovna	VERB
ajrhss-2194	1	38	a	a	DET
ajrhss-2194	1	39	b	b	PROPN
ajrhss-2194	1	40	s	s	ADP
ajrhss-2194	1	41	t	t	PROPN
ajrhss-2194	1	42	r	r	NOUN
ajrhss-2194	1	43	a	a	DET
ajrhss-2194	1	44	c	c	NOUN
ajrhss-2194	1	45	t	t	NOUN
ajrhss-2194	1	46	k	k	X
ajrhss-2194	1	47	e	e	PROPN
ajrhss-2194	1	48	y	y	PROPN
ajrhss-2194	1	49	w	w	NOUN
ajrhss-2194	1	50	o	o	NOUN
ajrhss-2194	1	51	r	r	NOUN
ajrhss-2194	1	52	d	d	X
ajrhss-2194	1	53	s	s	X
ajrhss-2194	1	54	geometric	geometric	ADJ
ajrhss-2194	1	55	functions	function	NOUN
ajrhss-2194	1	56	play	play	VERB
ajrhss-2194	1	57	a	a	DET
ajrhss-2194	1	58	fundamental	fundamental	ADJ
ajrhss-2194	1	59	role	role	NOUN
ajrhss-2194	1	60	in	in	ADP
ajrhss-2194	1	61	various	various	ADJ
ajrhss-2194	1	62	branches	branch	NOUN
ajrhss-2194	1	63	of	of	ADP
ajrhss-2194	1	64	geometry	geometry	NOUN
ajrhss-2194	1	65	,	,	PUNCT
ajrhss-2194	1	66	providing	provide	VERB
ajrhss-2194	1	67	precise	precise	ADJ
ajrhss-2194	1	68	mathematical	mathematical	ADJ
ajrhss-2194	1	69	descriptions	description	NOUN
ajrhss-2194	1	70	of	of	ADP
ajrhss-2194	1	71	shapes	shape	NOUN
ajrhss-2194	1	72	,	,	PUNCT
ajrhss-2194	1	73	transformations	transformation	NOUN
ajrhss-2194	1	74	,	,	PUNCT
ajrhss-2194	1	75	and	and	CCONJ
ajrhss-2194	1	76	spatial	spatial	ADJ
ajrhss-2194	1	77	relationships	relationship	NOUN
ajrhss-2194	1	78	.	.	PUNCT
ajrhss-2194	2	1	these	these	DET
ajrhss-2194	2	2	functions	function	NOUN
ajrhss-2194	2	3	are	be	AUX
ajrhss-2194	2	4	essential	essential	ADJ
ajrhss-2194	2	5	in	in	ADP
ajrhss-2194	2	6	defining	define	VERB
ajrhss-2194	2	7	geometric	geometric	ADJ
ajrhss-2194	2	8	objects	object	NOUN
ajrhss-2194	2	9	,	,	PUNCT
ajrhss-2194	2	10	performing	perform	VERB
ajrhss-2194	2	11	transformations	transformation	NOUN
ajrhss-2194	2	12	,	,	PUNCT
ajrhss-2194	2	13	analyzing	analyze	VERB
ajrhss-2194	2	14	properties	property	NOUN
ajrhss-2194	2	15	,	,	PUNCT
ajrhss-2194	2	16	modeling	model	VERB
ajrhss-2194	2	17	physical	physical	ADJ
ajrhss-2194	2	18	systems	system	NOUN
ajrhss-2194	2	19	,	,	PUNCT
ajrhss-2194	2	20	and	and	CCONJ
ajrhss-2194	2	21	solving	solve	VERB
ajrhss-2194	2	22	geometric	geometric	ADJ
ajrhss-2194	2	23	problems	problem	NOUN
ajrhss-2194	2	24	.	.	PUNCT
ajrhss-2194	3	1	in	in	ADP
ajrhss-2194	3	2	complex	complex	ADJ
ajrhss-2194	3	3	analysis	analysis	NOUN
ajrhss-2194	3	4	,	,	PUNCT
ajrhss-2194	3	5	geometric	geometric	ADJ
ajrhss-2194	3	6	functions	function	NOUN
ajrhss-2194	3	7	such	such	ADJ
ajrhss-2194	3	8	as	as	ADP
ajrhss-2194	3	9	conformal	conformal	ADJ
ajrhss-2194	3	10	mappings	mapping	NOUN
ajrhss-2194	3	11	preserve	preserve	VERB
ajrhss-2194	3	12	angles	angle	NOUN
ajrhss-2194	3	13	and	and	CCONJ
ajrhss-2194	3	14	shapes	shape	NOUN
ajrhss-2194	3	15	,	,	PUNCT
ajrhss-2194	3	16	while	while	SCONJ
ajrhss-2194	3	17	in	in	ADP
ajrhss-2194	3	18	algebraic	algebraic	ADJ
ajrhss-2194	3	19	geometry	geometry	NOUN
ajrhss-2194	3	20	,	,	PUNCT
ajrhss-2194	3	21	they	they	PRON
ajrhss-2194	3	22	define	define	VERB
ajrhss-2194	3	23	curves	curve	NOUN
ajrhss-2194	3	24	and	and	CCONJ
ajrhss-2194	3	25	surfaces	surface	NOUN
ajrhss-2194	3	26	through	through	ADP
ajrhss-2194	3	27	algebraic	algebraic	ADJ
ajrhss-2194	3	28	equations	equation	NOUN
ajrhss-2194	3	29	.	.	PUNCT
ajrhss-2194	4	1	transformations	transformation	NOUN
ajrhss-2194	4	2	like	like	ADP
ajrhss-2194	4	3	rotations	rotation	NOUN
ajrhss-2194	4	4	and	and	CCONJ
ajrhss-2194	4	5	translations	translation	NOUN
ajrhss-2194	4	6	are	be	AUX
ajrhss-2194	4	7	described	describe	VERB
ajrhss-2194	4	8	using	use	VERB
ajrhss-2194	4	9	matrix	matrix	NOUN
ajrhss-2194	4	10	functions	function	NOUN
ajrhss-2194	4	11	.	.	PUNCT
ajrhss-2194	5	1	trigonometric	trigonometric	ADJ
ajrhss-2194	5	2	functions	function	NOUN
ajrhss-2194	5	3	relate	relate	VERB
ajrhss-2194	5	4	angles	angle	NOUN
ajrhss-2194	5	5	to	to	ADP
ajrhss-2194	5	6	side	side	NOUN
ajrhss-2194	5	7	lengths	length	NOUN
ajrhss-2194	5	8	in	in	ADP
ajrhss-2194	5	9	right	right	ADJ
ajrhss-2194	5	10	triangles	triangle	NOUN
ajrhss-2194	5	11	and	and	CCONJ
ajrhss-2194	5	12	are	be	AUX
ajrhss-2194	5	13	crucial	crucial	ADJ
ajrhss-2194	5	14	in	in	ADP
ajrhss-2194	5	15	geometry	geometry	NOUN
ajrhss-2194	5	16	.	.	PUNCT
ajrhss-2194	6	1	geometric	geometric	ADJ
ajrhss-2194	6	2	functions	function	NOUN
ajrhss-2194	6	3	also	also	ADV
ajrhss-2194	6	4	find	find	VERB
ajrhss-2194	6	5	applications	application	NOUN
ajrhss-2194	6	6	in	in	ADP
ajrhss-2194	6	7	computer	computer	NOUN
ajrhss-2194	6	8	-	-	PUNCT
ajrhss-2194	6	9	aided	aid	VERB
ajrhss-2194	6	10	design	design	NOUN
ajrhss-2194	6	11	(	(	PUNCT
ajrhss-2194	6	12	cad	cad	NOUN
ajrhss-2194	6	13	)	)	PUNCT
ajrhss-2194	6	14	and	and	CCONJ
ajrhss-2194	6	15	differential	differential	ADJ
ajrhss-2194	6	16	geometry	geometry	NOUN
ajrhss-2194	6	17	,	,	PUNCT
ajrhss-2194	6	18	where	where	SCONJ
ajrhss-2194	6	19	they	they	PRON
ajrhss-2194	6	20	describe	describe	VERB
ajrhss-2194	6	21	shortest	short	ADJ
ajrhss-2194	6	22	paths	path	NOUN
ajrhss-2194	6	23	and	and	CCONJ
ajrhss-2194	6	24	curvature	curvature	NOUN
ajrhss-2194	6	25	.	.	PUNCT
ajrhss-2194	7	1	this	this	DET
ajrhss-2194	7	2	comprehensive	comprehensive	ADJ
ajrhss-2194	7	3	overview	overview	NOUN
ajrhss-2194	7	4	highlights	highlight	NOUN
ajrhss-2194	7	5	the	the	DET
ajrhss-2194	7	6	diverse	diverse	ADJ
ajrhss-2194	7	7	applications	application	NOUN
ajrhss-2194	7	8	and	and	CCONJ
ajrhss-2194	7	9	significance	significance	NOUN
ajrhss-2194	7	10	of	of	ADP
ajrhss-2194	7	11	geometric	geometric	ADJ
ajrhss-2194	7	12	functions	function	NOUN
ajrhss-2194	7	13	in	in	ADP
ajrhss-2194	7	14	mathematics	mathematic	NOUN
ajrhss-2194	7	15	and	and	CCONJ
ajrhss-2194	7	16	its	its	PRON
ajrhss-2194	7	17	applications	application	NOUN
ajrhss-2194	7	18	.	.	PUNCT
ajrhss-2194	8	1	geometric	geometric	ADJ
ajrhss-2194	8	2	function	function	NOUN
ajrhss-2194	8	3	,	,	PUNCT
ajrhss-2194	8	4	complex	complex	ADJ
ajrhss-2194	8	5	analysis	analysis	NOUN
ajrhss-2194	8	6	,	,	PUNCT
ajrhss-2194	8	7	conformal	conformal	NOUN
ajrhss-2194	8	8	mappings	mapping	NOUN
ajrhss-2194	8	9	,	,	PUNCT
ajrhss-2194	8	10	algebraic	algebraic	ADJ
ajrhss-2194	8	11	geometry	geometry	NOUN
ajrhss-2194	8	12	,	,	PUNCT
ajrhss-2194	8	13	curves	curve	NOUN
ajrhss-2194	8	14	and	and	CCONJ
ajrhss-2194	8	15	surfaces	surface	NOUN
ajrhss-2194	8	16	,	,	PUNCT
ajrhss-2194	8	17	geometric	geometric	ADJ
ajrhss-2194	8	18	transformations	transformation	NOUN
ajrhss-2194	8	19	,	,	PUNCT
ajrhss-2194	8	20	affine	affine	NOUN
ajrhss-2194	8	21	transformations	transformation	NOUN
ajrhss-2194	8	22	,	,	PUNCT
ajrhss-2194	8	23	trigonometric	trigonometric	ADJ
ajrhss-2194	8	24	functions	function	NOUN
ajrhss-2194	8	25	,	,	PUNCT
ajrhss-2194	8	26	distance	distance	NOUN
ajrhss-2194	8	27	formula	formula	NOUN
ajrhss-2194	8	28	,	,	PUNCT
ajrhss-2194	8	29	area	area	NOUN
ajrhss-2194	8	30	and	and	CCONJ
ajrhss-2194	8	31	volume	volume	NOUN
ajrhss-2194	8	32	calculations	calculation	NOUN
ajrhss-2194	8	33	introduction	introduction	NOUN
ajrhss-2194	8	34	a	a	DET
ajrhss-2194	8	35	geometric	geometric	ADJ
ajrhss-2194	8	36	function	function	NOUN
ajrhss-2194	8	37	is	be	AUX
ajrhss-2194	8	38	a	a	DET
ajrhss-2194	8	39	type	type	NOUN
ajrhss-2194	8	40	of	of	ADP
ajrhss-2194	8	41	mathematical	mathematical	ADJ
ajrhss-2194	8	42	function	function	NOUN
ajrhss-2194	8	43	that	that	PRON
ajrhss-2194	8	44	describes	describe	VERB
ajrhss-2194	8	45	a	a	DET
ajrhss-2194	8	46	relationship	relationship	NOUN
ajrhss-2194	8	47	between	between	ADP
ajrhss-2194	8	48	variables	variable	NOUN
ajrhss-2194	8	49	using	use	VERB
ajrhss-2194	8	50	geometric	geometric	ADJ
ajrhss-2194	8	51	principles	principle	NOUN
ajrhss-2194	8	52	or	or	CCONJ
ajrhss-2194	8	53	structures	structure	NOUN
ajrhss-2194	8	54	.	.	PUNCT
ajrhss-2194	9	1	it	it	PRON
ajrhss-2194	9	2	can	can	AUX
ajrhss-2194	9	3	involve	involve	VERB
ajrhss-2194	9	4	various	various	ADJ
ajrhss-2194	9	5	aspects	aspect	NOUN
ajrhss-2194	9	6	of	of	ADP
ajrhss-2194	9	7	geometry	geometry	NOUN
ajrhss-2194	9	8	,	,	PUNCT
ajrhss-2194	9	9	such	such	ADJ
ajrhss-2194	9	10	as	as	ADP
ajrhss-2194	9	11	distances	distance	NOUN
ajrhss-2194	9	12	,	,	PUNCT
ajrhss-2194	9	13	angles	angle	NOUN
ajrhss-2194	9	14	,	,	PUNCT
ajrhss-2194	9	15	shapes	shape	NOUN
ajrhss-2194	9	16	,	,	PUNCT
ajrhss-2194	9	17	and	and	CCONJ
ajrhss-2194	9	18	transformations	transformation	NOUN
ajrhss-2194	9	19	.	.	PUNCT
ajrhss-2194	10	1	here	here	ADV
ajrhss-2194	10	2	are	be	AUX
ajrhss-2194	10	3	a	a	DET
ajrhss-2194	10	4	few	few	ADJ
ajrhss-2194	10	5	contexts	context	NOUN
ajrhss-2194	10	6	in	in	ADP
ajrhss-2194	10	7	which	which	PRON
ajrhss-2194	10	8	the	the	DET
ajrhss-2194	10	9	term	term	NOUN
ajrhss-2194	10	10	"	"	PUNCT
ajrhss-2194	10	11	geometric	geometric	ADJ
ajrhss-2194	10	12	function	function	NOUN
ajrhss-2194	10	13	"	"	PUNCT
ajrhss-2194	10	14	might	might	AUX
ajrhss-2194	10	15	be	be	AUX
ajrhss-2194	10	16	used	use	VERB
ajrhss-2194	10	17	:	:	PUNCT
ajrhss-2194	10	18	1	1	X
ajrhss-2194	10	19	.	.	PUNCT
ajrhss-2194	10	20	complex	complex	ADJ
ajrhss-2194	10	21	analysis	analysis	NOUN
ajrhss-2194	10	22	:	:	PUNCT
ajrhss-2194	10	23	in	in	ADP
ajrhss-2194	10	24	this	this	DET
ajrhss-2194	10	25	context	context	NOUN
ajrhss-2194	10	26	,	,	PUNCT
ajrhss-2194	10	27	geometric	geometric	ADJ
ajrhss-2194	10	28	functions	function	NOUN
ajrhss-2194	10	29	are	be	AUX
ajrhss-2194	10	30	often	often	ADV
ajrhss-2194	10	31	functions	function	NOUN
ajrhss-2194	10	32	of	of	ADP
ajrhss-2194	10	33	a	a	DET
ajrhss-2194	10	34	complex	complex	ADJ
ajrhss-2194	10	35	variable	variable	NOUN
ajrhss-2194	10	36	that	that	PRON
ajrhss-2194	10	37	exhibit	exhibit	VERB
ajrhss-2194	10	38	specific	specific	ADJ
ajrhss-2194	10	39	geometric	geometric	ADJ
ajrhss-2194	10	40	properties	property	NOUN
ajrhss-2194	10	41	.	.	PUNCT
ajrhss-2194	11	1	for	for	ADP
ajrhss-2194	11	2	example	example	NOUN
ajrhss-2194	11	3	,	,	PUNCT
ajrhss-2194	11	4	conformal	conformal	ADJ
ajrhss-2194	11	5	mappings	mapping	NOUN
ajrhss-2194	11	6	preserve	preserve	VERB
ajrhss-2194	11	7	angles	angle	NOUN
ajrhss-2194	11	8	and	and	CCONJ
ajrhss-2194	11	9	are	be	AUX
ajrhss-2194	11	10	an	an	DET
ajrhss-2194	11	11	important	important	ADJ
ajrhss-2194	11	12	type	type	NOUN
ajrhss-2194	11	13	of	of	ADP
ajrhss-2194	11	14	geometric	geometric	ADJ
ajrhss-2194	11	15	function	function	NOUN
ajrhss-2194	11	16	in	in	ADP
ajrhss-2194	11	17	this	this	DET
ajrhss-2194	11	18	field	field	NOUN
ajrhss-2194	11	19	.	.	PUNCT
ajrhss-2194	12	1	2	2	X
ajrhss-2194	12	2	.	.	X
ajrhss-2194	12	3	algebraic	algebraic	ADJ
ajrhss-2194	12	4	geometry	geometry	NOUN
ajrhss-2194	12	5	:	:	PUNCT
ajrhss-2194	12	6	geometric	geometric	ADJ
ajrhss-2194	12	7	functions	function	NOUN
ajrhss-2194	12	8	can	can	AUX
ajrhss-2194	12	9	describe	describe	VERB
ajrhss-2194	12	10	algebraic	algebraic	ADJ
ajrhss-2194	12	11	curves	curve	NOUN
ajrhss-2194	12	12	and	and	CCONJ
ajrhss-2194	12	13	surfaces	surface	NOUN
ajrhss-2194	12	14	.	.	PUNCT
ajrhss-2194	13	1	these	these	DET
ajrhss-2194	13	2	functions	function	NOUN
ajrhss-2194	13	3	define	define	VERB
ajrhss-2194	13	4	geometric	geometric	ADJ
ajrhss-2194	13	5	objects	object	NOUN
ajrhss-2194	13	6	and	and	CCONJ
ajrhss-2194	13	7	their	their	PRON
ajrhss-2194	13	8	properties	property	NOUN
ajrhss-2194	13	9	in	in	ADP
ajrhss-2194	13	10	terms	term	NOUN
ajrhss-2194	13	11	of	of	ADP
ajrhss-2194	13	12	algebraic	algebraic	ADJ
ajrhss-2194	13	13	equations	equation	NOUN
ajrhss-2194	13	14	.	.	PUNCT
ajrhss-2194	14	1	3	3	X
ajrhss-2194	14	2	.	.	X
ajrhss-2194	14	3	transformations	transformation	NOUN
ajrhss-2194	14	4	:	:	PUNCT
ajrhss-2194	14	5	functions	function	NOUN
ajrhss-2194	14	6	that	that	PRON
ajrhss-2194	14	7	perform	perform	VERB
ajrhss-2194	14	8	geometric	geometric	ADJ
ajrhss-2194	14	9	transformations	transformation	NOUN
ajrhss-2194	14	10	,	,	PUNCT
ajrhss-2194	14	11	such	such	ADJ
ajrhss-2194	14	12	as	as	ADP
ajrhss-2194	14	13	rotations	rotation	NOUN
ajrhss-2194	14	14	,	,	PUNCT
ajrhss-2194	14	15	translations	translation	NOUN
ajrhss-2194	14	16	,	,	PUNCT
ajrhss-2194	14	17	scalings	scaling	NOUN
ajrhss-2194	14	18	,	,	PUNCT
ajrhss-2194	14	19	and	and	CCONJ
ajrhss-2194	14	20	reflections	reflection	NOUN
ajrhss-2194	14	21	.	.	PUNCT
ajrhss-2194	15	1	these	these	PRON
ajrhss-2194	15	2	are	be	AUX
ajrhss-2194	15	3	often	often	ADV
ajrhss-2194	15	4	represented	represent	VERB
ajrhss-2194	15	5	using	use	VERB
ajrhss-2194	15	6	matrices	matrix	NOUN
ajrhss-2194	15	7	and	and	CCONJ
ajrhss-2194	15	8	are	be	AUX
ajrhss-2194	15	9	fundamental	fundamental	ADJ
ajrhss-2194	15	10	in	in	ADP
ajrhss-2194	15	11	fields	field	NOUN
ajrhss-2194	15	12	like	like	ADP
ajrhss-2194	15	13	computer	computer	NOUN
ajrhss-2194	15	14	graphics	graphic	NOUN
ajrhss-2194	15	15	and	and	CCONJ
ajrhss-2194	15	16	robotics	robotic	NOUN
ajrhss-2194	15	17	.	.	PUNCT
ajrhss-2194	16	1	4	4	X
ajrhss-2194	16	2	.	.	X
ajrhss-2194	16	3	geometric	geometric	ADJ
ajrhss-2194	16	4	sequences	sequence	NOUN
ajrhss-2194	16	5	and	and	CCONJ
ajrhss-2194	16	6	series	series	NOUN
ajrhss-2194	16	7	:	:	PUNCT
ajrhss-2194	16	8	sometimes	sometimes	ADV
ajrhss-2194	16	9	,	,	PUNCT
ajrhss-2194	16	10	the	the	DET
ajrhss-2194	16	11	term	term	NOUN
ajrhss-2194	16	12	may	may	AUX
ajrhss-2194	16	13	loosely	loosely	ADV
ajrhss-2194	16	14	refer	refer	VERB
ajrhss-2194	16	15	to	to	ADP
ajrhss-2194	16	16	sequences	sequence	NOUN
ajrhss-2194	16	17	or	or	CCONJ
ajrhss-2194	16	18	series	series	NOUN
ajrhss-2194	16	19	where	where	SCONJ
ajrhss-2194	16	20	each	each	DET
ajrhss-2194	16	21	term	term	NOUN
ajrhss-2194	16	22	is	be	AUX
ajrhss-2194	16	23	a	a	DET
ajrhss-2194	16	24	constant	constant	ADJ
ajrhss-2194	16	25	multiple	multiple	NOUN
ajrhss-2194	16	26	of	of	ADP
ajrhss-2194	16	27	the	the	DET
ajrhss-2194	16	28	previous	previous	ADJ
ajrhss-2194	16	29	one	one	NUM
ajrhss-2194	16	30	.	.	PUNCT
ajrhss-2194	17	1	while	while	SCONJ
ajrhss-2194	17	2	not	not	PART
ajrhss-2194	17	3	strictly	strictly	ADV
ajrhss-2194	17	4	"	"	PUNCT
ajrhss-2194	17	5	functions	function	NOUN
ajrhss-2194	17	6	"	"	PUNCT
ajrhss-2194	17	7	in	in	ADP
ajrhss-2194	17	8	the	the	DET
ajrhss-2194	17	9	conventional	conventional	ADJ
ajrhss-2194	17	10	sense	sense	NOUN
ajrhss-2194	17	11	,	,	PUNCT
ajrhss-2194	17	12	they	they	PRON
ajrhss-2194	17	13	exhibit	exhibit	VERB
ajrhss-2194	17	14	geometric	geometric	ADJ
ajrhss-2194	17	15	growth	growth	NOUN
ajrhss-2194	17	16	properties	property	NOUN
ajrhss-2194	17	17	.	.	PUNCT
ajrhss-2194	18	1	american	american	ADJ
ajrhss-2194	18	2	journal	journal	PROPN
ajrhss-2194	18	3	of	of	ADP
ajrhss-2194	18	4	research	research	NOUN
ajrhss-2194	18	5	in	in	ADP
ajrhss-2194	18	6	humanities	humanity	NOUN
ajrhss-2194	18	7	and	and	CCONJ
ajrhss-2194	18	8	social	social	ADJ
ajrhss-2194	18	9	sciences	science	NOUN
ajrhss-2194	18	10	volume	volume	NOUN
ajrhss-2194	18	11	25	25	NUM
ajrhss-2194	18	12	june	june	PROPN
ajrhss-2194	18	13	2024	2024	NUM
ajrhss-2194	18	14	p	p	NOUN
ajrhss-2194	18	15	a	a	DET
ajrhss-2194	18	16	g	g	NOUN
ajrhss-2194	18	17	e	e	NOUN
ajrhss-2194	19	1	|	|	ADV
ajrhss-2194	19	2	42	42	NUM
ajrhss-2194	19	3	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2194	19	4	5	5	NUM
ajrhss-2194	19	5	.	.	PUNCT
ajrhss-2194	19	6	trigonometric	trigonometric	ADJ
ajrhss-2194	19	7	functions	function	NOUN
ajrhss-2194	19	8	:	:	PUNCT
ajrhss-2194	19	9	these	these	DET
ajrhss-2194	19	10	functions	function	NOUN
ajrhss-2194	19	11	(	(	PUNCT
ajrhss-2194	19	12	sine	sine	NOUN
ajrhss-2194	19	13	,	,	PUNCT
ajrhss-2194	19	14	cosine	cosine	NOUN
ajrhss-2194	19	15	,	,	PUNCT
ajrhss-2194	19	16	tangent	tangent	NOUN
ajrhss-2194	19	17	,	,	PUNCT
ajrhss-2194	19	18	etc	etc	X
ajrhss-2194	19	19	.	.	X
ajrhss-2194	19	20	)	)	PUNCT
ajrhss-2194	19	21	are	be	AUX
ajrhss-2194	19	22	also	also	ADV
ajrhss-2194	19	23	sometimes	sometimes	ADV
ajrhss-2194	19	24	considered	consider	VERB
ajrhss-2194	19	25	geometric	geometric	ADJ
ajrhss-2194	19	26	functions	function	NOUN
ajrhss-2194	19	27	because	because	SCONJ
ajrhss-2194	19	28	they	they	PRON
ajrhss-2194	19	29	relate	relate	VERB
ajrhss-2194	19	30	angles	angle	NOUN
ajrhss-2194	19	31	of	of	ADP
ajrhss-2194	19	32	a	a	DET
ajrhss-2194	19	33	triangle	triangle	NOUN
ajrhss-2194	19	34	to	to	ADP
ajrhss-2194	19	35	the	the	DET
ajrhss-2194	19	36	ratios	ratio	NOUN
ajrhss-2194	19	37	of	of	ADP
ajrhss-2194	19	38	its	its	PRON
ajrhss-2194	19	39	sides	side	NOUN
ajrhss-2194	19	40	and	and	CCONJ
ajrhss-2194	19	41	have	have	VERB
ajrhss-2194	19	42	important	important	ADJ
ajrhss-2194	19	43	applications	application	NOUN
ajrhss-2194	19	44	in	in	ADP
ajrhss-2194	19	45	geometry	geometry	NOUN
ajrhss-2194	19	46	.	.	PUNCT
ajrhss-2194	20	1	overall	overall	ADV
ajrhss-2194	20	2	,	,	PUNCT
ajrhss-2194	20	3	a	a	DET
ajrhss-2194	20	4	geometric	geometric	ADJ
ajrhss-2194	20	5	function	function	NOUN
ajrhss-2194	20	6	is	be	AUX
ajrhss-2194	20	7	any	any	DET
ajrhss-2194	20	8	function	function	NOUN
ajrhss-2194	20	9	that	that	PRON
ajrhss-2194	20	10	inherently	inherently	ADV
ajrhss-2194	20	11	relies	rely	VERB
ajrhss-2194	20	12	on	on	ADP
ajrhss-2194	20	13	or	or	CCONJ
ajrhss-2194	20	14	describes	describe	VERB
ajrhss-2194	20	15	geometric	geometric	ADJ
ajrhss-2194	20	16	concepts	concept	NOUN
ajrhss-2194	20	17	and	and	CCONJ
ajrhss-2194	20	18	relationships	relationship	NOUN
ajrhss-2194	20	19	.	.	PUNCT
ajrhss-2194	21	1	here	here	ADV
ajrhss-2194	21	2	are	be	AUX
ajrhss-2194	21	3	more	more	ADV
ajrhss-2194	21	4	detailed	detailed	ADJ
ajrhss-2194	21	5	explanations	explanation	NOUN
ajrhss-2194	21	6	and	and	CCONJ
ajrhss-2194	21	7	examples	example	NOUN
ajrhss-2194	21	8	of	of	ADP
ajrhss-2194	21	9	various	various	ADJ
ajrhss-2194	21	10	types	type	NOUN
ajrhss-2194	21	11	of	of	ADP
ajrhss-2194	21	12	geometric	geometric	ADJ
ajrhss-2194	21	13	functions	function	NOUN
ajrhss-2194	21	14	:	:	PUNCT
ajrhss-2194	21	15	1	1	X
ajrhss-2194	21	16	.	.	PUNCT
ajrhss-2194	21	17	complex	complex	ADJ
ajrhss-2194	21	18	analysis	analysis	NOUN
ajrhss-2194	21	19	in	in	ADP
ajrhss-2194	21	20	complex	complex	ADJ
ajrhss-2194	21	21	analysis	analysis	NOUN
ajrhss-2194	21	22	,	,	PUNCT
ajrhss-2194	21	23	geometric	geometric	ADJ
ajrhss-2194	21	24	functions	function	NOUN
ajrhss-2194	21	25	are	be	AUX
ajrhss-2194	21	26	often	often	ADV
ajrhss-2194	21	27	used	use	VERB
ajrhss-2194	21	28	to	to	PART
ajrhss-2194	21	29	describe	describe	VERB
ajrhss-2194	21	30	and	and	CCONJ
ajrhss-2194	21	31	analyze	analyze	VERB
ajrhss-2194	21	32	the	the	DET
ajrhss-2194	21	33	properties	property	NOUN
ajrhss-2194	21	34	of	of	ADP
ajrhss-2194	21	35	complex	complex	ADJ
ajrhss-2194	21	36	variables	variable	NOUN
ajrhss-2194	21	37	.	.	PUNCT
ajrhss-2194	22	1	one	one	NUM
ajrhss-2194	22	2	of	of	ADP
ajrhss-2194	22	3	the	the	DET
ajrhss-2194	22	4	key	key	ADJ
ajrhss-2194	22	5	types	type	NOUN
ajrhss-2194	22	6	of	of	ADP
ajrhss-2194	22	7	geometric	geometric	ADJ
ajrhss-2194	22	8	functions	function	NOUN
ajrhss-2194	22	9	here	here	ADV
ajrhss-2194	22	10	is	be	AUX
ajrhss-2194	22	11	conformal	conformal	ADJ
ajrhss-2194	22	12	mappings	mapping	NOUN
ajrhss-2194	22	13	:	:	PUNCT
ajrhss-2194	22	14	conformal	conformal	NOUN
ajrhss-2194	22	15	mappings	mapping	NOUN
ajrhss-2194	22	16	:	:	PUNCT
ajrhss-2194	22	17	these	these	PRON
ajrhss-2194	22	18	are	be	AUX
ajrhss-2194	22	19	functions	function	NOUN
ajrhss-2194	22	20	that	that	PRON
ajrhss-2194	22	21	locally	locally	ADV
ajrhss-2194	22	22	preserve	preserve	VERB
ajrhss-2194	22	23	angles	angle	NOUN
ajrhss-2194	22	24	and	and	CCONJ
ajrhss-2194	22	25	shapes	shape	NOUN
ajrhss-2194	22	26	.	.	PUNCT
ajrhss-2194	23	1	an	an	DET
ajrhss-2194	23	2	example	example	NOUN
ajrhss-2194	23	3	is	be	AUX
ajrhss-2194	23	4	the	the	DET
ajrhss-2194	23	5	complex	complex	ADJ
ajrhss-2194	23	6	function	function	NOUN
ajrhss-2194	23	7	\	\	PROPN
ajrhss-2194	23	8	(	(	PUNCT
ajrhss-2194	23	9	f(z	f(z	PROPN
ajrhss-2194	23	10	)	)	PUNCT
ajrhss-2194	23	11	=	=	PUNCT
ajrhss-2194	24	1	z^2	z^2	NUM
ajrhss-2194	24	2	\	\	PROPN
ajrhss-2194	24	3	)	)	PUNCT
ajrhss-2194	24	4	,	,	PUNCT
ajrhss-2194	24	5	which	which	PRON
ajrhss-2194	24	6	maps	map	VERB
ajrhss-2194	24	7	a	a	DET
ajrhss-2194	24	8	complex	complex	ADJ
ajrhss-2194	24	9	number	number	NOUN
ajrhss-2194	24	10	\	\	PROPN
ajrhss-2194	24	11	(	(	PUNCT
ajrhss-2194	24	12	z	z	NOUN
ajrhss-2194	24	13	\	\	NOUN
ajrhss-2194	24	14	)	)	PUNCT
ajrhss-2194	24	15	to	to	ADP
ajrhss-2194	24	16	its	its	PRON
ajrhss-2194	24	17	square	square	NOUN
ajrhss-2194	24	18	.	.	PUNCT
ajrhss-2194	25	1	conformal	conformal	ADJ
ajrhss-2194	25	2	mappings	mapping	NOUN
ajrhss-2194	25	3	are	be	AUX
ajrhss-2194	25	4	used	use	VERB
ajrhss-2194	25	5	in	in	ADP
ajrhss-2194	25	6	fields	field	NOUN
ajrhss-2194	25	7	like	like	ADP
ajrhss-2194	25	8	fluid	fluid	NOUN
ajrhss-2194	25	9	dynamics	dynamic	NOUN
ajrhss-2194	25	10	and	and	CCONJ
ajrhss-2194	25	11	electrostatics	electrostatic	NOUN
ajrhss-2194	25	12	to	to	PART
ajrhss-2194	25	13	solve	solve	VERB
ajrhss-2194	25	14	problems	problem	NOUN
ajrhss-2194	25	15	by	by	ADP
ajrhss-2194	25	16	transforming	transform	VERB
ajrhss-2194	25	17	complex	complex	ADJ
ajrhss-2194	25	18	shapes	shape	NOUN
ajrhss-2194	25	19	into	into	ADP
ajrhss-2194	25	20	simpler	simple	ADJ
ajrhss-2194	25	21	ones	one	NOUN
ajrhss-2194	25	22	.	.	PUNCT
ajrhss-2194	26	1	2	2	X
ajrhss-2194	26	2	.	.	X
ajrhss-2194	26	3	algebraic	algebraic	ADJ
ajrhss-2194	26	4	geometry	geometry	NOUN
ajrhss-2194	26	5	in	in	ADP
ajrhss-2194	26	6	algebraic	algebraic	ADJ
ajrhss-2194	26	7	geometry	geometry	NOUN
ajrhss-2194	26	8	,	,	PUNCT
ajrhss-2194	26	9	geometric	geometric	ADJ
ajrhss-2194	26	10	functions	function	NOUN
ajrhss-2194	26	11	are	be	AUX
ajrhss-2194	26	12	used	use	VERB
ajrhss-2194	26	13	to	to	PART
ajrhss-2194	26	14	define	define	VERB
ajrhss-2194	26	15	algebraic	algebraic	ADJ
ajrhss-2194	26	16	varieties	variety	NOUN
ajrhss-2194	26	17	,	,	PUNCT
ajrhss-2194	26	18	which	which	PRON
ajrhss-2194	26	19	are	be	AUX
ajrhss-2194	26	20	the	the	DET
ajrhss-2194	26	21	sets	set	NOUN
ajrhss-2194	26	22	of	of	ADP
ajrhss-2194	26	23	solutions	solution	NOUN
ajrhss-2194	26	24	to	to	ADP
ajrhss-2194	26	25	systems	system	NOUN
ajrhss-2194	26	26	of	of	ADP
ajrhss-2194	26	27	polynomial	polynomial	ADJ
ajrhss-2194	26	28	equations	equation	NOUN
ajrhss-2194	26	29	.	.	PUNCT
ajrhss-2194	27	1	algebraic	algebraic	ADJ
ajrhss-2194	27	2	curves	curve	NOUN
ajrhss-2194	27	3	and	and	CCONJ
ajrhss-2194	27	4	surfaces	surface	NOUN
ajrhss-2194	27	5	:	:	PUNCT
ajrhss-2194	27	6	for	for	ADP
ajrhss-2194	27	7	example	example	NOUN
ajrhss-2194	27	8	,	,	PUNCT
ajrhss-2194	27	9	the	the	DET
ajrhss-2194	27	10	function	function	NOUN
ajrhss-2194	27	11	\	\	PROPN
ajrhss-2194	27	12	(	(	PUNCT
ajrhss-2194	27	13	f(x	f(x	PROPN
ajrhss-2194	27	14	,	,	PUNCT
ajrhss-2194	27	15	y	y	PROPN
ajrhss-2194	27	16	)	)	PUNCT
ajrhss-2194	27	17	=	=	PUNCT
ajrhss-2194	28	1	x^2	x^2	PROPN
ajrhss-2194	29	1	+	+	CCONJ
ajrhss-2194	29	2	y^2	y^2	VERB
ajrhss-2194	29	3	1	1	NUM
ajrhss-2194	29	4	\	\	NOUN
ajrhss-2194	29	5	)	)	PUNCT
ajrhss-2194	29	6	defines	define	VERB
ajrhss-2194	29	7	a	a	DET
ajrhss-2194	29	8	circle	circle	NOUN
ajrhss-2194	29	9	in	in	ADP
ajrhss-2194	29	10	the	the	DET
ajrhss-2194	29	11	plane	plane	NOUN
ajrhss-2194	29	12	.	.	PUNCT
ajrhss-2194	30	1	in	in	ADP
ajrhss-2194	30	2	three	three	NUM
ajrhss-2194	30	3	dimensions	dimension	NOUN
ajrhss-2194	30	4	,	,	PUNCT
ajrhss-2194	30	5	a	a	DET
ajrhss-2194	30	6	function	function	NOUN
ajrhss-2194	30	7	like	like	ADP
ajrhss-2194	30	8	\	\	PROPN
ajrhss-2194	30	9	(	(	PUNCT
ajrhss-2194	30	10	g(x	g(x	PROPN
ajrhss-2194	30	11	,	,	PUNCT
ajrhss-2194	30	12	y	y	PROPN
ajrhss-2194	30	13	,	,	PUNCT
ajrhss-2194	30	14	z	z	NOUN
ajrhss-2194	30	15	)	)	PUNCT
ajrhss-2194	30	16	=	=	SYM
ajrhss-2194	31	1	x^2	x^2	PROPN
ajrhss-2194	32	1	+	+	CCONJ
ajrhss-2194	32	2	y^2	y^2	PROPN
ajrhss-2194	32	3	+	+	CCONJ
ajrhss-2194	32	4	z^2	z^2	PROPN
ajrhss-2194	32	5	1	1	NUM
ajrhss-2194	32	6	\	\	NOUN
ajrhss-2194	32	7	)	)	PUNCT
ajrhss-2194	32	8	defines	define	VERB
ajrhss-2194	32	9	a	a	DET
ajrhss-2194	32	10	sphere	sphere	NOUN
ajrhss-2194	32	11	.	.	PUNCT
ajrhss-2194	33	1	these	these	DET
ajrhss-2194	33	2	geometric	geometric	ADJ
ajrhss-2194	33	3	objects	object	NOUN
ajrhss-2194	33	4	are	be	AUX
ajrhss-2194	33	5	studied	study	VERB
ajrhss-2194	33	6	to	to	PART
ajrhss-2194	33	7	understand	understand	VERB
ajrhss-2194	33	8	their	their	PRON
ajrhss-2194	33	9	properties	property	NOUN
ajrhss-2194	33	10	and	and	CCONJ
ajrhss-2194	33	11	relationships	relationship	NOUN
ajrhss-2194	33	12	.	.	PUNCT
ajrhss-2194	34	1	3	3	X
ajrhss-2194	34	2	.	.	X
ajrhss-2194	34	3	geometric	geometric	ADJ
ajrhss-2194	34	4	transformations	transformation	NOUN
ajrhss-2194	34	5	functions	function	NOUN
ajrhss-2194	34	6	that	that	PRON
ajrhss-2194	34	7	perform	perform	VERB
ajrhss-2194	34	8	geometric	geometric	ADJ
ajrhss-2194	34	9	transformations	transformation	NOUN
ajrhss-2194	34	10	are	be	AUX
ajrhss-2194	34	11	fundamental	fundamental	ADJ
ajrhss-2194	34	12	in	in	ADP
ajrhss-2194	34	13	many	many	ADJ
ajrhss-2194	34	14	areas	area	NOUN
ajrhss-2194	34	15	,	,	PUNCT
ajrhss-2194	34	16	especially	especially	ADV
ajrhss-2194	34	17	in	in	ADP
ajrhss-2194	34	18	computer	computer	NOUN
ajrhss-2194	34	19	graphics	graphic	NOUN
ajrhss-2194	34	20	and	and	CCONJ
ajrhss-2194	34	21	robotics	robotic	NOUN
ajrhss-2194	34	22	.	.	PUNCT
ajrhss-2194	35	1	transformation	transformation	NOUN
ajrhss-2194	35	2	matrices	matrix	NOUN
ajrhss-2194	35	3	:	:	PUNCT
ajrhss-2194	35	4	a	a	DET
ajrhss-2194	35	5	rotation	rotation	NOUN
ajrhss-2194	35	6	of	of	ADP
ajrhss-2194	35	7	a	a	DET
ajrhss-2194	35	8	point	point	NOUN
ajrhss-2194	35	9	\	\	PROPN
ajrhss-2194	35	10	(	(	PUNCT
ajrhss-2194	35	11	(	(	PUNCT
ajrhss-2194	35	12	x	x	NOUN
ajrhss-2194	35	13	,	,	PUNCT
ajrhss-2194	35	14	y	y	NOUN
ajrhss-2194	35	15	)	)	PUNCT
ajrhss-2194	35	16	\	\	NOUN
ajrhss-2194	35	17	)	)	PUNCT
ajrhss-2194	35	18	in	in	ADP
ajrhss-2194	35	19	the	the	DET
ajrhss-2194	35	20	plane	plane	NOUN
ajrhss-2194	35	21	by	by	ADP
ajrhss-2194	35	22	an	an	DET
ajrhss-2194	35	23	angle	angle	NOUN
ajrhss-2194	35	24	\	\	PROPN
ajrhss-2194	35	25	(	(	PUNCT
ajrhss-2194	35	26	\theta	\theta	PROPN
ajrhss-2194	35	27	\	\	PROPN
ajrhss-2194	35	28	)	)	PUNCT
ajrhss-2194	35	29	can	can	AUX
ajrhss-2194	35	30	be	be	AUX
ajrhss-2194	35	31	represented	represent	VERB
ajrhss-2194	35	32	by	by	ADP
ajrhss-2194	35	33	the	the	DET
ajrhss-2194	35	34	function	function	NOUN
ajrhss-2194	35	35	:	:	PUNCT
ajrhss-2194	36	1	\	\	PROPN
ajrhss-2194	36	2	[	[	PUNCT
ajrhss-2194	36	3	\begin{pmatrix	\begin{pmatrix	NOUN
ajrhss-2194	36	4	}	}	PUNCT
ajrhss-2194	36	5	x	x	NOUN
ajrhss-2194	36	6	'	'	PART
ajrhss-2194	36	7	\\	\\	NOUN
ajrhss-2194	36	8	y	y	NOUN
ajrhss-2194	36	9	'	'	PUNCT
ajrhss-2194	36	10	\end{pmatrix	\end{pmatrix	NOUN
ajrhss-2194	36	11	}	}	PUNCT
ajrhss-2194	36	12	=	=	SYM
ajrhss-2194	36	13	\begin{pmatrix	\begin{pmatrix	ADJ
ajrhss-2194	36	14	}	}	PUNCT
ajrhss-2194	36	15	\cos	\cos	PROPN
ajrhss-2194	36	16	\theta	\theta	PROPN
ajrhss-2194	36	17	&	&	CCONJ
ajrhss-2194	36	18	-\sin	-\sin	PROPN
ajrhss-2194	36	19	\theta	\theta	PROPN
ajrhss-2194	36	20	\\	\\	NOUN
ajrhss-2194	36	21	\sin	\sin	PROPN
ajrhss-2194	36	22	\theta	\theta	PROPN
ajrhss-2194	36	23	&	&	CCONJ
ajrhss-2194	36	24	\cos	\cos	PROPN
ajrhss-2194	36	25	\theta\end{pmatrix	\theta\end{pmatrix	PROPN
ajrhss-2194	36	26	}	}	PUNCT
ajrhss-2194	36	27	\begin{pmatrix	\begin{pmatrix	NOUN
ajrhss-2194	36	28	}	}	PUNCT
ajrhss-2194	36	29	x	x	SYM
ajrhss-2194	36	30	\\	\\	NOUN
ajrhss-2194	36	31	y	y	PROPN
ajrhss-2194	36	32	\end{pmatrix}\	\end{pmatrix}\	PROPN
ajrhss-2194	36	33	]	]	PUNCT
ajrhss-2194	36	34	this	this	DET
ajrhss-2194	36	35	matrix	matrix	NOUN
ajrhss-2194	36	36	function	function	NOUN
ajrhss-2194	36	37	rotates	rotate	VERB
ajrhss-2194	36	38	the	the	DET
ajrhss-2194	36	39	point	point	NOUN
ajrhss-2194	36	40	around	around	ADP
ajrhss-2194	36	41	the	the	DET
ajrhss-2194	36	42	origin	origin	NOUN
ajrhss-2194	36	43	by	by	ADP
ajrhss-2194	36	44	\	\	PROPN
ajrhss-2194	36	45	(	(	PUNCT
ajrhss-2194	36	46	\theta	\theta	PROPN
ajrhss-2194	36	47	\	\	ADJ
ajrhss-2194	36	48	)	)	PUNCT
ajrhss-2194	36	49	degrees	degree	NOUN
ajrhss-2194	36	50	.	.	PUNCT
ajrhss-2194	37	1	4	4	X
ajrhss-2194	37	2	.	.	X
ajrhss-2194	37	3	geometric	geometric	ADJ
ajrhss-2194	37	4	sequences	sequence	NOUN
ajrhss-2194	37	5	and	and	CCONJ
ajrhss-2194	37	6	series	series	NOUN
ajrhss-2194	37	7	while	while	SCONJ
ajrhss-2194	37	8	not	not	PART
ajrhss-2194	37	9	strictly	strictly	ADV
ajrhss-2194	37	10	functions	function	NOUN
ajrhss-2194	37	11	in	in	ADP
ajrhss-2194	37	12	the	the	DET
ajrhss-2194	37	13	conventional	conventional	ADJ
ajrhss-2194	37	14	sense	sense	NOUN
ajrhss-2194	37	15	,	,	PUNCT
ajrhss-2194	37	16	geometric	geometric	ADJ
ajrhss-2194	37	17	sequences	sequence	NOUN
ajrhss-2194	37	18	and	and	CCONJ
ajrhss-2194	37	19	series	series	NOUN
ajrhss-2194	37	20	exhibit	exhibit	NOUN
ajrhss-2194	37	21	properties	property	NOUN
ajrhss-2194	37	22	that	that	PRON
ajrhss-2194	37	23	can	can	AUX
ajrhss-2194	37	24	be	be	AUX
ajrhss-2194	37	25	described	describe	VERB
ajrhss-2194	37	26	functionally	functionally	ADV
ajrhss-2194	37	27	.	.	PUNCT
ajrhss-2194	38	1	geometric	geometric	ADJ
ajrhss-2194	38	2	sequences	sequence	NOUN
ajrhss-2194	38	3	:	:	PUNCT
ajrhss-2194	38	4	a	a	DET
ajrhss-2194	38	5	geometric	geometric	ADJ
ajrhss-2194	38	6	sequence	sequence	NOUN
ajrhss-2194	38	7	is	be	AUX
ajrhss-2194	38	8	a	a	DET
ajrhss-2194	38	9	sequence	sequence	NOUN
ajrhss-2194	38	10	of	of	ADP
ajrhss-2194	38	11	numbers	number	NOUN
ajrhss-2194	38	12	where	where	SCONJ
ajrhss-2194	38	13	each	each	DET
ajrhss-2194	38	14	term	term	NOUN
ajrhss-2194	38	15	after	after	SCONJ
ajrhss-2194	38	16	the	the	DET
ajrhss-2194	38	17	first	first	ADJ
ajrhss-2194	38	18	is	be	AUX
ajrhss-2194	38	19	found	find	VERB
ajrhss-2194	38	20	by	by	ADP
ajrhss-2194	38	21	multiplying	multiply	VERB
ajrhss-2194	38	22	the	the	DET
ajrhss-2194	38	23	previous	previous	ADJ
ajrhss-2194	38	24	term	term	NOUN
ajrhss-2194	38	25	by	by	ADP
ajrhss-2194	38	26	a	a	DET
ajrhss-2194	38	27	constant	constant	ADJ
ajrhss-2194	38	28	ratio	ratio	NOUN
ajrhss-2194	38	29	\	\	PROPN
ajrhss-2194	38	30	(	(	PUNCT
ajrhss-2194	38	31	r	r	NOUN
ajrhss-2194	38	32	\	\	PROPN
ajrhss-2194	38	33	)	)	PUNCT
ajrhss-2194	38	34	.	.	PUNCT
ajrhss-2194	39	1	for	for	ADP
ajrhss-2194	39	2	example	example	NOUN
ajrhss-2194	39	3	,	,	PUNCT
ajrhss-2194	39	4	the	the	DET
ajrhss-2194	39	5	sequence	sequence	NOUN
ajrhss-2194	39	6	\	\	PROPN
ajrhss-2194	39	7	(	(	PUNCT
ajrhss-2194	39	8	a	a	PRON
ajrhss-2194	39	9	,	,	PUNCT
ajrhss-2194	39	10	ar	ar	NOUN
ajrhss-2194	39	11	,	,	PUNCT
ajrhss-2194	39	12	ar^2	ar^2	NOUN
ajrhss-2194	39	13	,	,	PUNCT
ajrhss-2194	39	14	ar^3	ar^3	NOUN
ajrhss-2194	39	15	,	,	PUNCT
ajrhss-2194	39	16	\	\	PROPN
ajrhss-2194	39	17	...	...	PUNCT
ajrhss-2194	39	18	\	\	NOUN
ajrhss-2194	39	19	)	)	PUNCT
ajrhss-2194	39	20	can	can	AUX
ajrhss-2194	39	21	be	be	AUX
ajrhss-2194	39	22	described	describe	VERB
ajrhss-2194	39	23	by	by	ADP
ajrhss-2194	39	24	the	the	DET
ajrhss-2194	39	25	function	function	NOUN
ajrhss-2194	39	26	\	\	PROPN
ajrhss-2194	39	27	(	(	PUNCT
ajrhss-2194	39	28	a_n	a_n	SYM
ajrhss-2194	39	29	=	=	SYM
ajrhss-2194	39	30	ar^{n-1	ar^{n-1	NOUN
ajrhss-2194	39	31	}	}	PUNCT
ajrhss-2194	39	32	\	\	NOUN
ajrhss-2194	39	33	)	)	PUNCT
ajrhss-2194	39	34	,	,	PUNCT
ajrhss-2194	39	35	where	where	SCONJ
ajrhss-2194	39	36	\	\	PROPN
ajrhss-2194	39	37	(	(	PUNCT
ajrhss-2194	39	38	a	a	DET
ajrhss-2194	39	39	\	\	NOUN
ajrhss-2194	39	40	)	)	PUNCT
ajrhss-2194	39	41	is	be	AUX
ajrhss-2194	39	42	the	the	DET
ajrhss-2194	39	43	initial	initial	ADJ
ajrhss-2194	39	44	term	term	NOUN
ajrhss-2194	39	45	and	and	CCONJ
ajrhss-2194	39	46	\	\	PROPN
ajrhss-2194	39	47	(	(	PUNCT
ajrhss-2194	39	48	r	r	NOUN
ajrhss-2194	39	49	\	\	NOUN
ajrhss-2194	39	50	)	)	PUNCT
ajrhss-2194	39	51	is	be	AUX
ajrhss-2194	39	52	the	the	DET
ajrhss-2194	39	53	common	common	ADJ
ajrhss-2194	39	54	ratio	ratio	NOUN
ajrhss-2194	39	55	.	.	PUNCT
ajrhss-2194	40	1	5	5	X
ajrhss-2194	40	2	.	.	X
ajrhss-2194	40	3	trigonometric	trigonometric	NOUN
ajrhss-2194	40	4	functions	function	NOUN
ajrhss-2194	40	5	trigonometric	trigonometric	ADJ
ajrhss-2194	40	6	functions	function	NOUN
ajrhss-2194	40	7	,	,	PUNCT
ajrhss-2194	40	8	which	which	PRON
ajrhss-2194	40	9	relate	relate	VERB
ajrhss-2194	40	10	angles	angle	NOUN
ajrhss-2194	40	11	to	to	ADP
ajrhss-2194	40	12	side	side	NOUN
ajrhss-2194	40	13	lengths	length	NOUN
ajrhss-2194	40	14	in	in	ADP
ajrhss-2194	40	15	right	right	ADJ
ajrhss-2194	40	16	triangles	triangle	NOUN
ajrhss-2194	40	17	,	,	PUNCT
ajrhss-2194	40	18	are	be	AUX
ajrhss-2194	40	19	essential	essential	ADJ
ajrhss-2194	40	20	in	in	ADP
ajrhss-2194	40	21	geometry	geometry	NOUN
ajrhss-2194	40	22	.	.	PUNCT
ajrhss-2194	41	1	sine	sine	NOUN
ajrhss-2194	41	2	,	,	PUNCT
ajrhss-2194	41	3	cosine	cosine	NOUN
ajrhss-2194	41	4	,	,	PUNCT
ajrhss-2194	41	5	and	and	CCONJ
ajrhss-2194	41	6	tangent	tangent	NOUN
ajrhss-2194	41	7	:	:	PUNCT
ajrhss-2194	41	8	these	these	DET
ajrhss-2194	41	9	functions	function	NOUN
ajrhss-2194	41	10	are	be	AUX
ajrhss-2194	41	11	defined	define	VERB
ajrhss-2194	41	12	as	as	ADP
ajrhss-2194	41	13	ratios	ratio	NOUN
ajrhss-2194	41	14	of	of	ADP
ajrhss-2194	41	15	sides	side	NOUN
ajrhss-2194	41	16	in	in	ADP
ajrhss-2194	41	17	a	a	DET
ajrhss-2194	41	18	right	right	ADJ
ajrhss-2194	41	19	-	-	PUNCT
ajrhss-2194	41	20	angled	angle	VERB
ajrhss-2194	41	21	triangle	triangle	NOUN
ajrhss-2194	41	22	.	.	PUNCT
ajrhss-2194	42	1	for	for	ADP
ajrhss-2194	42	2	an	an	DET
ajrhss-2194	42	3	angle	angle	NOUN
ajrhss-2194	42	4	\	\	PROPN
ajrhss-2194	42	5	(	(	PUNCT
ajrhss-2194	42	6	\theta	\theta	PROPN
ajrhss-2194	42	7	\	\	PROPN
ajrhss-2194	42	8	):	):	PUNCT
ajrhss-2194	42	9	american	american	PROPN
ajrhss-2194	42	10	journal	journal	PROPN
ajrhss-2194	42	11	of	of	ADP
ajrhss-2194	42	12	research	research	NOUN
ajrhss-2194	42	13	in	in	ADP
ajrhss-2194	42	14	humanities	humanity	NOUN
ajrhss-2194	42	15	and	and	CCONJ
ajrhss-2194	42	16	social	social	ADJ
ajrhss-2194	42	17	sciences	science	NOUN
ajrhss-2194	42	18	volume	volume	NOUN
ajrhss-2194	42	19	25	25	NUM
ajrhss-2194	42	20	june	june	PROPN
ajrhss-2194	42	21	2024	2024	NUM
ajrhss-2194	42	22	p	p	NOUN
ajrhss-2194	42	23	a	a	DET
ajrhss-2194	42	24	g	g	NOUN
ajrhss-2194	42	25	e	e	NOUN
ajrhss-2194	42	26	|	|	ADV
ajrhss-2194	42	27	43	43	NUM
ajrhss-2194	42	28	www.americanjournal.org	www.americanjournal.org	PROPN
ajrhss-2194	42	29	\	\	PROPN
ajrhss-2194	42	30	(	(	PUNCT
ajrhss-2194	42	31	\sin(\theta	\sin(\theta	NUM
ajrhss-2194	42	32	)	)	PUNCT
ajrhss-2194	42	33	\	\	NOUN
ajrhss-2194	42	34	)	)	PUNCT
ajrhss-2194	42	35	is	be	AUX
ajrhss-2194	42	36	the	the	DET
ajrhss-2194	42	37	ratio	ratio	NOUN
ajrhss-2194	42	38	of	of	ADP
ajrhss-2194	42	39	the	the	DET
ajrhss-2194	42	40	length	length	NOUN
ajrhss-2194	42	41	of	of	ADP
ajrhss-2194	42	42	the	the	DET
ajrhss-2194	42	43	opposite	opposite	ADJ
ajrhss-2194	42	44	side	side	NOUN
ajrhss-2194	42	45	to	to	ADP
ajrhss-2194	42	46	the	the	DET
ajrhss-2194	42	47	hypotenuse	hypotenuse	NOUN
ajrhss-2194	42	48	.	.	PUNCT
ajrhss-2194	43	1	\	\	PROPN
ajrhss-2194	43	2	(	(	PUNCT
ajrhss-2194	43	3	\cos(\theta	\cos(\theta	NUM
ajrhss-2194	43	4	)	)	PUNCT
ajrhss-2194	43	5	\	\	NOUN
ajrhss-2194	43	6	)	)	PUNCT
ajrhss-2194	43	7	is	be	AUX
ajrhss-2194	43	8	the	the	DET
ajrhss-2194	43	9	ratio	ratio	NOUN
ajrhss-2194	43	10	of	of	ADP
ajrhss-2194	43	11	the	the	DET
ajrhss-2194	43	12	length	length	NOUN
ajrhss-2194	43	13	of	of	ADP
ajrhss-2194	43	14	the	the	DET
ajrhss-2194	43	15	adjacent	adjacent	ADJ
ajrhss-2194	43	16	side	side	NOUN
ajrhss-2194	43	17	to	to	ADP
ajrhss-2194	43	18	the	the	DET
ajrhss-2194	43	19	hypotenuse	hypotenuse	NOUN
ajrhss-2194	43	20	.	.	PUNCT
ajrhss-2194	44	1	\	\	PROPN
ajrhss-2194	44	2	(	(	PUNCT
ajrhss-2194	44	3	\tan(\theta	\tan(\theta	NUM
ajrhss-2194	44	4	)	)	PUNCT
ajrhss-2194	44	5	\	\	NOUN
ajrhss-2194	44	6	)	)	PUNCT
ajrhss-2194	44	7	is	be	AUX
ajrhss-2194	44	8	the	the	DET
ajrhss-2194	44	9	ratio	ratio	NOUN
ajrhss-2194	44	10	of	of	ADP
ajrhss-2194	44	11	the	the	DET
ajrhss-2194	44	12	length	length	NOUN
ajrhss-2194	44	13	of	of	ADP
ajrhss-2194	44	14	the	the	DET
ajrhss-2194	44	15	opposite	opposite	ADJ
ajrhss-2194	44	16	side	side	NOUN
ajrhss-2194	44	17	to	to	ADP
ajrhss-2194	44	18	the	the	DET
ajrhss-2194	44	19	adjacent	adjacent	ADJ
ajrhss-2194	44	20	side	side	NOUN
ajrhss-2194	44	21	.	.	PUNCT
ajrhss-2194	45	1	these	these	DET
ajrhss-2194	45	2	functions	function	NOUN
ajrhss-2194	45	3	are	be	AUX
ajrhss-2194	45	4	critical	critical	ADJ
ajrhss-2194	45	5	in	in	ADP
ajrhss-2194	45	6	various	various	ADJ
ajrhss-2194	45	7	applications	application	NOUN
ajrhss-2194	45	8	,	,	PUNCT
ajrhss-2194	45	9	including	include	VERB
ajrhss-2194	45	10	signal	signal	NOUN
ajrhss-2194	45	11	processing	processing	NOUN
ajrhss-2194	45	12	,	,	PUNCT
ajrhss-2194	45	13	wave	wave	NOUN
ajrhss-2194	45	14	analysis	analysis	NOUN
ajrhss-2194	45	15	,	,	PUNCT
ajrhss-2194	45	16	and	and	CCONJ
ajrhss-2194	45	17	circular	circular	ADJ
ajrhss-2194	45	18	motion	motion	NOUN
ajrhss-2194	45	19	.	.	PUNCT
ajrhss-2194	46	1	6	6	X
ajrhss-2194	46	2	.	.	X
ajrhss-2194	46	3	geometry	geometry	NOUN
ajrhss-2194	46	4	of	of	ADP
ajrhss-2194	46	5	differential	differential	ADJ
ajrhss-2194	46	6	equations	equation	NOUN
ajrhss-2194	46	7	some	some	DET
ajrhss-2194	46	8	geometric	geometric	ADJ
ajrhss-2194	46	9	functions	function	NOUN
ajrhss-2194	46	10	arise	arise	VERB
ajrhss-2194	46	11	in	in	ADP
ajrhss-2194	46	12	the	the	DET
ajrhss-2194	46	13	study	study	NOUN
ajrhss-2194	46	14	of	of	ADP
ajrhss-2194	46	15	differential	differential	ADJ
ajrhss-2194	46	16	equations	equation	NOUN
ajrhss-2194	46	17	,	,	PUNCT
ajrhss-2194	46	18	particularly	particularly	ADV
ajrhss-2194	46	19	in	in	ADP
ajrhss-2194	46	20	the	the	DET
ajrhss-2194	46	21	context	context	NOUN
ajrhss-2194	46	22	of	of	ADP
ajrhss-2194	46	23	dynamical	dynamical	ADJ
ajrhss-2194	46	24	systems	system	NOUN
ajrhss-2194	46	25	and	and	CCONJ
ajrhss-2194	46	26	differential	differential	ADJ
ajrhss-2194	46	27	geometry	geometry	NOUN
ajrhss-2194	46	28	.	.	PUNCT
ajrhss-2194	47	1	geodesics	geodesic	NOUN
ajrhss-2194	47	2	and	and	CCONJ
ajrhss-2194	47	3	curvature	curvature	NOUN
ajrhss-2194	47	4	:	:	PUNCT
ajrhss-2194	47	5	functions	function	NOUN
ajrhss-2194	47	6	that	that	PRON
ajrhss-2194	47	7	describe	describe	VERB
ajrhss-2194	47	8	the	the	DET
ajrhss-2194	47	9	shortest	short	ADJ
ajrhss-2194	47	10	paths	path	NOUN
ajrhss-2194	47	11	between	between	ADP
ajrhss-2194	47	12	points	point	NOUN
ajrhss-2194	47	13	on	on	ADP
ajrhss-2194	47	14	curved	curved	ADJ
ajrhss-2194	47	15	surfaces	surface	NOUN
ajrhss-2194	47	16	(	(	PUNCT
ajrhss-2194	47	17	geodesics	geodesic	NOUN
ajrhss-2194	47	18	)	)	PUNCT
ajrhss-2194	47	19	and	and	CCONJ
ajrhss-2194	47	20	the	the	DET
ajrhss-2194	47	21	curvature	curvature	NOUN
ajrhss-2194	47	22	of	of	ADP
ajrhss-2194	47	23	surfaces	surface	NOUN
ajrhss-2194	47	24	are	be	AUX
ajrhss-2194	47	25	examples	example	NOUN
ajrhss-2194	47	26	of	of	ADP
ajrhss-2194	47	27	geometric	geometric	ADJ
ajrhss-2194	47	28	functions	function	NOUN
ajrhss-2194	47	29	in	in	ADP
ajrhss-2194	47	30	differential	differential	ADJ
ajrhss-2194	47	31	geometry	geometry	NOUN
ajrhss-2194	47	32	.	.	PUNCT
ajrhss-2194	48	1	geometric	geometric	ADJ
ajrhss-2194	48	2	functions	function	NOUN
ajrhss-2194	48	3	play	play	VERB
ajrhss-2194	48	4	a	a	DET
ajrhss-2194	48	5	crucial	crucial	ADJ
ajrhss-2194	48	6	role	role	NOUN
ajrhss-2194	48	7	in	in	ADP
ajrhss-2194	48	8	various	various	ADJ
ajrhss-2194	48	9	branches	branch	NOUN
ajrhss-2194	48	10	of	of	ADP
ajrhss-2194	48	11	geometry	geometry	NOUN
ajrhss-2194	48	12	by	by	ADP
ajrhss-2194	48	13	providing	provide	VERB
ajrhss-2194	48	14	precise	precise	ADJ
ajrhss-2194	48	15	mathematical	mathematical	ADJ
ajrhss-2194	48	16	descriptions	description	NOUN
ajrhss-2194	48	17	of	of	ADP
ajrhss-2194	48	18	shapes	shape	NOUN
ajrhss-2194	48	19	,	,	PUNCT
ajrhss-2194	48	20	transformations	transformation	NOUN
ajrhss-2194	48	21	,	,	PUNCT
ajrhss-2194	48	22	and	and	CCONJ
ajrhss-2194	48	23	spatial	spatial	ADJ
ajrhss-2194	48	24	relationships	relationship	NOUN
ajrhss-2194	48	25	.	.	PUNCT
ajrhss-2194	49	1	here	here	ADV
ajrhss-2194	49	2	are	be	AUX
ajrhss-2194	49	3	some	some	DET
ajrhss-2194	49	4	specific	specific	ADJ
ajrhss-2194	49	5	roles	role	NOUN
ajrhss-2194	49	6	and	and	CCONJ
ajrhss-2194	49	7	applications	application	NOUN
ajrhss-2194	49	8	of	of	ADP
ajrhss-2194	49	9	geometric	geometric	ADJ
ajrhss-2194	49	10	functions	function	NOUN
ajrhss-2194	49	11	in	in	ADP
ajrhss-2194	49	12	geometry	geometry	NOUN
ajrhss-2194	49	13	:	:	PUNCT
ajrhss-2194	49	14	1	1	X
ajrhss-2194	49	15	.	.	X
ajrhss-2194	49	16	defining	define	VERB
ajrhss-2194	49	17	shapes	shape	NOUN
ajrhss-2194	49	18	and	and	CCONJ
ajrhss-2194	49	19	curves	curve	NOUN
ajrhss-2194	49	20	geometric	geometric	ADJ
ajrhss-2194	49	21	functions	function	NOUN
ajrhss-2194	49	22	are	be	AUX
ajrhss-2194	49	23	used	use	VERB
ajrhss-2194	49	24	to	to	PART
ajrhss-2194	49	25	define	define	VERB
ajrhss-2194	49	26	and	and	CCONJ
ajrhss-2194	49	27	describe	describe	VERB
ajrhss-2194	49	28	the	the	DET
ajrhss-2194	49	29	properties	property	NOUN
ajrhss-2194	49	30	of	of	ADP
ajrhss-2194	49	31	geometric	geometric	ADJ
ajrhss-2194	49	32	shapes	shape	NOUN
ajrhss-2194	49	33	and	and	CCONJ
ajrhss-2194	49	34	curves	curve	NOUN
ajrhss-2194	49	35	.	.	PUNCT
ajrhss-2194	50	1	this	this	PRON
ajrhss-2194	50	2	is	be	AUX
ajrhss-2194	50	3	fundamental	fundamental	ADJ
ajrhss-2194	50	4	in	in	ADP
ajrhss-2194	50	5	both	both	CCONJ
ajrhss-2194	50	6	euclidean	euclidean	ADJ
ajrhss-2194	50	7	and	and	CCONJ
ajrhss-2194	50	8	non	non	ADJ
ajrhss-2194	50	9	-	-	ADJ
ajrhss-2194	50	10	euclidean	euclidean	ADJ
ajrhss-2194	50	11	geometry	geometry	NOUN
ajrhss-2194	50	12	.	.	PUNCT
ajrhss-2194	51	1	implicit	implicit	ADJ
ajrhss-2194	51	2	and	and	CCONJ
ajrhss-2194	51	3	parametric	parametric	ADJ
ajrhss-2194	51	4	equations	equation	NOUN
ajrhss-2194	51	5	:	:	PUNCT
ajrhss-2194	51	6	geometric	geometric	ADJ
ajrhss-2194	51	7	functions	function	NOUN
ajrhss-2194	51	8	can	can	AUX
ajrhss-2194	51	9	define	define	VERB
ajrhss-2194	51	10	curves	curve	NOUN
ajrhss-2194	51	11	and	and	CCONJ
ajrhss-2194	51	12	surfaces	surface	NOUN
ajrhss-2194	51	13	.	.	PUNCT
ajrhss-2194	52	1	for	for	ADP
ajrhss-2194	52	2	example	example	NOUN
ajrhss-2194	52	3	,	,	PUNCT
ajrhss-2194	52	4	the	the	DET
ajrhss-2194	52	5	circle	circle	NOUN
ajrhss-2194	52	6	\	\	PROPN
ajrhss-2194	52	7	(	(	PUNCT
ajrhss-2194	52	8	x^2	x^2	PROPN
ajrhss-2194	52	9	+	+	PROPN
ajrhss-2194	52	10	y^2	y^2	PROPN
ajrhss-2194	52	11	=	=	SYM
ajrhss-2194	52	12	1	1	NUM
ajrhss-2194	52	13	\	\	NOUN
ajrhss-2194	52	14	)	)	PUNCT
ajrhss-2194	52	15	can	can	AUX
ajrhss-2194	52	16	be	be	AUX
ajrhss-2194	52	17	described	describe	VERB
ajrhss-2194	52	18	parametrically	parametrically	ADV
ajrhss-2194	52	19	by	by	ADP
ajrhss-2194	52	20	\	\	PROPN
ajrhss-2194	52	21	(	(	PUNCT
ajrhss-2194	52	22	x	x	SYM
ajrhss-2194	52	23	=	=	SYM
ajrhss-2194	52	24	\cos(t	\cos(t	ADJ
ajrhss-2194	52	25	)	)	PUNCT
ajrhss-2194	52	26	\	\	NOUN
ajrhss-2194	52	27	)	)	PUNCT
ajrhss-2194	52	28	and	and	CCONJ
ajrhss-2194	53	1	\	\	PROPN
ajrhss-2194	53	2	(	(	PUNCT
ajrhss-2194	53	3	y	y	NOUN
ajrhss-2194	53	4	=	=	PUNCT
ajrhss-2194	53	5	\sin(t	\sin(t	PROPN
ajrhss-2194	53	6	)	)	PUNCT
ajrhss-2194	53	7	\	\	NOUN
ajrhss-2194	53	8	)	)	PUNCT
ajrhss-2194	53	9	.	.	PUNCT
ajrhss-2194	54	1	bezier	bezier	NOUN
ajrhss-2194	54	2	curves	curve	NOUN
ajrhss-2194	54	3	and	and	CCONJ
ajrhss-2194	54	4	splines	spline	NOUN
ajrhss-2194	54	5	:	:	PUNCT
ajrhss-2194	54	6	these	these	PRON
ajrhss-2194	54	7	are	be	AUX
ajrhss-2194	54	8	used	use	VERB
ajrhss-2194	54	9	in	in	ADP
ajrhss-2194	54	10	computer	computer	NOUN
ajrhss-2194	54	11	graphics	graphic	NOUN
ajrhss-2194	54	12	to	to	PART
ajrhss-2194	54	13	model	model	VERB
ajrhss-2194	54	14	smooth	smooth	ADJ
ajrhss-2194	54	15	curves	curve	NOUN
ajrhss-2194	54	16	.	.	PUNCT
ajrhss-2194	55	1	a	a	DET
ajrhss-2194	55	2	quadratic	quadratic	ADJ
ajrhss-2194	55	3	bezier	bezier	NOUN
ajrhss-2194	55	4	curve	curve	NOUN
ajrhss-2194	55	5	,	,	PUNCT
ajrhss-2194	55	6	for	for	ADP
ajrhss-2194	55	7	instance	instance	NOUN
ajrhss-2194	55	8	,	,	PUNCT
ajrhss-2194	55	9	is	be	AUX
ajrhss-2194	55	10	defined	define	VERB
ajrhss-2194	55	11	by	by	ADP
ajrhss-2194	55	12	the	the	DET
ajrhss-2194	55	13	function	function	NOUN
ajrhss-2194	55	14	\	\	PROPN
ajrhss-2194	55	15	(	(	PUNCT
ajrhss-2194	55	16	b(t	b(t	NOUN
ajrhss-2194	55	17	)	)	PUNCT
ajrhss-2194	55	18	=	=	PUNCT
ajrhss-2194	56	1	(	(	PUNCT
ajrhss-2194	56	2	1	1	NUM
ajrhss-2194	56	3	-	-	PUNCT
ajrhss-2194	56	4	t)^2	t)^2	PROPN
ajrhss-2194	56	5	p_0	p_0	NOUN
ajrhss-2194	56	6	+	+	CCONJ
ajrhss-2194	56	7	2(1	2(1	NUM
ajrhss-2194	56	8	-	-	PUNCT
ajrhss-2194	56	9	t)t	t)t	ADJ
ajrhss-2194	56	10	p_1	p_1	NOUN
ajrhss-2194	56	11	+	+	CCONJ
ajrhss-2194	56	12	t^2	t^2	NOUN
ajrhss-2194	56	13	p_2	p_2	X
ajrhss-2194	56	14	\	\	PROPN
ajrhss-2194	56	15	)	)	PUNCT
ajrhss-2194	56	16	,	,	PUNCT
ajrhss-2194	56	17	where	where	SCONJ
ajrhss-2194	56	18	\	\	PROPN
ajrhss-2194	56	19	(	(	PUNCT
ajrhss-2194	56	20	p_0	p_0	NOUN
ajrhss-2194	56	21	,	,	PUNCT
ajrhss-2194	56	22	p_1	p_1	NOUN
ajrhss-2194	56	23	,	,	PUNCT
ajrhss-2194	56	24	\	\	PROPN
ajrhss-2194	56	25	)	)	PUNCT
ajrhss-2194	56	26	and	and	CCONJ
ajrhss-2194	56	27	\	\	PROPN
ajrhss-2194	56	28	(	(	PUNCT
ajrhss-2194	56	29	p_2	p_2	PROPN
ajrhss-2194	56	30	\	\	PROPN
ajrhss-2194	56	31	)	)	PUNCT
ajrhss-2194	56	32	are	be	AUX
ajrhss-2194	56	33	control	control	NOUN
ajrhss-2194	56	34	points	point	NOUN
ajrhss-2194	56	35	.	.	PUNCT
ajrhss-2194	57	1	2	2	X
ajrhss-2194	57	2	.	.	X
ajrhss-2194	57	3	describing	describe	VERB
ajrhss-2194	57	4	transformations	transformation	NOUN
ajrhss-2194	57	5	geometric	geometric	ADJ
ajrhss-2194	57	6	functions	function	NOUN
ajrhss-2194	57	7	are	be	AUX
ajrhss-2194	57	8	essential	essential	ADJ
ajrhss-2194	57	9	for	for	ADP
ajrhss-2194	57	10	describing	describe	VERB
ajrhss-2194	57	11	transformations	transformation	NOUN
ajrhss-2194	57	12	that	that	PRON
ajrhss-2194	57	13	change	change	VERB
ajrhss-2194	57	14	the	the	DET
ajrhss-2194	57	15	position	position	NOUN
ajrhss-2194	57	16	,	,	PUNCT
ajrhss-2194	57	17	size	size	NOUN
ajrhss-2194	57	18	,	,	PUNCT
ajrhss-2194	57	19	and	and	CCONJ
ajrhss-2194	57	20	orientation	orientation	NOUN
ajrhss-2194	57	21	of	of	ADP
ajrhss-2194	57	22	geometric	geometric	ADJ
ajrhss-2194	57	23	objects	object	NOUN
ajrhss-2194	57	24	.	.	PUNCT
ajrhss-2194	58	1	affine	affine	NOUN
ajrhss-2194	58	2	transformations	transformation	NOUN
ajrhss-2194	58	3	:	:	PUNCT
ajrhss-2194	58	4	these	these	PRON
ajrhss-2194	58	5	include	include	VERB
ajrhss-2194	58	6	translations	translation	NOUN
ajrhss-2194	58	7	,	,	PUNCT
ajrhss-2194	58	8	rotations	rotation	NOUN
ajrhss-2194	58	9	,	,	PUNCT
ajrhss-2194	58	10	scalings	scaling	NOUN
ajrhss-2194	58	11	,	,	PUNCT
ajrhss-2194	58	12	and	and	CCONJ
ajrhss-2194	58	13	shears	shear	NOUN
ajrhss-2194	58	14	.	.	PUNCT
ajrhss-2194	59	1	they	they	PRON
ajrhss-2194	59	2	can	can	AUX
ajrhss-2194	59	3	be	be	AUX
ajrhss-2194	59	4	represented	represent	VERB
ajrhss-2194	59	5	using	use	VERB
ajrhss-2194	59	6	matrices	matrix	NOUN
ajrhss-2194	59	7	and	and	CCONJ
ajrhss-2194	59	8	applied	apply	VERB
ajrhss-2194	59	9	to	to	ADP
ajrhss-2194	59	10	points	point	NOUN
ajrhss-2194	59	11	and	and	CCONJ
ajrhss-2194	59	12	vectors	vector	NOUN
ajrhss-2194	59	13	in	in	ADP
ajrhss-2194	59	14	space	space	NOUN
ajrhss-2194	59	15	.	.	PUNCT
ajrhss-2194	60	1	\[\begin{pmatrix	\[\begin{pmatrix	VERB
ajrhss-2194	60	2	}	}	PUNCT
ajrhss-2194	61	1	x	x	X
ajrhss-2194	61	2	'	'	PART
ajrhss-2194	61	3	\\	\\	NOUN
ajrhss-2194	61	4	y'\end{pmatrix	y'\end{pmatrix	VERB
ajrhss-2194	61	5	}	}	PUNCT
ajrhss-2194	61	6	=	=	SYM
ajrhss-2194	61	7	\begin{pmatrix	\begin{pmatrix	NOUN
ajrhss-2194	61	8	}	}	PUNCT
ajrhss-2194	61	9	a	a	PRON
ajrhss-2194	61	10	&	&	CCONJ
ajrhss-2194	61	11	b	b	PROPN
ajrhss-2194	61	12	\\	\\	PROPN
ajrhss-2194	61	13	c	c	PROPN
ajrhss-2194	61	14	&	&	CCONJ
ajrhss-2194	61	15	d	d	PROPN
ajrhss-2194	61	16	\end{pmatrix	\end{pmatrix	PROPN
ajrhss-2194	61	17	}	}	PUNCT
ajrhss-2194	61	18	\begin{pmatrix}x	\begin{pmatrix}x	PROPN
ajrhss-2194	61	19	\\	\\	NOUN
ajrhss-2194	61	20	y\end{pmatrix}+\begin{pmatrix	y\end{pmatrix}+\begin{pmatrix	VERB
ajrhss-2194	61	21	}	}	PUNCT
ajrhss-2194	61	22	e	e	X
ajrhss-2194	61	23	\\f	\\f	PROPN
ajrhss-2194	61	24	\end{pmatrix}\	\end{pmatrix}\	NOUN
ajrhss-2194	61	25	]	]	X
ajrhss-2194	61	26	3	3	X
ajrhss-2194	61	27	.	.	X
ajrhss-2194	61	28	analyzing	analyze	VERB
ajrhss-2194	61	29	geometric	geometric	ADJ
ajrhss-2194	61	30	properties	property	NOUN
ajrhss-2194	61	31	geometric	geometric	ADJ
ajrhss-2194	61	32	functions	function	NOUN
ajrhss-2194	61	33	help	help	AUX
ajrhss-2194	61	34	analyze	analyze	VERB
ajrhss-2194	61	35	and	and	CCONJ
ajrhss-2194	61	36	compute	compute	VERB
ajrhss-2194	61	37	properties	property	NOUN
ajrhss-2194	61	38	such	such	ADJ
ajrhss-2194	61	39	as	as	ADP
ajrhss-2194	61	40	distance	distance	NOUN
ajrhss-2194	61	41	,	,	PUNCT
ajrhss-2194	61	42	angle	angle	NOUN
ajrhss-2194	61	43	,	,	PUNCT
ajrhss-2194	61	44	area	area	NOUN
ajrhss-2194	61	45	,	,	PUNCT
ajrhss-2194	61	46	and	and	CCONJ
ajrhss-2194	61	47	volume	volume	NOUN
ajrhss-2194	61	48	.	.	PUNCT
ajrhss-2194	62	1	distance	distance	NOUN
ajrhss-2194	62	2	formula	formula	NOUN
ajrhss-2194	62	3	:	:	PUNCT
ajrhss-2194	62	4	the	the	DET
ajrhss-2194	62	5	euclidean	euclidean	ADJ
ajrhss-2194	62	6	distance	distance	NOUN
ajrhss-2194	62	7	between	between	ADP
ajrhss-2194	62	8	two	two	NUM
ajrhss-2194	62	9	points	point	NOUN
ajrhss-2194	62	10	\	\	PROPN
ajrhss-2194	62	11	(	(	PUNCT
ajrhss-2194	62	12	(	(	PUNCT
ajrhss-2194	62	13	x_1	x_1	X
ajrhss-2194	62	14	,	,	PUNCT
ajrhss-2194	62	15	y_1	y_1	PROPN
ajrhss-2194	62	16	)	)	PUNCT
ajrhss-2194	62	17	\	\	NOUN
ajrhss-2194	62	18	)	)	PUNCT
ajrhss-2194	62	19	and	and	CCONJ
ajrhss-2194	62	20	\	\	PROPN
ajrhss-2194	62	21	(	(	PUNCT
ajrhss-2194	62	22	(	(	PUNCT
ajrhss-2194	62	23	x_2	x_2	PROPN
ajrhss-2194	62	24	,	,	PUNCT
ajrhss-2194	62	25	y_2	y_2	PROPN
ajrhss-2194	62	26	)	)	PUNCT
ajrhss-2194	62	27	\	\	NOUN
ajrhss-2194	62	28	)	)	PUNCT
ajrhss-2194	62	29	is	be	AUX
ajrhss-2194	62	30	given	give	VERB
ajrhss-2194	62	31	by	by	ADP
ajrhss-2194	62	32	the	the	DET
ajrhss-2194	62	33	function	function	NOUN
ajrhss-2194	62	34	\	\	PROPN
ajrhss-2194	62	35	(	(	PUNCT
ajrhss-2194	62	36	d	d	NOUN
ajrhss-2194	62	37	=	=	SYM
ajrhss-2194	62	38	\sqrt{(x_2	\sqrt{(x_2	NOUN
ajrhss-2194	62	39	x_1)^2	x_1)^2	PROPN
ajrhss-2194	62	40	+	+	CCONJ
ajrhss-2194	62	41	(	(	PUNCT
ajrhss-2194	62	42	y_2	y_2	PROPN
ajrhss-2194	62	43	y_1)^2	y_1)^2	PROPN
ajrhss-2194	62	44	}	}	PUNCT
ajrhss-2194	62	45	\	\	PROPN
ajrhss-2194	62	46	)	)	PUNCT
ajrhss-2194	62	47	.	.	PUNCT
ajrhss-2194	63	1	area	area	NOUN
ajrhss-2194	63	2	and	and	CCONJ
ajrhss-2194	63	3	volume	volume	NOUN
ajrhss-2194	63	4	calculations	calculation	NOUN
ajrhss-2194	63	5	:	:	PUNCT
ajrhss-2194	63	6	functions	function	NOUN
ajrhss-2194	63	7	can	can	AUX
ajrhss-2194	63	8	describe	describe	VERB
ajrhss-2194	63	9	how	how	SCONJ
ajrhss-2194	63	10	to	to	PART
ajrhss-2194	63	11	compute	compute	VERB
ajrhss-2194	63	12	the	the	DET
ajrhss-2194	63	13	area	area	NOUN
ajrhss-2194	63	14	of	of	ADP
ajrhss-2194	63	15	complex	complex	ADJ
ajrhss-2194	63	16	shapes	shape	NOUN
ajrhss-2194	63	17	or	or	CCONJ
ajrhss-2194	63	18	the	the	DET
ajrhss-2194	63	19	volume	volume	NOUN
ajrhss-2194	63	20	of	of	ADP
ajrhss-2194	63	21	solids	solid	NOUN
ajrhss-2194	63	22	.	.	PUNCT
ajrhss-2194	64	1	for	for	ADP
ajrhss-2194	64	2	example	example	NOUN
ajrhss-2194	64	3	,	,	PUNCT
ajrhss-2194	64	4	the	the	DET
ajrhss-2194	64	5	area	area	NOUN
ajrhss-2194	64	6	of	of	ADP
ajrhss-2194	64	7	a	a	DET
ajrhss-2194	64	8	triangle	triangle	NOUN
ajrhss-2194	64	9	with	with	ADP
ajrhss-2194	64	10	vertices	vertex	NOUN
ajrhss-2194	64	11	at	at	ADP
ajrhss-2194	64	12	\	\	PROPN
ajrhss-2194	64	13	(	(	PUNCT
ajrhss-2194	64	14	(	(	PUNCT
ajrhss-2194	64	15	x_1	x_1	X
ajrhss-2194	64	16	,	,	PUNCT
ajrhss-2194	64	17	y_1	y_1	PROPN
ajrhss-2194	64	18	)	)	PUNCT
ajrhss-2194	64	19	\	\	PROPN
ajrhss-2194	64	20	)	)	PUNCT
ajrhss-2194	64	21	,	,	PUNCT
ajrhss-2194	64	22	\	\	PROPN
ajrhss-2194	64	23	(	(	PUNCT
ajrhss-2194	64	24	(	(	PUNCT
ajrhss-2194	64	25	x_2	x_2	PROPN
ajrhss-2194	64	26	,	,	PUNCT
ajrhss-2194	64	27	y_2	y_2	PROPN
ajrhss-2194	64	28	)	)	PUNCT
ajrhss-2194	64	29	\	\	NOUN
ajrhss-2194	64	30	)	)	PUNCT
ajrhss-2194	64	31	,	,	PUNCT
ajrhss-2194	64	32	and	and	CCONJ
ajrhss-2194	64	33	\	\	PROPN
ajrhss-2194	64	34	(	(	PUNCT
ajrhss-2194	64	35	(	(	PUNCT
ajrhss-2194	64	36	x_3	x_3	PROPN
ajrhss-2194	64	37	,	,	PUNCT
ajrhss-2194	64	38	y_3	y_3	PROPN
ajrhss-2194	64	39	)	)	PUNCT
ajrhss-2194	64	40	\	\	NOUN
ajrhss-2194	64	41	)	)	PUNCT
ajrhss-2194	64	42	can	can	AUX
ajrhss-2194	64	43	be	be	AUX
ajrhss-2194	64	44	found	find	VERB
ajrhss-2194	64	45	using	use	VERB
ajrhss-2194	64	46	the	the	DET
ajrhss-2194	64	47	function	function	NOUN
ajrhss-2194	64	48	:	:	PUNCT
ajrhss-2194	64	49	\[\text{area	\[\text{area	PROPN
ajrhss-2194	64	50	}	}	PUNCT
ajrhss-2194	64	51	=	=	SYM
ajrhss-2194	64	52	\frac{1}{2	\frac{1}{2	NOUN
ajrhss-2194	64	53	}	}	PUNCT
ajrhss-2194	64	54	\left|	\left|	NOUN
ajrhss-2194	64	55	x_1(y_2	x_1(y_2	PROPN
ajrhss-2194	64	56	y_3	y_3	PROPN
ajrhss-2194	64	57	)	)	PUNCT
ajrhss-2194	65	1	+	+	CCONJ
ajrhss-2194	65	2	x_2(y_3	x_2(y_3	PROPN
ajrhss-2194	65	3	y_1	y_1	PROPN
ajrhss-2194	65	4	)	)	PUNCT
ajrhss-2194	66	1	+	+	CCONJ
ajrhss-2194	66	2	x_3(y_1	x_3(y_1	PROPN
ajrhss-2194	66	3	y_2	y_2	NOUN
ajrhss-2194	66	4	)	)	PUNCT
ajrhss-2194	66	5	\right|\	\right|\	PROPN
ajrhss-2194	66	6	]	]	PUNCT
ajrhss-2194	66	7	american	american	PROPN
ajrhss-2194	66	8	journal	journal	PROPN
ajrhss-2194	66	9	of	of	ADP
ajrhss-2194	66	10	research	research	NOUN
ajrhss-2194	66	11	in	in	ADP
ajrhss-2194	66	12	humanities	humanity	NOUN
ajrhss-2194	66	13	and	and	CCONJ
ajrhss-2194	66	14	social	social	ADJ
ajrhss-2194	66	15	sciences	science	NOUN
ajrhss-2194	66	16	volume	volume	NOUN
ajrhss-2194	66	17	25	25	NUM
ajrhss-2194	66	18	june	june	PROPN
ajrhss-2194	66	19	2024	2024	NUM
ajrhss-2194	66	20	p	p	NOUN
ajrhss-2194	66	21	a	a	DET
ajrhss-2194	66	22	g	g	NOUN
ajrhss-2194	66	23	e	e	NOUN
ajrhss-2194	66	24	|	|	ADV
ajrhss-2194	66	25	44	44	NUM
ajrhss-2194	66	26	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2194	66	27	4	4	NUM
ajrhss-2194	66	28	.	.	PUNCT
ajrhss-2194	66	29	modeling	model	VERB
ajrhss-2194	66	30	physical	physical	ADJ
ajrhss-2194	66	31	systems	system	NOUN
ajrhss-2194	66	32	in	in	ADP
ajrhss-2194	66	33	physics	physics	NOUN
ajrhss-2194	66	34	and	and	CCONJ
ajrhss-2194	66	35	engineering	engineering	NOUN
ajrhss-2194	66	36	,	,	PUNCT
ajrhss-2194	66	37	geometric	geometric	ADJ
ajrhss-2194	66	38	functions	function	NOUN
ajrhss-2194	66	39	are	be	AUX
ajrhss-2194	66	40	used	use	VERB
ajrhss-2194	66	41	to	to	PART
ajrhss-2194	66	42	model	model	VERB
ajrhss-2194	66	43	and	and	CCONJ
ajrhss-2194	66	44	solve	solve	VERB
ajrhss-2194	66	45	problems	problem	NOUN
ajrhss-2194	66	46	involving	involve	VERB
ajrhss-2194	66	47	physical	physical	ADJ
ajrhss-2194	66	48	systems	system	NOUN
ajrhss-2194	66	49	.	.	PUNCT
ajrhss-2194	67	1	projectile	projectile	NOUN
ajrhss-2194	67	2	motion	motion	NOUN
ajrhss-2194	67	3	:	:	PUNCT
ajrhss-2194	67	4	the	the	DET
ajrhss-2194	67	5	trajectory	trajectory	NOUN
ajrhss-2194	67	6	of	of	ADP
ajrhss-2194	67	7	a	a	DET
ajrhss-2194	67	8	projectile	projectile	NOUN
ajrhss-2194	67	9	can	can	AUX
ajrhss-2194	67	10	be	be	AUX
ajrhss-2194	67	11	described	describe	VERB
ajrhss-2194	67	12	by	by	ADP
ajrhss-2194	67	13	parametric	parametric	ADJ
ajrhss-2194	67	14	equations	equation	NOUN
ajrhss-2194	67	15	involving	involve	VERB
ajrhss-2194	67	16	geometric	geometric	ADJ
ajrhss-2194	67	17	functions	function	NOUN
ajrhss-2194	67	18	.	.	PUNCT
ajrhss-2194	68	1	for	for	ADP
ajrhss-2194	68	2	instance	instance	NOUN
ajrhss-2194	68	3	,	,	PUNCT
ajrhss-2194	68	4	the	the	DET
ajrhss-2194	68	5	horizontal	horizontal	ADJ
ajrhss-2194	68	6	and	and	CCONJ
ajrhss-2194	68	7	vertical	vertical	ADJ
ajrhss-2194	68	8	positions	position	NOUN
ajrhss-2194	68	9	\	\	PROPN
ajrhss-2194	68	10	(	(	PUNCT
ajrhss-2194	68	11	(	(	PUNCT
ajrhss-2194	68	12	x(t	x(t	PROPN
ajrhss-2194	68	13	)	)	PUNCT
ajrhss-2194	68	14	,	,	PUNCT
ajrhss-2194	68	15	y(t	y(t	NOUN
ajrhss-2194	68	16	)	)	PUNCT
ajrhss-2194	68	17	)	)	PUNCT
ajrhss-2194	69	1	\	\	NOUN
ajrhss-2194	69	2	)	)	PUNCT
ajrhss-2194	69	3	can	can	AUX
ajrhss-2194	69	4	be	be	AUX
ajrhss-2194	69	5	given	give	VERB
ajrhss-2194	69	6	by	by	ADP
ajrhss-2194	69	7	:	:	PUNCT
ajrhss-2194	69	8	\	\	PROPN
ajrhss-2194	69	9	[	[	PUNCT
ajrhss-2194	69	10	x(t	x(t	PROPN
ajrhss-2194	69	11	)	)	PUNCT
ajrhss-2194	69	12	=	=	SYM
ajrhss-2194	69	13	v_0	v_0	PROPN
ajrhss-2194	69	14	\cos(\theta	\cos(\theta	NUM
ajrhss-2194	69	15	)	)	PUNCT
ajrhss-2194	69	16	t	t	PROPN
ajrhss-2194	69	17	]	]	PUNCT
ajrhss-2194	69	18	\	\	PROPN
ajrhss-2194	69	19	[	[	PUNCT
ajrhss-2194	69	20	y(t	y(t	NOUN
ajrhss-2194	69	21	)	)	PUNCT
ajrhss-2194	69	22	=	=	SYM
ajrhss-2194	69	23	v_0	v_0	PROPN
ajrhss-2194	69	24	\sin(\theta	\sin(\theta	NUM
ajrhss-2194	69	25	)	)	PUNCT
ajrhss-2194	69	26	t	t	NOUN
ajrhss-2194	69	27	\frac{1}{2	\frac{1}{2	NOUN
ajrhss-2194	69	28	}	}	PUNCT
ajrhss-2194	69	29	g	g	NOUN
ajrhss-2194	69	30	t^2	t^2	NOUN
ajrhss-2194	69	31	\	\	NOUN
ajrhss-2194	69	32	]	]	PUNCT
ajrhss-2194	69	33	where	where	SCONJ
ajrhss-2194	69	34	\	\	PROPN
ajrhss-2194	69	35	(	(	PUNCT
ajrhss-2194	69	36	v_0	v_0	PROPN
ajrhss-2194	69	37	\	\	PROPN
ajrhss-2194	69	38	)	)	PUNCT
ajrhss-2194	69	39	is	be	AUX
ajrhss-2194	69	40	the	the	DET
ajrhss-2194	69	41	initial	initial	ADJ
ajrhss-2194	69	42	velocity	velocity	NOUN
ajrhss-2194	69	43	,	,	PUNCT
ajrhss-2194	69	44	\	\	PROPN
ajrhss-2194	69	45	(	(	PUNCT
ajrhss-2194	69	46	\theta	\theta	PROPN
ajrhss-2194	69	47	\	\	PROPN
ajrhss-2194	69	48	)	)	PUNCT
ajrhss-2194	69	49	is	be	AUX
ajrhss-2194	69	50	the	the	DET
ajrhss-2194	69	51	angle	angle	NOUN
ajrhss-2194	69	52	of	of	ADP
ajrhss-2194	69	53	launch	launch	NOUN
ajrhss-2194	69	54	,	,	PUNCT
ajrhss-2194	69	55	and	and	CCONJ
ajrhss-2194	69	56	\	\	PROPN
ajrhss-2194	69	57	(	(	PUNCT
ajrhss-2194	69	58	g	g	PROPN
ajrhss-2194	69	59	\	\	PROPN
ajrhss-2194	69	60	)	)	PUNCT
ajrhss-2194	69	61	is	be	AUX
ajrhss-2194	69	62	the	the	DET
ajrhss-2194	69	63	acceleration	acceleration	NOUN
ajrhss-2194	69	64	due	due	ADP
ajrhss-2194	69	65	to	to	ADP
ajrhss-2194	69	66	gravity	gravity	NOUN
ajrhss-2194	69	67	.	.	PUNCT
ajrhss-2194	70	1	5	5	X
ajrhss-2194	70	2	.	.	X
ajrhss-2194	70	3	solving	solve	VERB
ajrhss-2194	70	4	geometric	geometric	ADJ
ajrhss-2194	70	5	problems	problem	NOUN
ajrhss-2194	70	6	geometric	geometric	ADJ
ajrhss-2194	70	7	functions	function	NOUN
ajrhss-2194	70	8	enable	enable	VERB
ajrhss-2194	70	9	the	the	DET
ajrhss-2194	70	10	solution	solution	NOUN
ajrhss-2194	70	11	of	of	ADP
ajrhss-2194	70	12	various	various	ADJ
ajrhss-2194	70	13	geometric	geometric	ADJ
ajrhss-2194	70	14	problems	problem	NOUN
ajrhss-2194	70	15	,	,	PUNCT
ajrhss-2194	70	16	including	include	VERB
ajrhss-2194	70	17	optimization	optimization	NOUN
ajrhss-2194	70	18	and	and	CCONJ
ajrhss-2194	70	19	intersection	intersection	NOUN
ajrhss-2194	70	20	.	.	PUNCT
ajrhss-2194	71	1	optimization	optimization	NOUN
ajrhss-2194	71	2	problems	problem	NOUN
ajrhss-2194	71	3	:	:	PUNCT
ajrhss-2194	71	4	functions	function	NOUN
ajrhss-2194	71	5	can	can	AUX
ajrhss-2194	71	6	describe	describe	VERB
ajrhss-2194	71	7	constraints	constraint	NOUN
ajrhss-2194	71	8	and	and	CCONJ
ajrhss-2194	71	9	objectives	objective	NOUN
ajrhss-2194	71	10	in	in	ADP
ajrhss-2194	71	11	optimization	optimization	NOUN
ajrhss-2194	71	12	problems	problem	NOUN
ajrhss-2194	71	13	.	.	PUNCT
ajrhss-2194	72	1	for	for	ADP
ajrhss-2194	72	2	example	example	NOUN
ajrhss-2194	72	3	,	,	PUNCT
ajrhss-2194	72	4	finding	find	VERB
ajrhss-2194	72	5	the	the	DET
ajrhss-2194	72	6	shortest	short	ADJ
ajrhss-2194	72	7	path	path	NOUN
ajrhss-2194	72	8	between	between	ADP
ajrhss-2194	72	9	two	two	NUM
ajrhss-2194	72	10	points	point	NOUN
ajrhss-2194	72	11	involves	involve	VERB
ajrhss-2194	72	12	using	use	VERB
ajrhss-2194	72	13	functions	function	NOUN
ajrhss-2194	72	14	that	that	PRON
ajrhss-2194	72	15	describe	describe	VERB
ajrhss-2194	72	16	distances	distance	NOUN
ajrhss-2194	72	17	.	.	PUNCT
ajrhss-2194	73	1	intersection	intersection	NOUN
ajrhss-2194	73	2	of	of	ADP
ajrhss-2194	73	3	curves	curve	NOUN
ajrhss-2194	73	4	and	and	CCONJ
ajrhss-2194	73	5	surfaces	surface	NOUN
ajrhss-2194	73	6	:	:	PUNCT
ajrhss-2194	73	7	functions	function	NOUN
ajrhss-2194	73	8	can	can	AUX
ajrhss-2194	73	9	be	be	AUX
ajrhss-2194	73	10	used	use	VERB
ajrhss-2194	73	11	to	to	PART
ajrhss-2194	73	12	determine	determine	VERB
ajrhss-2194	73	13	the	the	DET
ajrhss-2194	73	14	points	point	NOUN
ajrhss-2194	73	15	of	of	ADP
ajrhss-2194	73	16	intersection	intersection	NOUN
ajrhss-2194	73	17	between	between	ADP
ajrhss-2194	73	18	curves	curve	NOUN
ajrhss-2194	73	19	and	and	CCONJ
ajrhss-2194	73	20	surfaces	surface	NOUN
ajrhss-2194	73	21	.	.	PUNCT
ajrhss-2194	74	1	for	for	ADP
ajrhss-2194	74	2	example	example	NOUN
ajrhss-2194	74	3	,	,	PUNCT
ajrhss-2194	74	4	solving	solve	VERB
ajrhss-2194	74	5	\	\	PROPN
ajrhss-2194	74	6	(	(	PUNCT
ajrhss-2194	74	7	x^2	x^2	PROPN
ajrhss-2194	74	8	+	+	PROPN
ajrhss-2194	74	9	y^2	y^2	PROPN
ajrhss-2194	74	10	=	=	SYM
ajrhss-2194	74	11	1	1	NUM
ajrhss-2194	74	12	\	\	NOUN
ajrhss-2194	74	13	)	)	PUNCT
ajrhss-2194	74	14	and	and	CCONJ
ajrhss-2194	74	15	\	\	PROPN
ajrhss-2194	74	16	(	(	PUNCT
ajrhss-2194	74	17	y	y	NOUN
ajrhss-2194	74	18	=	=	SYM
ajrhss-2194	74	19	x	x	SYM
ajrhss-2194	74	20	\	\	NOUN
ajrhss-2194	74	21	)	)	PUNCT
ajrhss-2194	74	22	simultaneously	simultaneously	ADV
ajrhss-2194	74	23	finds	find	VERB
ajrhss-2194	74	24	the	the	DET
ajrhss-2194	74	25	points	point	NOUN
ajrhss-2194	74	26	where	where	SCONJ
ajrhss-2194	74	27	a	a	DET
ajrhss-2194	74	28	circle	circle	NOUN
ajrhss-2194	74	29	intersects	intersect	VERB
ajrhss-2194	74	30	a	a	DET
ajrhss-2194	74	31	line	line	NOUN
ajrhss-2194	74	32	.	.	PUNCT
ajrhss-2194	75	1	6	6	X
ajrhss-2194	75	2	.	.	X
ajrhss-2194	75	3	geometric	geometric	ADJ
ajrhss-2194	75	4	construction	construction	NOUN
ajrhss-2194	75	5	and	and	CCONJ
ajrhss-2194	75	6	design	design	NOUN
ajrhss-2194	75	7	in	in	ADP
ajrhss-2194	75	8	fields	field	NOUN
ajrhss-2194	75	9	such	such	ADJ
ajrhss-2194	75	10	as	as	ADP
ajrhss-2194	75	11	architecture	architecture	NOUN
ajrhss-2194	75	12	and	and	CCONJ
ajrhss-2194	75	13	industrial	industrial	ADJ
ajrhss-2194	75	14	design	design	NOUN
ajrhss-2194	75	15	,	,	PUNCT
ajrhss-2194	75	16	geometric	geometric	ADJ
ajrhss-2194	75	17	functions	function	NOUN
ajrhss-2194	75	18	facilitate	facilitate	VERB
ajrhss-2194	75	19	the	the	DET
ajrhss-2194	75	20	precise	precise	ADJ
ajrhss-2194	75	21	construction	construction	NOUN
ajrhss-2194	75	22	and	and	CCONJ
ajrhss-2194	75	23	design	design	NOUN
ajrhss-2194	75	24	of	of	ADP
ajrhss-2194	75	25	objects	object	NOUN
ajrhss-2194	75	26	.	.	PUNCT
ajrhss-2194	76	1	computer	computer	NOUN
ajrhss-2194	76	2	-	-	PUNCT
ajrhss-2194	76	3	aided	aid	VERB
ajrhss-2194	76	4	design	design	NOUN
ajrhss-2194	76	5	(	(	PUNCT
ajrhss-2194	76	6	cad	cad	NOUN
ajrhss-2194	76	7	):	):	PUNCT
ajrhss-2194	76	8	geometric	geometric	ADJ
ajrhss-2194	76	9	functions	function	NOUN
ajrhss-2194	76	10	are	be	AUX
ajrhss-2194	76	11	used	use	VERB
ajrhss-2194	76	12	in	in	ADP
ajrhss-2194	76	13	cad	cad	PROPN
ajrhss-2194	76	14	software	software	NOUN
ajrhss-2194	76	15	to	to	AUX
ajrhss-2194	76	16	model	model	VERB
ajrhss-2194	76	17	complex	complex	ADJ
ajrhss-2194	76	18	shapes	shape	NOUN
ajrhss-2194	76	19	and	and	CCONJ
ajrhss-2194	76	20	structures	structure	NOUN
ajrhss-2194	76	21	.	.	PUNCT
ajrhss-2194	77	1	for	for	ADP
ajrhss-2194	77	2	example	example	NOUN
ajrhss-2194	77	3	,	,	PUNCT
ajrhss-2194	77	4	nurbs	nurbs	X
ajrhss-2194	77	5	(	(	PUNCT
ajrhss-2194	77	6	non	non	ADJ
ajrhss-2194	77	7	-	-	ADJ
ajrhss-2194	77	8	uniform	uniform	ADJ
ajrhss-2194	77	9	rational	rational	ADJ
ajrhss-2194	77	10	b	b	NOUN
ajrhss-2194	77	11	-	-	PUNCT
ajrhss-2194	77	12	splines	spline	NOUN
ajrhss-2194	77	13	)	)	PUNCT
ajrhss-2194	77	14	are	be	AUX
ajrhss-2194	77	15	a	a	DET
ajrhss-2194	77	16	type	type	NOUN
ajrhss-2194	77	17	of	of	ADP
ajrhss-2194	77	18	geometric	geometric	ADJ
ajrhss-2194	77	19	function	function	NOUN
ajrhss-2194	77	20	used	use	VERB
ajrhss-2194	77	21	to	to	PART
ajrhss-2194	77	22	represent	represent	VERB
ajrhss-2194	77	23	curves	curve	NOUN
ajrhss-2194	77	24	and	and	CCONJ
ajrhss-2194	77	25	surfaces	surface	NOUN
ajrhss-2194	77	26	in	in	ADP
ajrhss-2194	77	27	cad	cad	PROPN
ajrhss-2194	77	28	systems	system	NOUN
ajrhss-2194	77	29	.	.	PUNCT
ajrhss-2194	78	1	summary	summary	NOUN
ajrhss-2194	78	2	geometric	geometric	ADJ
ajrhss-2194	78	3	functions	function	NOUN
ajrhss-2194	78	4	are	be	AUX
ajrhss-2194	78	5	indispensable	indispensable	ADJ
ajrhss-2194	78	6	in	in	ADP
ajrhss-2194	78	7	geometry	geometry	NOUN
ajrhss-2194	78	8	for	for	ADP
ajrhss-2194	78	9	defining	define	VERB
ajrhss-2194	78	10	shapes	shape	NOUN
ajrhss-2194	78	11	,	,	PUNCT
ajrhss-2194	78	12	describing	describe	VERB
ajrhss-2194	78	13	transformations	transformation	NOUN
ajrhss-2194	78	14	,	,	PUNCT
ajrhss-2194	78	15	analyzing	analyze	VERB
ajrhss-2194	78	16	properties	property	NOUN
ajrhss-2194	78	17	,	,	PUNCT
ajrhss-2194	78	18	modeling	model	VERB
ajrhss-2194	78	19	physical	physical	ADJ
ajrhss-2194	78	20	systems	system	NOUN
ajrhss-2194	78	21	,	,	PUNCT
ajrhss-2194	78	22	solving	solve	VERB
ajrhss-2194	78	23	geometric	geometric	ADJ
ajrhss-2194	78	24	problems	problem	NOUN
ajrhss-2194	78	25	,	,	PUNCT
ajrhss-2194	78	26	and	and	CCONJ
ajrhss-2194	78	27	aiding	aid	VERB
ajrhss-2194	78	28	in	in	ADP
ajrhss-2194	78	29	construction	construction	NOUN
ajrhss-2194	78	30	and	and	CCONJ
ajrhss-2194	78	31	design	design	NOUN
ajrhss-2194	78	32	.	.	PUNCT
ajrhss-2194	79	1	they	they	PRON
ajrhss-2194	79	2	provide	provide	VERB
ajrhss-2194	79	3	a	a	DET
ajrhss-2194	79	4	powerful	powerful	ADJ
ajrhss-2194	79	5	mathematical	mathematical	ADJ
ajrhss-2194	79	6	framework	framework	NOUN
ajrhss-2194	79	7	that	that	PRON
ajrhss-2194	79	8	supports	support	VERB
ajrhss-2194	79	9	a	a	DET
ajrhss-2194	79	10	wide	wide	ADJ
ajrhss-2194	79	11	range	range	NOUN
ajrhss-2194	79	12	of	of	ADP
ajrhss-2194	79	13	applications	application	NOUN
ajrhss-2194	79	14	in	in	ADP
ajrhss-2194	79	15	science	science	NOUN
ajrhss-2194	79	16	,	,	PUNCT
ajrhss-2194	79	17	engineering	engineering	NOUN
ajrhss-2194	79	18	,	,	PUNCT
ajrhss-2194	79	19	and	and	CCONJ
ajrhss-2194	79	20	technology	technology	NOUN
ajrhss-2194	79	21	.	.	PUNCT
ajrhss-2194	80	1	references	reference	NOUN
ajrhss-2194	80	2	1	1	NUM
ajrhss-2194	80	3	.	.	PUNCT
ajrhss-2194	80	4	complex	complex	ADJ
ajrhss-2194	80	5	analysis	analysis	NOUN
ajrhss-2194	80	6	and	and	CCONJ
ajrhss-2194	80	7	conformal	conformal	NOUN
ajrhss-2194	80	8	mappings	mapping	NOUN
ajrhss-2194	80	9	:	:	PUNCT
ajrhss-2194	80	10	ahlfors	ahlfor	NOUN
ajrhss-2194	80	11	,	,	PUNCT
ajrhss-2194	80	12	lars	lars	PROPN
ajrhss-2194	80	13	v.	v.	PROPN
ajrhss-2194	80	14	"	"	PUNCT
ajrhss-2194	80	15	complex	complex	ADJ
ajrhss-2194	80	16	analysis	analysis	NOUN
ajrhss-2194	80	17	:	:	PUNCT
ajrhss-2194	80	18	an	an	DET
ajrhss-2194	80	19	introduction	introduction	NOUN
ajrhss-2194	80	20	to	to	ADP
ajrhss-2194	80	21	the	the	DET
ajrhss-2194	80	22	theory	theory	NOUN
ajrhss-2194	80	23	of	of	ADP
ajrhss-2194	80	24	analytic	analytic	ADJ
ajrhss-2194	80	25	functions	function	NOUN
ajrhss-2194	80	26	of	of	ADP
ajrhss-2194	80	27	one	one	NUM
ajrhss-2194	80	28	complex	complex	ADJ
ajrhss-2194	80	29	variable	variable	NOUN
ajrhss-2194	80	30	.	.	PUNCT
ajrhss-2194	80	31	"	"	PUNCT
ajrhss-2194	80	32	mcgraw	mcgraw	PROPN
ajrhss-2194	80	33	-	-	PUNCT
ajrhss-2194	80	34	hill	hill	NOUN
ajrhss-2194	80	35	education	education	NOUN
ajrhss-2194	80	36	,	,	PUNCT
ajrhss-2194	80	37	1979	1979	NUM
ajrhss-2194	80	38	.	.	PUNCT
ajrhss-2194	81	1	2	2	X
ajrhss-2194	81	2	.	.	X
ajrhss-2194	81	3	algebraic	algebraic	ADJ
ajrhss-2194	81	4	geometry	geometry	NOUN
ajrhss-2194	81	5	:	:	PUNCT
ajrhss-2194	81	6	hartshorne	hartshorne	PROPN
ajrhss-2194	81	7	,	,	PUNCT
ajrhss-2194	81	8	robin	robin	PROPN
ajrhss-2194	81	9	.	.	PUNCT
ajrhss-2194	82	1	"	"	PUNCT
ajrhss-2194	82	2	algebraic	algebraic	ADJ
ajrhss-2194	82	3	geometry	geometry	NOUN
ajrhss-2194	82	4	.	.	PUNCT
ajrhss-2194	82	5	"	"	PUNCT
ajrhss-2194	82	6	springer	springer	NOUN
ajrhss-2194	82	7	,	,	PUNCT
ajrhss-2194	82	8	1977	1977	NUM
ajrhss-2194	82	9	.	.	PUNCT
ajrhss-2194	83	1	3	3	X
ajrhss-2194	83	2	.	.	X
ajrhss-2194	83	3	geometric	geometric	ADJ
ajrhss-2194	83	4	transformations	transformation	NOUN
ajrhss-2194	83	5	:	:	PUNCT
ajrhss-2194	83	6	yaglom	yaglom	NOUN
ajrhss-2194	83	7	,	,	PUNCT
ajrhss-2194	83	8	isaak	isaak	PROPN
ajrhss-2194	83	9	m.	m.	NOUN
ajrhss-2194	83	10	"	"	PUNCT
ajrhss-2194	83	11	geometric	geometric	ADJ
ajrhss-2194	83	12	transformations	transformation	NOUN
ajrhss-2194	83	13	i.	i.	NOUN
ajrhss-2194	83	14	"	"	PUNCT
ajrhss-2194	83	15	mathematical	mathematical	ADJ
ajrhss-2194	83	16	association	association	PROPN
ajrhss-2194	83	17	of	of	ADP
ajrhss-2194	83	18	america	america	PROPN
ajrhss-2194	83	19	,	,	PUNCT
ajrhss-2194	83	20	1962	1962	NUM
ajrhss-2194	83	21	.	.	PUNCT
ajrhss-2194	84	1	goldstein	goldstein	PROPN
ajrhss-2194	84	2	,	,	PUNCT
ajrhss-2194	84	3	herbert	herbert	PROPN
ajrhss-2194	84	4	,	,	PUNCT
ajrhss-2194	84	5	poole	poole	PROPN
ajrhss-2194	84	6	,	,	PUNCT
ajrhss-2194	84	7	charles	charles	PROPN
ajrhss-2194	84	8	p.	p.	PROPN
ajrhss-2194	84	9	,	,	PUNCT
ajrhss-2194	84	10	and	and	CCONJ
ajrhss-2194	84	11	safko	safko	NOUN
ajrhss-2194	84	12	,	,	PUNCT
ajrhss-2194	84	13	john	john	PROPN
ajrhss-2194	84	14	l.	l.	PROPN
ajrhss-2194	84	15	"	"	PUNCT
ajrhss-2194	84	16	classical	classical	ADJ
ajrhss-2194	84	17	mechanics	mechanic	NOUN
ajrhss-2194	84	18	.	.	PUNCT
ajrhss-2194	84	19	"	"	PUNCT
ajrhss-2194	85	1	addison	addison	PROPN
ajrhss-2194	85	2	-	-	PUNCT
ajrhss-2194	85	3	wesley	wesley	PROPN
ajrhss-2194	85	4	,	,	PUNCT
ajrhss-2194	85	5	2002	2002	NUM
ajrhss-2194	85	6	.	.	PUNCT
ajrhss-2194	86	1	4	4	X
ajrhss-2194	86	2	.	.	X
ajrhss-2194	86	3	geometric	geometric	ADJ
ajrhss-2194	86	4	sequences	sequence	NOUN
ajrhss-2194	86	5	and	and	CCONJ
ajrhss-2194	86	6	series	series	NOUN
ajrhss-2194	86	7	:	:	PUNCT
ajrhss-2194	86	8	stewart	stewart	PROPN
ajrhss-2194	86	9	,	,	PUNCT
ajrhss-2194	86	10	james	james	PROPN
ajrhss-2194	86	11	.	.	PUNCT
ajrhss-2194	87	1	"	"	PUNCT
ajrhss-2194	87	2	calculus	calculus	NOUN
ajrhss-2194	87	3	:	:	PUNCT
ajrhss-2194	87	4	early	early	ADJ
ajrhss-2194	87	5	transcendentals	transcendental	NOUN
ajrhss-2194	87	6	.	.	PUNCT
ajrhss-2194	87	7	"	"	PUNCT
ajrhss-2194	88	1	cengage	cengage	PROPN
ajrhss-2194	88	2	learning	learning	PROPN
ajrhss-2194	88	3	,	,	PUNCT
ajrhss-2194	88	4	2015	2015	NUM
ajrhss-2194	88	5	.	.	PUNCT
ajrhss-2194	89	1	5	5	NUM
ajrhss-2194	89	2	.	.	PUNCT
ajrhss-2194	89	3	trigonometric	trigonometric	ADJ
ajrhss-2194	89	4	functions	function	NOUN
ajrhss-2194	89	5	:	:	PUNCT
ajrhss-2194	89	6	simmons	simmon	NOUN
ajrhss-2194	89	7	,	,	PUNCT
ajrhss-2194	89	8	george	george	PROPN
ajrhss-2194	89	9	f.	f.	PROPN
ajrhss-2194	90	1	"	"	PUNCT
ajrhss-2194	90	2	precalculus	precalculus	ADJ
ajrhss-2194	90	3	mathematics	mathematic	NOUN
ajrhss-2194	90	4	in	in	ADP
ajrhss-2194	90	5	a	a	DET
ajrhss-2194	90	6	nutshell	nutshell	NOUN
ajrhss-2194	90	7	:	:	PUNCT
ajrhss-2194	90	8	geometry	geometry	NOUN
ajrhss-2194	90	9	,	,	PUNCT
ajrhss-2194	90	10	algebra	algebra	NOUN
ajrhss-2194	90	11	,	,	PUNCT
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ajrhss-2194	90	13	.	.	PUNCT
ajrhss-2194	90	14	"	"	PUNCT
ajrhss-2194	91	1	wipf	wipf	NOUN
ajrhss-2194	91	2	and	and	CCONJ
ajrhss-2194	91	3	stock	stock	NOUN
ajrhss-2194	91	4	publishers	publisher	NOUN
ajrhss-2194	91	5	,	,	PUNCT
ajrhss-2194	91	6	2003	2003	NUM
ajrhss-2194	91	7	.	.	PUNCT
ajrhss-2194	92	1	american	american	PROPN
ajrhss-2194	92	2	journal	journal	PROPN
ajrhss-2194	92	3	of	of	ADP
ajrhss-2194	92	4	research	research	NOUN
ajrhss-2194	92	5	in	in	ADP
ajrhss-2194	92	6	humanities	humanity	NOUN
ajrhss-2194	92	7	and	and	CCONJ
ajrhss-2194	92	8	social	social	ADJ
ajrhss-2194	92	9	sciences	science	NOUN
ajrhss-2194	92	10	volume	volume	NOUN
ajrhss-2194	92	11	25	25	NUM
ajrhss-2194	92	12	june	june	PROPN
ajrhss-2194	92	13	2024	2024	NUM
ajrhss-2194	92	14	p	p	NOUN
ajrhss-2194	92	15	a	a	PRON
ajrhss-2194	92	16	g	g	NOUN
ajrhss-2194	92	17	e	e	NOUN
ajrhss-2194	93	1	|	|	ADV
ajrhss-2194	93	2	45	45	NUM
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ajrhss-2194	93	4	6	6	NUM
ajrhss-2194	93	5	.	.	PUNCT
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ajrhss-2194	93	7	geometry	geometry	NOUN
ajrhss-2194	93	8	:	:	PUNCT
ajrhss-2194	94	1	do	do	VERB
ajrhss-2194	94	2	carmo	carmo	PROPN
ajrhss-2194	94	3	,	,	PUNCT
ajrhss-2194	94	4	manfredo	manfredo	PROPN
ajrhss-2194	94	5	.	.	PUNCT
ajrhss-2194	95	1	"	"	PUNCT
ajrhss-2194	95	2	differential	differential	ADJ
ajrhss-2194	95	3	geometry	geometry	NOUN
ajrhss-2194	95	4	of	of	ADP
ajrhss-2194	95	5	curves	curve	NOUN
ajrhss-2194	95	6	and	and	CCONJ
ajrhss-2194	95	7	surfaces	surface	NOUN
ajrhss-2194	95	8	.	.	PUNCT
ajrhss-2194	95	9	"	"	PUNCT
ajrhss-2194	95	10	prentice	prentice	NOUN
ajrhss-2194	95	11	-	-	PUNCT
ajrhss-2194	95	12	hall	hall	NOUN
ajrhss-2194	95	13	,	,	PUNCT
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ajrhss-2194	95	15	.	.	PUNCT
ajrhss-2194	96	1	7	7	X
ajrhss-2194	96	2	.	.	X
ajrhss-2194	96	3	geometric	geometric	ADJ
ajrhss-2194	96	4	functions	function	NOUN
ajrhss-2194	96	5	in	in	ADP
ajrhss-2194	96	6	physics	physics	NOUN
ajrhss-2194	96	7	and	and	CCONJ
ajrhss-2194	96	8	engineering	engineering	PROPN
ajrhss-2194	96	9	:	:	PUNCT
ajrhss-2194	96	10	marion	marion	PROPN
ajrhss-2194	96	11	,	,	PUNCT
ajrhss-2194	96	12	jerry	jerry	PROPN
ajrhss-2194	96	13	b.	b.	PROPN
ajrhss-2194	96	14	,	,	PUNCT
ajrhss-2194	96	15	and	and	CCONJ
ajrhss-2194	96	16	thornton	thornton	PROPN
ajrhss-2194	96	17	,	,	PUNCT
ajrhss-2194	96	18	stephen	stephen	NOUN
ajrhss-2194	96	19	t.	t.	NOUN
ajrhss-2194	97	1	"	"	PUNCT
ajrhss-2194	97	2	classical	classical	ADJ
ajrhss-2194	97	3	dynamics	dynamic	NOUN
ajrhss-2194	97	4	of	of	ADP
ajrhss-2194	97	5	particles	particle	NOUN
ajrhss-2194	97	6	and	and	CCONJ
ajrhss-2194	97	7	systems	system	NOUN
ajrhss-2194	97	8	.	.	PUNCT
ajrhss-2194	97	9	"	"	PUNCT
ajrhss-2194	97	10	brooks	brooks	PROPN
ajrhss-2194	97	11	cole	cole	PROPN
ajrhss-2194	97	12	,	,	PUNCT
ajrhss-2194	97	13	2003	2003	NUM
ajrhss-2194	97	14	.	.	PUNCT
ajrhss-2194	98	1	hibbeler	hibbeler	NOUN
ajrhss-2194	98	2	,	,	PUNCT
ajrhss-2194	98	3	r.	r.	PROPN
ajrhss-2194	98	4	c.	c.	PROPN
ajrhss-2194	98	5	"	"	PUNCT
ajrhss-2194	98	6	engineering	engineering	NOUN
ajrhss-2194	98	7	mechanics	mechanic	NOUN
ajrhss-2194	98	8	:	:	PUNCT
ajrhss-2194	98	9	dynamics	dynamic	NOUN
ajrhss-2194	98	10	.	.	PUNCT
ajrhss-2194	98	11	"	"	PUNCT
ajrhss-2194	99	1	pearson	pearson	PROPN
ajrhss-2194	99	2	,	,	PUNCT
ajrhss-2194	99	3	2015	2015	NUM
ajrhss-2194	99	4	.	.	PUNCT
ajrhss-2194	100	1	8	8	NUM
ajrhss-2194	100	2	.	.	PUNCT
ajrhss-2194	101	1	geometric	geometric	ADJ
ajrhss-2194	101	2	construction	construction	NOUN
ajrhss-2194	101	3	and	and	CCONJ
ajrhss-2194	101	4	design	design	NOUN
ajrhss-2194	101	5	:	:	PUNCT
ajrhss-2194	101	6	mortenson	mortenson	PROPN
ajrhss-2194	101	7	,	,	PUNCT
ajrhss-2194	101	8	michael	michael	PROPN
ajrhss-2194	101	9	e.	e.	PROPN
ajrhss-2194	101	10	"	"	PUNCT
ajrhss-2194	101	11	mathematics	mathematics	PROPN
ajrhss-2194	101	12	for	for	ADP
ajrhss-2194	101	13	computer	computer	NOUN
ajrhss-2194	101	14	graphics	graphic	NOUN
ajrhss-2194	101	15	applications	application	NOUN
ajrhss-2194	101	16	.	.	PUNCT
ajrhss-2194	101	17	"	"	PUNCT
ajrhss-2194	102	1	industrial	industrial	ADJ
ajrhss-2194	102	2	press	press	NOUN
ajrhss-2194	102	3	,	,	PUNCT
ajrhss-2194	102	4	1999	1999	NUM
ajrhss-2194	102	5	.	.	PUNCT
ajrhss-2194	103	1	piegl	piegl	PROPN
ajrhss-2194	103	2	,	,	PUNCT
ajrhss-2194	103	3	les	le	NOUN
ajrhss-2194	103	4	,	,	PUNCT
ajrhss-2194	103	5	and	and	CCONJ
ajrhss-2194	103	6	tiller	tiller	NOUN
ajrhss-2194	103	7	,	,	PUNCT
ajrhss-2194	103	8	wayne	wayne	PROPN
ajrhss-2194	103	9	.	.	PUNCT
ajrhss-2194	104	1	"	"	PUNCT
ajrhss-2194	104	2	the	the	DET
ajrhss-2194	104	3	nurbs	nurbs	NOUN
ajrhss-2194	104	4	book	book	NOUN
ajrhss-2194	104	5	.	.	PUNCT
ajrhss-2194	104	6	"	"	PUNCT
ajrhss-2194	104	7	springer	springer	NOUN
ajrhss-2194	104	8	,	,	PUNCT
ajrhss-2194	104	9	1997	1997	NUM
ajrhss-2194	104	10	.	.	PUNCT