id	sid	tid	token	lemma	pos
cana-5309	1	1	communications	communication	NOUN
cana-5309	1	2	on	on	ADP
cana-5309	1	3	applied	apply	VERB
cana-5309	1	4	nonlinear	nonlinear	ADJ
cana-5309	1	5	analysis	analysis	NOUN
cana-5309	1	6	issn	issn	NOUN
cana-5309	1	7	:	:	PUNCT
cana-5309	1	8	1074	1074	NUM
cana-5309	1	9	-	-	PUNCT
cana-5309	1	10	133x	133x	NUM
cana-5309	1	11	vol	vol	NOUN
cana-5309	1	12	31	31	NUM
cana-5309	1	13	no	no	NOUN
cana-5309	1	14	.	.	PUNCT
cana-5309	2	1	4s	4s	NUM
cana-5309	2	2	(	(	PUNCT
cana-5309	2	3	2024	2024	NUM
cana-5309	2	4	)	)	PUNCT
cana-5309	2	5	672	672	NUM
cana-5309	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	2	7	geometric	geometric	ADJ
cana-5309	2	8	insights	insight	NOUN
cana-5309	2	9	into	into	ADP
cana-5309	2	10	non	non	ADJ
cana-5309	2	11	-	-	ADJ
cana-5309	2	12	euclidean	euclidean	ADJ
cana-5309	2	13	spaces	space	NOUN
cana-5309	2	14	:	:	PUNCT
cana-5309	2	15	a	a	DET
cana-5309	2	16	study	study	NOUN
cana-5309	2	17	in	in	ADP
cana-5309	2	18	hyperbolic	hyperbolic	ADJ
cana-5309	2	19	geometry	geometry	NOUN
cana-5309	2	20	1s	1s	X
cana-5309	2	21	.	.	PUNCT
cana-5309	3	1	balamuralitharan	balamuralitharan	PROPN
cana-5309	3	2	,	,	PUNCT
cana-5309	3	3	2a	2a	NUM
cana-5309	3	4	.	.	PUNCT
cana-5309	4	1	nasreen	nasreen	VERB
cana-5309	4	2	sultana	sultana	NOUN
cana-5309	4	3	,	,	PUNCT
cana-5309	4	4	3r	3r	NUM
cana-5309	4	5	.	.	PUNCT
cana-5309	5	1	arulprakasam	arulprakasam	PROPN
cana-5309	5	2	,	,	PUNCT
cana-5309	5	3	4v.venkata	4v.venkata	PROPN
cana-5309	5	4	kumar	kumar	PROPN
cana-5309	5	5	,	,	PUNCT
cana-5309	5	6	5r	5r	NUM
cana-5309	5	7	giri	giri	PROPN
cana-5309	5	8	prasad	prasad	PROPN
cana-5309	5	9	1adjunct	1adjunct	NUM
cana-5309	5	10	faculty	faculty	NOUN
cana-5309	5	11	,	,	PUNCT
cana-5309	5	12	department	department	NOUN
cana-5309	5	13	of	of	ADP
cana-5309	5	14	pure	pure	ADJ
cana-5309	5	15	and	and	CCONJ
cana-5309	5	16	applied	applied	ADJ
cana-5309	5	17	mathematics	mathematic	NOUN
cana-5309	5	18	,	,	PUNCT
cana-5309	5	19	saveetha	saveetha	PROPN
cana-5309	5	20	school	school	PROPN
cana-5309	5	21	of	of	ADP
cana-5309	5	22	engineering	engineering	NOUN
cana-5309	5	23	,	,	PUNCT
cana-5309	5	24	simats	simat	NOUN
cana-5309	5	25	,	,	PUNCT
cana-5309	5	26	chennai	chennai	PROPN
cana-5309	5	27	,	,	PUNCT
cana-5309	5	28	tamil	tamil	PROPN
cana-5309	5	29	nadu	nadu	PROPN
cana-5309	5	30	,	,	PUNCT
cana-5309	5	31	india	india	PROPN
cana-5309	5	32	email	email	NOUN
cana-5309	5	33	i	i	PROPN
cana-5309	5	34	d	d	PROPN
cana-5309	5	35	:	:	PUNCT
cana-5309	5	36	balamurali.maths@gmail.com	balamurali.maths@gmail.com	X
cana-5309	5	37	2assistant	2assistant	NUM
cana-5309	5	38	professor	professor	NOUN
cana-5309	5	39	,	,	PUNCT
cana-5309	5	40	department	department	NOUN
cana-5309	5	41	of	of	ADP
cana-5309	5	42	mathematics	mathematics	PROPN
cana-5309	5	43	,	,	PUNCT
cana-5309	5	44	s.d.n.b	s.d.n.b	PROPN
cana-5309	5	45	.	.	PUNCT
cana-5309	6	1	vaishnav	vaishnav	PROPN
cana-5309	6	2	college	college	PROPN
cana-5309	6	3	for	for	ADP
cana-5309	6	4	women	woman	NOUN
cana-5309	6	5	,	,	PUNCT
cana-5309	6	6	chennai	chennai	PROPN
cana-5309	6	7	,	,	PUNCT
cana-5309	6	8	tamil	tamil	PROPN
cana-5309	6	9	nadu	nadu	PROPN
cana-5309	6	10	,	,	PUNCT
cana-5309	6	11	india	india	PROPN
cana-5309	6	12	email	email	NOUN
cana-5309	6	13	:	:	PUNCT
cana-5309	7	1	a.nasreensultana@gmail.com	a.nasreensultana@gmail.com	PROPN
cana-5309	7	2	3department	3department	NUM
cana-5309	7	3	of	of	ADP
cana-5309	7	4	mathematics	mathematic	NOUN
cana-5309	7	5	,	,	PUNCT
cana-5309	7	6	college	college	NOUN
cana-5309	7	7	of	of	ADP
cana-5309	7	8	engineering	engineering	NOUN
cana-5309	7	9	and	and	CCONJ
cana-5309	7	10	technology	technology	NOUN
cana-5309	7	11	,	,	PUNCT
cana-5309	7	12	srm	srm	PROPN
cana-5309	7	13	institute	institute	PROPN
cana-5309	7	14	of	of	ADP
cana-5309	7	15	science	science	NOUN
cana-5309	7	16	and	and	CCONJ
cana-5309	7	17	technology	technology	NOUN
cana-5309	7	18	,	,	PUNCT
cana-5309	7	19	srm	srm	NOUN
cana-5309	7	20	nagar	nagar	NOUN
cana-5309	7	21	,	,	PUNCT
cana-5309	7	22	kattankulathur	kattankulathur	PROPN
cana-5309	7	23	603203	603203	NUM
cana-5309	7	24	,	,	PUNCT
cana-5309	7	25	chengalpattu	chengalpattu	ADJ
cana-5309	7	26	district	district	NOUN
cana-5309	7	27	,	,	PUNCT
cana-5309	7	28	tamilnadu	tamilnadu	PROPN
cana-5309	7	29	,	,	PUNCT
cana-5309	7	30	india	india	PROPN
cana-5309	7	31	r.aruljeeva@gmail.com	r.aruljeeva@gmail.com	PROPN
cana-5309	7	32	4associate	4associate	PROPN
cana-5309	7	33	professor	professor	NOUN
cana-5309	7	34	,	,	PUNCT
cana-5309	7	35	department	department	NOUN
cana-5309	7	36	of	of	ADP
cana-5309	7	37	mathematics	mathematics	PROPN
cana-5309	7	38	,	,	PUNCT
cana-5309	7	39	aditya	aditya	PROPN
cana-5309	7	40	university	university	PROPN
cana-5309	7	41	,	,	PUNCT
cana-5309	7	42	surampalem	surampalem	NOUN
cana-5309	7	43	,	,	PUNCT
cana-5309	7	44	india	india	PROPN
cana-5309	7	45	,	,	PUNCT
cana-5309	7	46	venkatakumar.v@aec.edu.in	venkatakumar.v@aec.edu.in	PROPN
cana-5309	8	1	5associate	5associate	PROPN
cana-5309	8	2	professor	professor	NOUN
cana-5309	8	3	,	,	PUNCT
cana-5309	8	4	dept	dept	NOUN
cana-5309	8	5	of	of	ADP
cana-5309	8	6	petroleum	petroleum	NOUN
cana-5309	8	7	technology	technology	NOUN
cana-5309	8	8	,	,	PUNCT
cana-5309	8	9	aditya	aditya	PROPN
cana-5309	8	10	university	university	PROPN
cana-5309	8	11	,	,	PUNCT
cana-5309	8	12	surampalem	surampalem	NOUN
cana-5309	8	13	,	,	PUNCT
cana-5309	8	14	india	india	PROPN
cana-5309	8	15	,	,	PUNCT
cana-5309	8	16	hod_pt@aec.edu.in	hod_pt@aec.edu.in	PROPN
cana-5309	8	17	article	article	NOUN
cana-5309	8	18	history	history	NOUN
cana-5309	8	19	:	:	PUNCT
cana-5309	8	20	received	receive	VERB
cana-5309	8	21	:	:	PUNCT
cana-5309	8	22	12	12	NUM
cana-5309	8	23	-	-	PUNCT
cana-5309	8	24	02	02	NUM
cana-5309	8	25	-	-	PUNCT
cana-5309	8	26	2024	2024	NUM
cana-5309	8	27	revised	revise	VERB
cana-5309	8	28	:	:	PUNCT
cana-5309	8	29	15	15	NUM
cana-5309	8	30	-	-	SYM
cana-5309	8	31	03	03	NUM
cana-5309	8	32	-	-	PUNCT
cana-5309	8	33	2024	2024	NUM
cana-5309	8	34	accepted	accept	VERB
cana-5309	8	35	:	:	PUNCT
cana-5309	8	36	19	19	NUM
cana-5309	8	37	-	-	PUNCT
cana-5309	8	38	04	04	NUM
cana-5309	8	39	-	-	PUNCT
cana-5309	8	40	2024	2024	NUM
cana-5309	8	41	abstract	abstract	NOUN
cana-5309	8	42	:	:	PUNCT
cana-5309	8	43	this	this	DET
cana-5309	8	44	work	work	NOUN
cana-5309	8	45	evaluates	evaluate	VERB
cana-5309	8	46	the	the	DET
cana-5309	8	47	core	core	ADJ
cana-5309	8	48	principles	principle	NOUN
cana-5309	8	49	along	along	ADP
cana-5309	8	50	with	with	ADP
cana-5309	8	51	models	model	NOUN
cana-5309	8	52	and	and	CCONJ
cana-5309	8	53	usages	usage	NOUN
cana-5309	8	54	of	of	ADP
cana-5309	8	55	hyperbolic	hyperbolic	ADJ
cana-5309	8	56	geometry	geometry	NOUN
cana-5309	8	57	which	which	PRON
cana-5309	8	58	establishes	establish	VERB
cana-5309	8	59	itself	itself	PRON
cana-5309	8	60	as	as	ADP
cana-5309	8	61	a	a	DET
cana-5309	8	62	main	main	ADJ
cana-5309	8	63	category	category	NOUN
cana-5309	8	64	of	of	ADP
cana-5309	8	65	non	non	ADJ
cana-5309	8	66	-	-	ADJ
cana-5309	8	67	euclidean	euclidean	ADJ
cana-5309	8	68	geometry	geometry	NOUN
cana-5309	8	69	because	because	SCONJ
cana-5309	8	70	it	it	PRON
cana-5309	8	71	lacks	lack	VERB
cana-5309	8	72	euclid	euclid	PROPN
cana-5309	8	73	's	's	PART
cana-5309	8	74	parallel	parallel	ADJ
cana-5309	8	75	axiom	axiom	NOUN
cana-5309	8	76	.	.	PUNCT
cana-5309	9	1	the	the	DET
cana-5309	9	2	analysis	analysis	NOUN
cana-5309	9	3	of	of	ADP
cana-5309	9	4	lines	line	NOUN
cana-5309	9	5	angles	angle	NOUN
cana-5309	9	6	and	and	CCONJ
cana-5309	9	7	triangles	triangle	NOUN
cana-5309	9	8	in	in	ADP
cana-5309	9	9	hyperbolic	hyperbolic	ADJ
cana-5309	9	10	spaces	space	NOUN
cana-5309	9	11	takes	take	VERB
cana-5309	9	12	place	place	NOUN
cana-5309	9	13	through	through	ADP
cana-5309	9	14	the	the	DET
cana-5309	9	15	study	study	NOUN
cana-5309	9	16	of	of	ADP
cana-5309	9	17	the	the	DET
cana-5309	9	18	poincaré	poincaré	PROPN
cana-5309	9	19	disk	disk	NOUN
cana-5309	9	20	model	model	NOUN
cana-5309	9	21	and	and	CCONJ
cana-5309	9	22	the	the	DET
cana-5309	9	23	hyperboloid	hyperboloid	ADJ
cana-5309	9	24	model	model	NOUN
cana-5309	9	25	.	.	PUNCT
cana-5309	10	1	this	this	DET
cana-5309	10	2	paper	paper	NOUN
cana-5309	10	3	examines	examine	VERB
cana-5309	10	4	the	the	DET
cana-5309	10	5	separation	separation	NOUN
cana-5309	10	6	between	between	ADP
cana-5309	10	7	euclidean	euclidean	ADJ
cana-5309	10	8	structures	structure	NOUN
cana-5309	10	9	and	and	CCONJ
cana-5309	10	10	hyperbolic	hyperbolic	ADJ
cana-5309	10	11	structures	structure	NOUN
cana-5309	10	12	while	while	SCONJ
cana-5309	10	13	showing	show	VERB
cana-5309	10	14	their	their	PRON
cana-5309	10	15	distinctive	distinctive	ADJ
cana-5309	10	16	properties	property	NOUN
cana-5309	10	17	which	which	PRON
cana-5309	10	18	affect	affect	VERB
cana-5309	10	19	mathematical	mathematical	ADJ
cana-5309	10	20	and	and	CCONJ
cana-5309	10	21	physical	physical	ADJ
cana-5309	10	22	applications	application	NOUN
cana-5309	10	23	.	.	PUNCT
cana-5309	11	1	keywords	keyword	NOUN
cana-5309	11	2	—	—	PUNCT
cana-5309	11	3	hyperbolic	hyperbolic	ADJ
cana-5309	11	4	geometry	geometry	NOUN
cana-5309	11	5	,	,	PUNCT
cana-5309	11	6	non	non	ADJ
cana-5309	11	7	-	-	ADJ
cana-5309	11	8	euclidean	euclidean	ADJ
cana-5309	11	9	space	space	NOUN
cana-5309	11	10	,	,	PUNCT
cana-5309	11	11	poincaré	poincaré	PROPN
cana-5309	11	12	disk	disk	NOUN
cana-5309	11	13	model	model	NOUN
cana-5309	11	14	,	,	PUNCT
cana-5309	11	15	hyperboloid	hyperboloid	ADJ
cana-5309	11	16	model	model	NOUN
cana-5309	11	17	,	,	PUNCT
cana-5309	11	18	parallel	parallel	ADJ
cana-5309	11	19	postulate	postulate	NOUN
cana-5309	11	20	,	,	PUNCT
cana-5309	11	21	curved	curved	ADJ
cana-5309	11	22	spaces	space	NOUN
cana-5309	11	23	,	,	PUNCT
cana-5309	11	24	geometric	geometric	ADJ
cana-5309	11	25	structures	structure	NOUN
cana-5309	11	26	i.	i.	NOUN
cana-5309	11	27	introduction	introduction	NOUN
cana-5309	11	28	through	through	ADP
cana-5309	11	29	geometry	geometry	NOUN
cana-5309	11	30	humans	human	NOUN
cana-5309	11	31	have	have	AUX
cana-5309	11	32	historically	historically	ADV
cana-5309	11	33	established	establish	VERB
cana-5309	11	34	one	one	NUM
cana-5309	11	35	of	of	ADP
cana-5309	11	36	their	their	PRON
cana-5309	11	37	first	first	ADJ
cana-5309	11	38	methods	method	NOUN
cana-5309	11	39	to	to	PART
cana-5309	11	40	understand	understand	VERB
cana-5309	11	41	the	the	DET
cana-5309	11	42	world	world	NOUN
cana-5309	11	43	since	since	SCONJ
cana-5309	11	44	ancient	ancient	ADJ
cana-5309	11	45	times	time	NOUN
cana-5309	11	46	.	.	PUNCT
cana-5309	12	1	the	the	DET
cana-5309	12	2	foundation	foundation	NOUN
cana-5309	12	3	for	for	ADP
cana-5309	12	4	understanding	understand	VERB
cana-5309	12	5	geometry	geometry	NOUN
cana-5309	12	6	as	as	ADP
cana-5309	12	7	a	a	DET
cana-5309	12	8	discipline	discipline	NOUN
cana-5309	12	9	which	which	PRON
cana-5309	12	10	developed	develop	VERB
cana-5309	12	11	by	by	ADP
cana-5309	12	12	euclid	euclid	PROPN
cana-5309	12	13	in	in	ADP
cana-5309	12	14	300	300	NUM
cana-5309	12	15	bce	bce	NOUN
cana-5309	12	16	originated	originate	VERB
cana-5309	12	17	first	first	ADV
cana-5309	12	18	from	from	ADP
cana-5309	12	19	useful	useful	ADJ
cana-5309	12	20	requirements	requirement	NOUN
cana-5309	12	21	like	like	ADP
cana-5309	12	22	engineering	engineering	NOUN
cana-5309	12	23	and	and	CCONJ
cana-5309	12	24	farming	farming	NOUN
cana-5309	12	25	land	land	NOUN
cana-5309	12	26	.	.	PUNCT
cana-5309	13	1	the	the	DET
cana-5309	13	2	system	system	NOUN
cana-5309	13	3	consisted	consist	VERB
cana-5309	13	4	of	of	ADP
cana-5309	13	5	five	five	NUM
cana-5309	13	6	postulates	postulate	NOUN
cana-5309	13	7	where	where	SCONJ
cana-5309	13	8	the	the	DET
cana-5309	13	9	fifth	fifth	ADJ
cana-5309	13	10	discussed	discuss	VERB
cana-5309	13	11	parallel	parallel	ADJ
cana-5309	13	12	lines	line	NOUN
cana-5309	13	13	specifically	specifically	ADV
cana-5309	13	14	stating	state	VERB
cana-5309	13	15	that	that	SCONJ
cana-5309	13	16	one	one	NUM
cana-5309	13	17	and	and	CCONJ
cana-5309	13	18	only	only	ADV
cana-5309	13	19	one	one	NUM
cana-5309	13	20	line	line	NOUN
cana-5309	13	21	exists	exist	VERB
cana-5309	13	22	parallel	parallel	ADJ
cana-5309	13	23	to	to	ADP
cana-5309	13	24	any	any	DET
cana-5309	13	25	given	give	VERB
cana-5309	13	26	line	line	NOUN
cana-5309	13	27	.	.	PUNCT
cana-5309	14	1	although	although	SCONJ
cana-5309	14	2	fundamental	fundamental	ADJ
cana-5309	14	3	to	to	ADP
cana-5309	14	4	the	the	DET
cana-5309	14	5	structure	structure	NOUN
cana-5309	14	6	the	the	DET
cana-5309	14	7	postulate	postulate	NOUN
cana-5309	14	8	presented	present	VERB
cana-5309	14	9	difficulties	difficulty	NOUN
cana-5309	14	10	which	which	PRON
cana-5309	14	11	made	make	VERB
cana-5309	14	12	it	it	PRON
cana-5309	14	13	seem	seem	VERB
cana-5309	14	14	more	more	ADV
cana-5309	14	15	complicated	complicated	ADJ
cana-5309	14	16	and	and	CCONJ
cana-5309	14	17	less	less	ADV
cana-5309	14	18	straightforward	straightforward	ADJ
cana-5309	14	19	than	than	ADP
cana-5309	14	20	other	other	ADJ
cana-5309	14	21	principles	principle	NOUN
cana-5309	14	22	.	.	PUNCT
cana-5309	15	1	mathematicians	mathematician	NOUN
cana-5309	15	2	fruitlessly	fruitlessly	ADV
cana-5309	15	3	tried	try	VERB
cana-5309	15	4	to	to	PART
cana-5309	15	5	demonstrate	demonstrate	VERB
cana-5309	15	6	the	the	DET
cana-5309	15	7	parallel	parallel	ADJ
cana-5309	15	8	postulate	postulate	NOUN
cana-5309	15	9	as	as	ADP
cana-5309	15	10	a	a	DET
cana-5309	15	11	theorem	theorem	NOUN
cana-5309	15	12	for	for	ADP
cana-5309	15	13	more	more	ADJ
cana-5309	15	14	than	than	ADP
cana-5309	15	15	two	two	NUM
cana-5309	15	16	thousand	thousand	NUM
cana-5309	15	17	communications	communication	NOUN
cana-5309	15	18	on	on	ADP
cana-5309	15	19	applied	apply	VERB
cana-5309	15	20	nonlinear	nonlinear	ADJ
cana-5309	15	21	analysis	analysis	NOUN
cana-5309	15	22	issn	issn	NOUN
cana-5309	15	23	:	:	PUNCT
cana-5309	15	24	1074	1074	NUM
cana-5309	15	25	-	-	PUNCT
cana-5309	15	26	133x	133x	NUM
cana-5309	15	27	vol	vol	NOUN
cana-5309	15	28	31	31	NUM
cana-5309	15	29	no	no	NOUN
cana-5309	15	30	.	.	PUNCT
cana-5309	16	1	4s	4s	NUM
cana-5309	16	2	(	(	PUNCT
cana-5309	16	3	2024	2024	NUM
cana-5309	16	4	)	)	PUNCT
cana-5309	16	5	673	673	NUM
cana-5309	16	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-5309	16	7	years	year	NOUN
cana-5309	16	8	while	while	SCONJ
cana-5309	16	9	treating	treat	VERB
cana-5309	16	10	it	it	PRON
cana-5309	16	11	as	as	ADP
cana-5309	16	12	a	a	DET
cana-5309	16	13	basic	basic	ADJ
cana-5309	16	14	starting	starting	NOUN
cana-5309	16	15	assumption	assumption	NOUN
cana-5309	16	16	.	.	PUNCT
cana-5309	17	1	the	the	DET
cana-5309	17	2	unsuccessful	unsuccessful	ADJ
cana-5309	17	3	attempts	attempt	NOUN
cana-5309	17	4	at	at	ADP
cana-5309	17	5	proving	prove	VERB
cana-5309	17	6	the	the	DET
cana-5309	17	7	parallel	parallel	ADJ
cana-5309	17	8	postulate	postulate	NOUN
cana-5309	17	9	throughout	throughout	ADP
cana-5309	17	10	history	history	NOUN
cana-5309	17	11	finally	finally	ADV
cana-5309	17	12	revealed	reveal	VERB
cana-5309	17	13	in	in	ADP
cana-5309	17	14	the	the	DET
cana-5309	17	15	19th	19th	ADJ
cana-5309	17	16	century	century	NOUN
cana-5309	17	17	that	that	PRON
cana-5309	17	18	geometries	geometry	NOUN
cana-5309	17	19	could	could	AUX
cana-5309	17	20	exist	exist	VERB
cana-5309	17	21	which	which	PRON
cana-5309	17	22	rejected	reject	VERB
cana-5309	17	23	this	this	DET
cana-5309	17	24	assumption	assumption	NOUN
cana-5309	17	25	[	[	X
cana-5309	17	26	1	1	NUM
cana-5309	17	27	-	-	SYM
cana-5309	17	28	4	4	NUM
cana-5309	17	29	]	]	PUNCT
cana-5309	17	30	.	.	PUNCT
cana-5309	18	1	mathematicians	mathematician	NOUN
cana-5309	18	2	discovered	discover	VERB
cana-5309	18	3	the	the	DET
cana-5309	18	4	new	new	ADJ
cana-5309	18	5	era	era	NOUN
cana-5309	18	6	of	of	ADP
cana-5309	18	7	non	non	ADJ
cana-5309	18	8	-	-	ADJ
cana-5309	18	9	euclidean	euclidean	ADJ
cana-5309	18	10	geometries	geometry	NOUN
cana-5309	18	11	because	because	SCONJ
cana-5309	18	12	of	of	ADP
cana-5309	18	13	which	which	PRON
cana-5309	18	14	hyperbolic	hyperbolic	ADJ
cana-5309	18	15	geometry	geometry	NOUN
cana-5309	18	16	became	become	VERB
cana-5309	18	17	a	a	DET
cana-5309	18	18	major	major	ADJ
cana-5309	18	19	research	research	NOUN
cana-5309	18	20	area	area	NOUN
cana-5309	18	21	.	.	PUNCT
cana-5309	19	1	from	from	ADP
cana-5309	19	2	a	a	DET
cana-5309	19	3	point	point	NOUN
cana-5309	19	4	outside	outside	ADP
cana-5309	19	5	a	a	DET
cana-5309	19	6	given	give	VERB
cana-5309	19	7	line	line	NOUN
cana-5309	19	8	in	in	ADP
cana-5309	19	9	hyperbolic	hyperbolic	ADJ
cana-5309	19	10	space	space	NOUN
cana-5309	19	11	an	an	DET
cana-5309	19	12	infinite	infinite	ADJ
cana-5309	19	13	number	number	NOUN
cana-5309	19	14	of	of	ADP
cana-5309	19	15	lines	line	NOUN
cana-5309	19	16	will	will	AUX
cana-5309	19	17	extend	extend	VERB
cana-5309	19	18	without	without	ADP
cana-5309	19	19	intersecting	intersect	VERB
cana-5309	19	20	the	the	DET
cana-5309	19	21	first	first	ADJ
cana-5309	19	22	line	line	NOUN
cana-5309	19	23	.	.	PUNCT
cana-5309	20	1	within	within	ADP
cana-5309	20	2	hyperbolic	hyperbolic	ADJ
cana-5309	20	3	geometry	geometry	NOUN
cana-5309	20	4	we	we	PRON
cana-5309	20	5	find	find	VERB
cana-5309	20	6	a	a	DET
cana-5309	20	7	world	world	NOUN
cana-5309	20	8	that	that	PRON
cana-5309	20	9	breaks	break	VERB
cana-5309	20	10	most	most	ADJ
cana-5309	20	11	of	of	ADP
cana-5309	20	12	the	the	DET
cana-5309	20	13	fundamental	fundamental	ADJ
cana-5309	20	14	principles	principle	NOUN
cana-5309	20	15	associated	associate	VERB
cana-5309	20	16	with	with	ADP
cana-5309	20	17	euclidean	euclidean	ADJ
cana-5309	20	18	space	space	NOUN
cana-5309	20	19	.	.	PUNCT
cana-5309	21	1	in	in	ADP
cana-5309	21	2	hyperbolic	hyperbolic	ADJ
cana-5309	21	3	geometry	geometry	NOUN
cana-5309	21	4	one	one	NUM
cana-5309	21	5	finds	find	VERB
cana-5309	21	6	the	the	DET
cana-5309	21	7	sum	sum	NOUN
cana-5309	21	8	of	of	ADP
cana-5309	21	9	triangle	triangle	NOUN
cana-5309	21	10	angles	angle	NOUN
cana-5309	21	11	to	to	PART
cana-5309	21	12	be	be	AUX
cana-5309	21	13	smaller	small	ADJ
cana-5309	21	14	than	than	ADP
cana-5309	21	15	180	180	NUM
cana-5309	21	16	degrees	degree	NOUN
cana-5309	21	17	and	and	CCONJ
cana-5309	21	18	the	the	DET
cana-5309	21	19	circumference	circumference	NOUN
cana-5309	21	20	-	-	PUNCT
cana-5309	21	21	to	to	ADP
cana-5309	21	22	-	-	PUNCT
cana-5309	21	23	diameter	diameter	NOUN
cana-5309	21	24	ratio	ratio	NOUN
cana-5309	21	25	surpasses	surpass	VERB
cana-5309	21	26	π	π	PROPN
cana-5309	21	27	while	while	SCONJ
cana-5309	21	28	geodesics	geodesic	NOUN
cana-5309	21	29	display	display	VERB
cana-5309	21	30	curvilinear	curvilinear	NOUN
cana-5309	21	31	shapes	shape	NOUN
cana-5309	21	32	based	base	VERB
cana-5309	21	33	on	on	ADP
cana-5309	21	34	visualization	visualization	NOUN
cana-5309	21	35	approaches	approach	NOUN
cana-5309	21	36	.	.	PUNCT
cana-5309	22	1	multiple	multiple	ADJ
cana-5309	22	2	methods	method	NOUN
cana-5309	22	3	exist	exist	VERB
cana-5309	22	4	for	for	ADP
cana-5309	22	5	displaying	display	VERB
cana-5309	22	6	hyperbolic	hyperbolic	ADJ
cana-5309	22	7	geometry	geometry	NOUN
cana-5309	22	8	features	feature	NOUN
cana-5309	22	9	which	which	PRON
cana-5309	22	10	enhance	enhance	VERB
cana-5309	22	11	its	its	PRON
cana-5309	22	12	difficult	difficult	ADJ
cana-5309	22	13	concepts	concept	NOUN
cana-5309	22	14	for	for	ADP
cana-5309	22	15	better	well	ADJ
cana-5309	22	16	understanding	understanding	NOUN
cana-5309	22	17	.	.	PUNCT
cana-5309	23	1	two	two	NUM
cana-5309	23	2	essential	essential	ADJ
cana-5309	23	3	models	model	NOUN
cana-5309	23	4	exist	exist	VERB
cana-5309	23	5	in	in	ADP
cana-5309	23	6	both	both	DET
cana-5309	23	7	poincaré	poincaré	NOUN
cana-5309	23	8	disk	disk	NOUN
cana-5309	23	9	and	and	CCONJ
cana-5309	23	10	hyperboloid	hyperboloid	NOUN
cana-5309	23	11	.	.	PUNCT
cana-5309	24	1	when	when	SCONJ
cana-5309	24	2	represented	represent	VERB
cana-5309	24	3	by	by	ADP
cana-5309	24	4	the	the	DET
cana-5309	24	5	poincaré	poincaré	PROPN
cana-5309	24	6	disk	disk	NOUN
cana-5309	24	7	model	model	NOUN
cana-5309	24	8	hyperbolic	hyperbolic	ADJ
cana-5309	24	9	space	space	NOUN
cana-5309	24	10	exists	exist	VERB
cana-5309	24	11	inside	inside	ADP
cana-5309	24	12	a	a	DET
cana-5309	24	13	euclidean	euclidean	ADJ
cana-5309	24	14	circle	circle	NOUN
cana-5309	24	15	and	and	CCONJ
cana-5309	24	16	its	its	PRON
cana-5309	24	17	lines	line	NOUN
cana-5309	24	18	exhibit	exhibit	VERB
cana-5309	24	19	circular	circular	ADJ
cana-5309	24	20	shapes	shape	NOUN
cana-5309	24	21	that	that	PRON
cana-5309	24	22	intersect	intersect	VERB
cana-5309	24	23	the	the	DET
cana-5309	24	24	border	border	NOUN
cana-5309	24	25	circle	circle	NOUN
cana-5309	24	26	while	while	SCONJ
cana-5309	24	27	maintaining	maintain	VERB
cana-5309	24	28	90	90	NUM
cana-5309	24	29	-	-	PUNCT
cana-5309	24	30	degree	degree	NOUN
cana-5309	24	31	angles	angle	NOUN
cana-5309	24	32	.	.	PUNCT
cana-5309	25	1	the	the	DET
cana-5309	25	2	model	model	NOUN
cana-5309	25	3	maintains	maintain	VERB
cana-5309	25	4	angle	angle	NOUN
cana-5309	25	5	relationships	relationship	NOUN
cana-5309	25	6	as	as	SCONJ
cana-5309	25	7	it	it	PRON
cana-5309	25	8	acts	act	VERB
cana-5309	25	9	conformally	conformally	ADV
cana-5309	25	10	but	but	CCONJ
cana-5309	25	11	results	result	NOUN
cana-5309	25	12	in	in	ADP
cana-5309	25	13	dramatic	dramatic	ADJ
cana-5309	25	14	distance	distance	NOUN
cana-5309	25	15	modifications	modification	NOUN
cana-5309	25	16	close	close	ADJ
cana-5309	25	17	to	to	ADP
cana-5309	25	18	its	its	PRON
cana-5309	25	19	boundary	boundary	NOUN
cana-5309	25	20	.	.	PUNCT
cana-5309	26	1	the	the	DET
cana-5309	26	2	hyperboloid	hyperboloid	ADJ
cana-5309	26	3	model	model	NOUN
cana-5309	26	4	of	of	ADP
cana-5309	26	5	hyperbolic	hyperbolic	ADJ
cana-5309	26	6	space	space	NOUN
cana-5309	26	7	implements	implement	VERB
cana-5309	26	8	a	a	DET
cana-5309	26	9	visualization	visualization	NOUN
cana-5309	26	10	through	through	ADP
cana-5309	26	11	a	a	DET
cana-5309	26	12	double	double	ADJ
cana-5309	26	13	-	-	PUNCT
cana-5309	26	14	horned	horn	VERB
cana-5309	26	15	hyperbolic	hyperbolic	ADJ
cana-5309	26	16	surface	surface	NOUN
cana-5309	26	17	which	which	PRON
cana-5309	26	18	exists	exist	VERB
cana-5309	26	19	inside	inside	ADP
cana-5309	26	20	minkowski	minkowski	ADJ
cana-5309	26	21	space	space	NOUN
cana-5309	26	22	to	to	PART
cana-5309	26	23	present	present	VERB
cana-5309	26	24	algebraic	algebraic	ADJ
cana-5309	26	25	connections	connection	NOUN
cana-5309	26	26	[	[	X
cana-5309	26	27	15	15	NUM
cana-5309	26	28	]	]	PUNCT
cana-5309	26	29	.	.	PUNCT
cana-5309	27	1	hyperbolic	hyperbolic	ADJ
cana-5309	27	2	geometry	geometry	NOUN
cana-5309	27	3	finds	find	VERB
cana-5309	27	4	its	its	PRON
cana-5309	27	5	use	use	NOUN
cana-5309	27	6	in	in	ADP
cana-5309	27	7	theoretical	theoretical	ADJ
cana-5309	27	8	subjects	subject	NOUN
cana-5309	27	9	as	as	ADV
cana-5309	27	10	well	well	ADV
cana-5309	27	11	as	as	ADP
cana-5309	27	12	practical	practical	ADJ
cana-5309	27	13	applications	application	NOUN
cana-5309	27	14	.	.	PUNCT
cana-5309	28	1	the	the	DET
cana-5309	28	2	physics	physics	NOUN
cana-5309	28	3	field	field	NOUN
cana-5309	28	4	uses	use	VERB
cana-5309	28	5	hyperbolic	hyperbolic	ADJ
cana-5309	28	6	geometry	geometry	NOUN
cana-5309	28	7	to	to	PART
cana-5309	28	8	explain	explain	VERB
cana-5309	28	9	the	the	DET
cana-5309	28	10	large	large	ADJ
cana-5309	28	11	-	-	PUNCT
cana-5309	28	12	scale	scale	NOUN
cana-5309	28	13	organization	organization	NOUN
cana-5309	28	14	of	of	ADP
cana-5309	28	15	the	the	DET
cana-5309	28	16	universe	universe	NOUN
cana-5309	28	17	so	so	ADV
cana-5309	28	18	long	long	ADV
cana-5309	28	19	as	as	SCONJ
cana-5309	28	20	the	the	DET
cana-5309	28	21	space	space	NOUN
cana-5309	28	22	has	have	VERB
cana-5309	28	23	negative	negative	ADJ
cana-5309	28	24	curvature	curvature	NOUN
cana-5309	28	25	.	.	PUNCT
cana-5309	29	1	complex	complex	ADJ
cana-5309	29	2	networks	network	NOUN
cana-5309	29	3	together	together	ADV
cana-5309	29	4	with	with	ADP
cana-5309	29	5	data	datum	NOUN
cana-5309	29	6	science	science	NOUN
cana-5309	29	7	use	use	VERB
cana-5309	29	8	hyperbolic	hyperbolic	ADJ
cana-5309	29	9	spaces	space	NOUN
cana-5309	29	10	to	to	PART
cana-5309	29	11	create	create	VERB
cana-5309	29	12	efficient	efficient	ADJ
cana-5309	29	13	representations	representation	NOUN
cana-5309	29	14	of	of	ADP
cana-5309	29	15	hierarchical	hierarchical	ADJ
cana-5309	29	16	structures	structure	NOUN
cana-5309	29	17	where	where	SCONJ
cana-5309	29	18	social	social	ADJ
cana-5309	29	19	networks	network	NOUN
cana-5309	29	20	or	or	CCONJ
cana-5309	29	21	knowledge	knowledge	NOUN
cana-5309	29	22	graphs	graph	NOUN
cana-5309	29	23	benefit	benefit	VERB
cana-5309	29	24	from	from	ADP
cana-5309	29	25	better	well	ADJ
cana-5309	29	26	expansion	expansion	NOUN
cana-5309	29	27	models	model	NOUN
cana-5309	29	28	than	than	ADP
cana-5309	29	29	euclidean	euclidean	ADJ
cana-5309	29	30	spaces	space	NOUN
cana-5309	29	31	.	.	PUNCT
cana-5309	30	1	the	the	DET
cana-5309	30	2	infinite	infinite	ADJ
cana-5309	30	3	imagery	imagery	NOUN
cana-5309	30	4	created	create	VERB
cana-5309	30	5	by	by	ADP
cana-5309	30	6	m.c	m.c	PROPN
cana-5309	30	7	.	.	PROPN
cana-5309	30	8	escher	escher	PROPN
cana-5309	30	9	became	become	VERB
cana-5309	30	10	possible	possible	ADJ
cana-5309	30	11	through	through	ADP
cana-5309	30	12	his	his	PRON
cana-5309	30	13	representation	representation	NOUN
cana-5309	30	14	of	of	ADP
cana-5309	30	15	hyperbolic	hyperbolic	ADJ
cana-5309	30	16	principles	principle	NOUN
cana-5309	30	17	in	in	ADP
cana-5309	30	18	artwork	artwork	NOUN
cana-5309	30	19	.	.	PUNCT
cana-5309	31	1	the	the	DET
cana-5309	31	2	enhanced	enhanced	ADJ
cana-5309	31	3	relevance	relevance	NOUN
cana-5309	31	4	of	of	ADP
cana-5309	31	5	hyperbolic	hyperbolic	ADJ
cana-5309	31	6	geometry	geometry	NOUN
cana-5309	31	7	makes	make	VERB
cana-5309	31	8	it	it	PRON
cana-5309	31	9	hard	hard	ADJ
cana-5309	31	10	to	to	PART
cana-5309	31	11	learn	learn	VERB
cana-5309	31	12	and	and	CCONJ
cana-5309	31	13	understand	understand	VERB
cana-5309	31	14	at	at	ADP
cana-5309	31	15	an	an	DET
cana-5309	31	16	intuitive	intuitive	ADJ
cana-5309	31	17	level	level	NOUN
cana-5309	31	18	since	since	SCONJ
cana-5309	31	19	it	it	PRON
cana-5309	31	20	violates	violate	VERB
cana-5309	31	21	our	our	PRON
cana-5309	31	22	natural	natural	ADJ
cana-5309	31	23	visual	visual	ADJ
cana-5309	31	24	-	-	PUNCT
cana-5309	31	25	spatial	spatial	ADJ
cana-5309	31	26	instincts	instinct	NOUN
cana-5309	31	27	which	which	PRON
cana-5309	31	28	developed	develop	VERB
cana-5309	31	29	from	from	ADP
cana-5309	31	30	living	live	VERB
cana-5309	31	31	in	in	ADP
cana-5309	31	32	a	a	DET
cana-5309	31	33	flat	flat	ADJ
cana-5309	31	34	euclidean	euclidean	ADJ
cana-5309	31	35	environment	environment	NOUN
cana-5309	31	36	.	.	PUNCT
cana-5309	32	1	the	the	DET
cana-5309	32	2	research	research	NOUN
cana-5309	32	3	community	community	NOUN
cana-5309	32	4	strives	strive	VERB
cana-5309	32	5	to	to	PART
cana-5309	32	6	accomplish	accomplish	VERB
cana-5309	32	7	two	two	NUM
cana-5309	32	8	essential	essential	ADJ
cana-5309	32	9	goals	goal	NOUN
cana-5309	32	10	:	:	PUNCT
cana-5309	32	11	enhanced	enhance	VERB
cana-5309	32	12	theoretical	theoretical	ADJ
cana-5309	32	13	insights	insight	NOUN
cana-5309	32	14	and	and	CCONJ
cana-5309	32	15	innovative	innovative	ADJ
cana-5309	32	16	approaches	approach	NOUN
cana-5309	32	17	towards	towards	ADP
cana-5309	32	18	visualization	visualization	NOUN
cana-5309	32	19	,	,	PUNCT
cana-5309	32	20	simulation	simulation	NOUN
cana-5309	32	21	as	as	ADV
cana-5309	32	22	well	well	ADV
cana-5309	32	23	as	as	ADP
cana-5309	32	24	application	application	NOUN
cana-5309	32	25	of	of	ADP
cana-5309	32	26	hyperbolic	hyperbolic	ADJ
cana-5309	32	27	spaces	space	NOUN
cana-5309	32	28	[	[	X
cana-5309	32	29	13	13	NUM
cana-5309	32	30	-	-	SYM
cana-5309	32	31	14	14	NUM
cana-5309	32	32	]	]	PUNCT
cana-5309	32	33	.	.	PUNCT
cana-5309	33	1	this	this	DET
cana-5309	33	2	research	research	NOUN
cana-5309	33	3	examines	examine	VERB
cana-5309	33	4	fundamental	fundamental	ADJ
cana-5309	33	5	hyperbolic	hyperbolic	ADJ
cana-5309	33	6	geometry	geometry	NOUN
cana-5309	33	7	models	model	NOUN
cana-5309	33	8	while	while	SCONJ
cana-5309	33	9	studying	study	VERB
cana-5309	33	10	essential	essential	ADJ
cana-5309	33	11	properties	property	NOUN
cana-5309	33	12	between	between	ADP
cana-5309	33	13	parallelism	parallelism	NOUN
cana-5309	33	14	and	and	CCONJ
cana-5309	33	15	triangle	triangle	NOUN
cana-5309	33	16	structure	structure	NOUN
cana-5309	33	17	and	and	CCONJ
cana-5309	33	18	distance	distance	NOUN
cana-5309	33	19	computation	computation	NOUN
cana-5309	33	20	along	along	ADP
cana-5309	33	21	with	with	ADP
cana-5309	33	22	their	their	PRON
cana-5309	33	23	visual	visual	ADJ
cana-5309	33	24	representation	representation	NOUN
cana-5309	33	25	through	through	ADP
cana-5309	33	26	computational	computational	ADJ
cana-5309	33	27	tools	tool	NOUN
cana-5309	33	28	.	.	PUNCT
cana-5309	34	1	we	we	PRON
cana-5309	34	2	use	use	VERB
cana-5309	34	3	hyperbolic	hyperbolic	ADJ
cana-5309	34	4	concepts	concept	NOUN
cana-5309	34	5	to	to	PART
cana-5309	34	6	contrast	contrast	VERB
cana-5309	34	7	with	with	ADP
cana-5309	34	8	euclidean	euclidean	ADJ
cana-5309	34	9	concepts	concept	NOUN
cana-5309	34	10	to	to	PART
cana-5309	34	11	build	build	VERB
cana-5309	34	12	both	both	CCONJ
cana-5309	34	13	conceptual	conceptual	ADJ
cana-5309	34	14	and	and	CCONJ
cana-5309	34	15	physical	physical	ADJ
cana-5309	34	16	understanding	understanding	NOUN
cana-5309	34	17	of	of	ADP
cana-5309	34	18	hyperbolic	hyperbolic	ADJ
cana-5309	34	19	spaces	space	NOUN
cana-5309	34	20	.	.	PUNCT
cana-5309	35	1	our	our	PRON
cana-5309	35	2	study	study	NOUN
cana-5309	35	3	includes	include	VERB
cana-5309	35	4	an	an	DET
cana-5309	35	5	investigation	investigation	NOUN
cana-5309	35	6	of	of	ADP
cana-5309	35	7	modern	modern	ADJ
cana-5309	35	8	scientific	scientific	ADJ
cana-5309	35	9	and	and	CCONJ
cana-5309	35	10	technological	technological	ADJ
cana-5309	35	11	areas	area	NOUN
cana-5309	35	12	where	where	SCONJ
cana-5309	35	13	hyperbolic	hyperbolic	ADJ
cana-5309	35	14	geometry	geometry	NOUN
cana-5309	35	15	proves	prove	VERB
cana-5309	35	16	crucial	crucial	ADJ
cana-5309	35	17	along	along	ADV
cana-5309	35	18	with	with	ADP
cana-5309	35	19	existing	exist	VERB
cana-5309	35	20	and	and	CCONJ
cana-5309	35	21	anticipated	anticipated	ADJ
cana-5309	35	22	applications	application	NOUN
cana-5309	35	23	.	.	PUNCT
cana-5309	36	1	communications	communication	NOUN
cana-5309	36	2	on	on	ADP
cana-5309	36	3	applied	apply	VERB
cana-5309	36	4	nonlinear	nonlinear	ADJ
cana-5309	36	5	analysis	analysis	NOUN
cana-5309	36	6	issn	issn	NOUN
cana-5309	36	7	:	:	PUNCT
cana-5309	36	8	1074	1074	NUM
cana-5309	36	9	-	-	PUNCT
cana-5309	36	10	133x	133x	NUM
cana-5309	36	11	vol	vol	NOUN
cana-5309	36	12	31	31	NUM
cana-5309	36	13	no	no	NOUN
cana-5309	36	14	.	.	PUNCT
cana-5309	37	1	4s	4s	NUM
cana-5309	37	2	(	(	PUNCT
cana-5309	37	3	2024	2024	NUM
cana-5309	37	4	)	)	PUNCT
cana-5309	37	5	674	674	NUM
cana-5309	37	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	37	7	novelty	novelty	NOUN
cana-5309	37	8	and	and	CCONJ
cana-5309	37	9	contribution	contribution	NOUN
cana-5309	37	10	the	the	DET
cana-5309	37	11	presented	present	VERB
cana-5309	37	12	research	research	NOUN
cana-5309	37	13	presents	present	VERB
cana-5309	37	14	modern	modern	ADJ
cana-5309	37	15	computational	computational	ADJ
cana-5309	37	16	methods	method	NOUN
cana-5309	37	17	connected	connect	VERB
cana-5309	37	18	to	to	ADP
cana-5309	37	19	classical	classical	ADJ
cana-5309	37	20	theory	theory	NOUN
cana-5309	37	21	for	for	ADP
cana-5309	37	22	studying	study	VERB
cana-5309	37	23	hyperbolic	hyperbolic	ADJ
cana-5309	37	24	geometry	geometry	NOUN
cana-5309	37	25	.	.	PUNCT
cana-5309	38	1	the	the	DET
cana-5309	38	2	fundamental	fundamental	ADJ
cana-5309	38	3	concepts	concept	NOUN
cana-5309	38	4	of	of	ADP
cana-5309	38	5	hyperbolic	hyperbolic	ADJ
cana-5309	38	6	geometry	geometry	NOUN
cana-5309	38	7	already	already	ADV
cana-5309	38	8	exist	exist	VERB
cana-5309	38	9	yet	yet	ADV
cana-5309	38	10	researchers	researcher	NOUN
cana-5309	38	11	have	have	AUX
cana-5309	38	12	not	not	PART
cana-5309	38	13	yet	yet	ADV
cana-5309	38	14	identified	identify	VERB
cana-5309	38	15	methods	method	NOUN
cana-5309	38	16	to	to	PART
cana-5309	38	17	link	link	VERB
cana-5309	38	18	formal	formal	ADJ
cana-5309	38	19	theories	theory	NOUN
cana-5309	38	20	to	to	ADP
cana-5309	38	21	visual	visual	ADJ
cana-5309	38	22	understandings	understanding	NOUN
cana-5309	38	23	together	together	ADV
cana-5309	38	24	with	with	ADP
cana-5309	38	25	applications	application	NOUN
cana-5309	38	26	across	across	ADP
cana-5309	38	27	different	different	ADJ
cana-5309	38	28	fields	field	NOUN
cana-5309	38	29	.	.	PUNCT
cana-5309	39	1	our	our	PRON
cana-5309	39	2	contribution	contribution	NOUN
cana-5309	39	3	is	be	AUX
cana-5309	39	4	threefold	threefold	ADJ
cana-5309	39	5	:	:	PUNCT
cana-5309	39	6	•	•	NUM
cana-5309	39	7	the	the	DET
cana-5309	39	8	first	first	ADJ
cana-5309	39	9	component	component	NOUN
cana-5309	39	10	analyzes	analyze	VERB
cana-5309	39	11	both	both	DET
cana-5309	39	12	major	major	ADJ
cana-5309	39	13	models	model	NOUN
cana-5309	39	14	including	include	VERB
cana-5309	39	15	poincaré	poincaré	PROPN
cana-5309	39	16	disk	disk	NOUN
cana-5309	39	17	and	and	CCONJ
cana-5309	39	18	hyperboloid	hyperboloid	ADJ
cana-5309	39	19	model	model	NOUN
cana-5309	39	20	through	through	ADP
cana-5309	39	21	precise	precise	ADJ
cana-5309	39	22	mathematical	mathematical	ADJ
cana-5309	39	23	comparison	comparison	NOUN
cana-5309	39	24	along	along	ADP
cana-5309	39	25	with	with	ADP
cana-5309	39	26	visualization	visualization	NOUN
cana-5309	39	27	and	and	CCONJ
cana-5309	39	28	practical	practical	ADJ
cana-5309	39	29	computational	computational	ADJ
cana-5309	39	30	assessments	assessment	NOUN
cana-5309	39	31	.	.	PUNCT
cana-5309	40	1	each	each	DET
cana-5309	40	2	model	model	NOUN
cana-5309	40	3	finds	find	VERB
cana-5309	40	4	unique	unique	ADJ
cana-5309	40	5	application	application	NOUN
cana-5309	40	6	requirements	requirement	NOUN
cana-5309	40	7	because	because	SCONJ
cana-5309	40	8	they	they	PRON
cana-5309	40	9	preserve	preserve	VERB
cana-5309	40	10	angles	angle	NOUN
cana-5309	40	11	differently	differently	ADV
cana-5309	40	12	from	from	ADP
cana-5309	40	13	maintaining	maintain	VERB
cana-5309	40	14	metric	metric	ADJ
cana-5309	40	15	linearity	linearity	NOUN
cana-5309	40	16	depending	depend	VERB
cana-5309	40	17	on	on	ADP
cana-5309	40	18	the	the	DET
cana-5309	40	19	given	give	VERB
cana-5309	40	20	task	task	NOUN
cana-5309	40	21	.	.	PUNCT
cana-5309	41	1	•	•	NUM
cana-5309	41	2	our	our	PRON
cana-5309	41	3	methodology	methodology	NOUN
cana-5309	41	4	utilizes	utilize	VERB
cana-5309	41	5	computer	computer	NOUN
cana-5309	41	6	-	-	PUNCT
cana-5309	41	7	generated	generate	VERB
cana-5309	41	8	visualizations	visualization	NOUN
cana-5309	41	9	that	that	PRON
cana-5309	41	10	enhance	enhance	VERB
cana-5309	41	11	representation	representation	NOUN
cana-5309	41	12	of	of	ADP
cana-5309	41	13	hyperbolic	hyperbolic	ADJ
cana-5309	41	14	properties	property	NOUN
cana-5309	41	15	.	.	PUNCT
cana-5309	42	1	the	the	DET
cana-5309	42	2	utilization	utilization	NOUN
cana-5309	42	3	of	of	ADP
cana-5309	42	4	contemporary	contemporary	ADJ
cana-5309	42	5	software	software	NOUN
cana-5309	42	6	enables	enable	VERB
cana-5309	42	7	the	the	DET
cana-5309	42	8	demonstration	demonstration	NOUN
cana-5309	42	9	of	of	ADP
cana-5309	42	10	hyperbolic	hyperbolic	ADJ
cana-5309	42	11	lines	line	NOUN
cana-5309	42	12	and	and	CCONJ
cana-5309	42	13	geodesics	geodesic	NOUN
cana-5309	42	14	alongside	alongside	ADP
cana-5309	42	15	tessellations	tessellation	NOUN
cana-5309	42	16	through	through	ADP
cana-5309	42	17	visualizations	visualization	NOUN
cana-5309	42	18	which	which	PRON
cana-5309	42	19	simplify	simplify	VERB
cana-5309	42	20	the	the	DET
cana-5309	42	21	complex	complex	ADJ
cana-5309	42	22	nature	nature	NOUN
cana-5309	42	23	of	of	ADP
cana-5309	42	24	hyperbolic	hyperbolic	ADJ
cana-5309	42	25	space	space	NOUN
cana-5309	42	26	especially	especially	ADV
cana-5309	42	27	for	for	ADP
cana-5309	42	28	those	those	PRON
cana-5309	42	29	who	who	PRON
cana-5309	42	30	are	be	AUX
cana-5309	42	31	not	not	PART
cana-5309	42	32	mathematical	mathematical	ADJ
cana-5309	42	33	experts	expert	NOUN
cana-5309	42	34	.	.	PUNCT
cana-5309	43	1	the	the	DET
cana-5309	43	2	methodology	methodology	NOUN
cana-5309	43	3	creates	create	VERB
cana-5309	43	4	a	a	DET
cana-5309	43	5	connection	connection	NOUN
cana-5309	43	6	which	which	PRON
cana-5309	43	7	allows	allow	VERB
cana-5309	43	8	complex	complex	ADJ
cana-5309	43	9	theoretical	theoretical	ADJ
cana-5309	43	10	information	information	NOUN
cana-5309	43	11	to	to	PART
cana-5309	43	12	meet	meet	VERB
cana-5309	43	13	functional	functional	ADJ
cana-5309	43	14	learning	learning	NOUN
cana-5309	43	15	opportunities	opportunity	NOUN
cana-5309	43	16	.	.	PUNCT
cana-5309	44	1	•	•	NUM
cana-5309	44	2	the	the	DET
cana-5309	44	3	discussion	discussion	NOUN
cana-5309	44	4	about	about	ADP
cana-5309	44	5	hyperbolic	hyperbolic	ADJ
cana-5309	44	6	geometry	geometry	NOUN
cana-5309	44	7	includes	include	VERB
cana-5309	44	8	new	new	ADJ
cana-5309	44	9	applications	application	NOUN
cana-5309	44	10	which	which	PRON
cana-5309	44	11	cover	cover	VERB
cana-5309	44	12	network	network	NOUN
cana-5309	44	13	science	science	NOUN
cana-5309	44	14	and	and	CCONJ
cana-5309	44	15	data	datum	NOUN
cana-5309	44	16	visualization	visualization	NOUN
cana-5309	44	17	.	.	PUNCT
cana-5309	45	1	this	this	DET
cana-5309	45	2	study	study	NOUN
cana-5309	45	3	demonstrates	demonstrate	VERB
cana-5309	45	4	that	that	SCONJ
cana-5309	45	5	hyperbolic	hyperbolic	ADJ
cana-5309	45	6	geometry	geometry	NOUN
cana-5309	45	7	serves	serve	VERB
cana-5309	45	8	essential	essential	ADJ
cana-5309	45	9	practical	practical	ADJ
cana-5309	45	10	purposes	purpose	NOUN
cana-5309	45	11	because	because	SCONJ
cana-5309	45	12	we	we	PRON
cana-5309	45	13	can	can	AUX
cana-5309	45	14	use	use	VERB
cana-5309	45	15	its	its	PRON
cana-5309	45	16	properties	property	NOUN
cana-5309	45	17	to	to	PART
cana-5309	45	18	represent	represent	VERB
cana-5309	45	19	hierarchical	hierarchical	ADJ
cana-5309	45	20	datasets	dataset	NOUN
cana-5309	45	21	in	in	ADP
cana-5309	45	22	practical	practical	ADJ
cana-5309	45	23	applications	application	NOUN
cana-5309	45	24	.	.	PUNCT
cana-5309	46	1	our	our	PRON
cana-5309	46	2	research	research	NOUN
cana-5309	46	3	seeks	seek	VERB
cana-5309	46	4	dual	dual	ADJ
cana-5309	46	5	contributions	contribution	NOUN
cana-5309	46	6	to	to	ADP
cana-5309	46	7	studies	study	NOUN
cana-5309	46	8	about	about	ADP
cana-5309	46	9	hyperbolic	hyperbolic	ADJ
cana-5309	46	10	geometry	geometry	NOUN
cana-5309	46	11	while	while	SCONJ
cana-5309	46	12	working	work	VERB
cana-5309	46	13	to	to	PART
cana-5309	46	14	establish	establish	VERB
cana-5309	46	15	its	its	PRON
cana-5309	46	16	widespread	widespread	ADJ
cana-5309	46	17	use	use	NOUN
cana-5309	46	18	across	across	ADP
cana-5309	46	19	academic	academic	ADJ
cana-5309	46	20	fields	field	NOUN
cana-5309	46	21	and	and	CCONJ
cana-5309	46	22	practical	practical	ADJ
cana-5309	46	23	applications	application	NOUN
cana-5309	46	24	.	.	PUNCT
cana-5309	47	1	we	we	PRON
cana-5309	47	2	recommend	recommend	VERB
cana-5309	47	3	using	use	VERB
cana-5309	47	4	an	an	DET
cana-5309	47	5	integrated	integrate	VERB
cana-5309	47	6	strategy	strategy	NOUN
cana-5309	47	7	to	to	PART
cana-5309	47	8	explain	explain	VERB
cana-5309	47	9	hyperbolic	hyperbolic	ADJ
cana-5309	47	10	spaces	space	NOUN
cana-5309	47	11	while	while	SCONJ
cana-5309	47	12	opening	open	VERB
cana-5309	47	13	new	new	ADJ
cana-5309	47	14	directions	direction	NOUN
cana-5309	47	15	for	for	ADP
cana-5309	47	16	scientific	scientific	ADJ
cana-5309	47	17	development	development	NOUN
cana-5309	47	18	and	and	CCONJ
cana-5309	47	19	research	research	NOUN
cana-5309	47	20	[	[	X
cana-5309	47	21	7	7	NUM
cana-5309	47	22	]	]	PUNCT
cana-5309	47	23	.	.	PUNCT
cana-5309	48	1	ii	ii	PROPN
cana-5309	48	2	.	.	PROPN
cana-5309	48	3	related	relate	VERB
cana-5309	48	4	works	work	NOUN
cana-5309	48	5	in	in	ADP
cana-5309	48	6	2020	2020	NUM
cana-5309	48	7	t.	t.	PROPN
cana-5309	48	8	novello	novello	PROPN
cana-5309	48	9	et.al	et.al	PROPN
cana-5309	48	10	.	.	PROPN
cana-5309	48	11	,	,	PUNCT
cana-5309	49	1	v.	v.	PROPN
cana-5309	49	2	da	da	PROPN
cana-5309	49	3	silva	silva	PROPN
cana-5309	49	4	et.al	et.al	PROPN
cana-5309	49	5	.	.	PROPN
cana-5309	49	6	,	,	PUNCT
cana-5309	49	7	and	and	CCONJ
cana-5309	49	8	l.	l.	PROPN
cana-5309	49	9	velho	velho	PROPN
cana-5309	49	10	et.al	et.al	PROPN
cana-5309	49	11	.	.	PUNCT
cana-5309	50	1	,	,	PUNCT
cana-5309	51	1	[	[	X
cana-5309	51	2	9	9	NUM
cana-5309	51	3	]	]	PUNCT
cana-5309	51	4	introduced	introduce	VERB
cana-5309	51	5	the	the	DET
cana-5309	51	6	field	field	NOUN
cana-5309	51	7	of	of	ADP
cana-5309	51	8	non	non	ADJ
cana-5309	51	9	-	-	ADJ
cana-5309	51	10	euclidean	euclidean	ADJ
cana-5309	51	11	geometries	geometry	NOUN
cana-5309	51	12	especially	especially	ADV
cana-5309	51	13	hyperbolic	hyperbolic	ADJ
cana-5309	51	14	geometry	geometry	NOUN
cana-5309	51	15	unfolded	unfold	VERB
cana-5309	51	16	after	after	ADP
cana-5309	51	17	its	its	PRON
cana-5309	51	18	initial	initial	ADJ
cana-5309	51	19	discovery	discovery	NOUN
cana-5309	51	20	.	.	PUNCT
cana-5309	52	1	research	research	NOUN
cana-5309	52	2	in	in	ADP
cana-5309	52	3	the	the	DET
cana-5309	52	4	early	early	ADJ
cana-5309	52	5	stages	stage	NOUN
cana-5309	52	6	functioned	function	VERB
cana-5309	52	7	to	to	PART
cana-5309	52	8	prove	prove	VERB
cana-5309	52	9	that	that	SCONJ
cana-5309	52	10	hyperbolic	hyperbolic	ADJ
cana-5309	52	11	geometry	geometry	NOUN
cana-5309	52	12	stands	stand	VERB
cana-5309	52	13	alone	alone	ADJ
cana-5309	52	14	as	as	ADP
cana-5309	52	15	a	a	DET
cana-5309	52	16	valid	valid	ADJ
cana-5309	52	17	structure	structure	NOUN
cana-5309	52	18	despite	despite	SCONJ
cana-5309	52	19	existing	exist	VERB
cana-5309	52	20	independently	independently	ADV
cana-5309	52	21	from	from	ADP
cana-5309	52	22	euclidean	euclidean	ADJ
cana-5309	52	23	foundations	foundation	NOUN
cana-5309	52	24	.	.	PUNCT
cana-5309	53	1	researches	research	NOUN
cana-5309	53	2	developed	develop	VERB
cana-5309	53	3	several	several	ADJ
cana-5309	53	4	models	model	NOUN
cana-5309	53	5	to	to	PART
cana-5309	53	6	study	study	VERB
cana-5309	53	7	and	and	CCONJ
cana-5309	53	8	visualize	visualize	VERB
cana-5309	53	9	hyperbolic	hyperbolic	ADJ
cana-5309	53	10	spaces	space	NOUN
cana-5309	53	11	during	during	ADP
cana-5309	53	12	which	which	PRON
cana-5309	53	13	the	the	DET
cana-5309	53	14	poincaré	poincaré	PROPN
cana-5309	53	15	disk	disk	NOUN
cana-5309	53	16	model	model	NOUN
cana-5309	53	17	received	receive	VERB
cana-5309	53	18	specific	specific	ADJ
cana-5309	53	19	attention	attention	NOUN
cana-5309	53	20	together	together	ADV
cana-5309	53	21	with	with	ADP
cana-5309	53	22	the	the	DET
cana-5309	53	23	poincaré	poincaré	PROPN
cana-5309	53	24	half	half	ADJ
cana-5309	53	25	-	-	PUNCT
cana-5309	53	26	plane	plane	NOUN
cana-5309	53	27	model	model	NOUN
cana-5309	53	28	and	and	CCONJ
cana-5309	53	29	the	the	DET
cana-5309	53	30	hyperboloid	hyperboloid	ADJ
cana-5309	53	31	model	model	NOUN
cana-5309	53	32	.	.	PUNCT
cana-5309	54	1	the	the	DET
cana-5309	54	2	separate	separate	ADJ
cana-5309	54	3	models	model	NOUN
cana-5309	54	4	of	of	ADP
cana-5309	54	5	hyperbolic	hyperbolic	ADJ
cana-5309	54	6	space	space	NOUN
cana-5309	54	7	contained	contain	VERB
cana-5309	54	8	unique	unique	ADJ
cana-5309	54	9	features	feature	NOUN
cana-5309	54	10	which	which	PRON
cana-5309	54	11	made	make	VERB
cana-5309	54	12	them	they	PRON
cana-5309	54	13	ideal	ideal	ADJ
cana-5309	54	14	for	for	ADP
cana-5309	54	15	different	different	ADJ
cana-5309	54	16	aspects	aspect	NOUN
cana-5309	54	17	of	of	ADP
cana-5309	54	18	analysis	analysis	NOUN
cana-5309	54	19	such	such	ADJ
cana-5309	54	20	as	as	ADP
cana-5309	54	21	angle	angle	NOUN
cana-5309	54	22	preservation	preservation	NOUN
cana-5309	54	23	,	,	PUNCT
cana-5309	54	24	conformity	conformity	NOUN
cana-5309	54	25	and	and	CCONJ
cana-5309	54	26	computational	computational	ADJ
cana-5309	54	27	simplicity	simplicity	NOUN
cana-5309	54	28	.	.	PUNCT
cana-5309	55	1	in	in	ADP
cana-5309	55	2	2020	2020	NUM
cana-5309	55	3	s.	s.	PROPN
cana-5309	55	4	huckemann	huckemann	PROPN
cana-5309	55	5	et.al	et.al	PROPN
cana-5309	55	6	.	.	PUNCT
cana-5309	56	1	and	and	CCONJ
cana-5309	56	2	b.	b.	PROPN
cana-5309	56	3	eltzner	eltzner	PROPN
cana-5309	56	4	et.al	et.al	PROPN
cana-5309	56	5	.	.	PUNCT
cana-5309	57	1	[	[	X
cana-5309	57	2	12	12	NUM
cana-5309	57	3	]	]	PUNCT
cana-5309	57	4	suggested	suggest	VERB
cana-5309	57	5	the	the	DET
cana-5309	57	6	researchers	researcher	NOUN
cana-5309	57	7	studied	study	VERB
cana-5309	57	8	fundamental	fundamental	ADJ
cana-5309	57	9	structures	structure	NOUN
cana-5309	57	10	in	in	ADP
cana-5309	57	11	hyperbolic	hyperbolic	ADJ
cana-5309	57	12	geometry	geometry	NOUN
cana-5309	57	13	that	that	PRON
cana-5309	57	14	included	include	VERB
cana-5309	57	15	hyperbolic	hyperbolic	ADJ
cana-5309	57	16	lines	line	NOUN
cana-5309	57	17	and	and	CCONJ
cana-5309	57	18	triangles	triangle	NOUN
cana-5309	57	19	as	as	ADV
cana-5309	57	20	well	well	ADV
cana-5309	57	21	as	as	ADP
cana-5309	57	22	circles	circle	NOUN
cana-5309	57	23	and	and	CCONJ
cana-5309	57	24	tessellations	tessellation	NOUN
cana-5309	57	25	.	.	PUNCT
cana-5309	58	1	doctors	doctor	NOUN
cana-5309	58	2	studied	study	VERB
cana-5309	58	3	in	in	ADP
cana-5309	58	4	detail	detail	NOUN
cana-5309	58	5	the	the	DET
cana-5309	58	6	transformation	transformation	NOUN
cana-5309	58	7	of	of	ADP
cana-5309	58	8	traditional	traditional	ADJ
cana-5309	58	9	communications	communication	NOUN
cana-5309	58	10	on	on	ADP
cana-5309	58	11	applied	apply	VERB
cana-5309	58	12	nonlinear	nonlinear	ADJ
cana-5309	58	13	analysis	analysis	NOUN
cana-5309	58	14	issn	issn	NOUN
cana-5309	58	15	:	:	PUNCT
cana-5309	58	16	1074	1074	NUM
cana-5309	58	17	-	-	PUNCT
cana-5309	58	18	133x	133x	NUM
cana-5309	58	19	vol	vol	NOUN
cana-5309	58	20	31	31	NUM
cana-5309	58	21	no	no	NOUN
cana-5309	58	22	.	.	PUNCT
cana-5309	59	1	4s	4s	NUM
cana-5309	59	2	(	(	PUNCT
cana-5309	59	3	2024	2024	NUM
cana-5309	59	4	)	)	PUNCT
cana-5309	59	5	675	675	NUM
cana-5309	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	59	7	euclidean	euclidean	ADJ
cana-5309	59	8	distance	distance	NOUN
cana-5309	59	9	measures	measure	NOUN
cana-5309	59	10	together	together	ADV
cana-5309	59	11	with	with	ADP
cana-5309	59	12	angle	angle	NOUN
cana-5309	59	13	summations	summation	NOUN
cana-5309	59	14	and	and	CCONJ
cana-5309	59	15	parallel	parallel	ADJ
cana-5309	59	16	concepts	concept	NOUN
cana-5309	59	17	in	in	ADP
cana-5309	59	18	hyperbolic	hyperbolic	ADJ
cana-5309	59	19	systems	system	NOUN
cana-5309	59	20	.	.	PUNCT
cana-5309	60	1	through	through	ADP
cana-5309	60	2	specific	specific	ADJ
cana-5309	60	3	studies	study	NOUN
cana-5309	60	4	of	of	ADP
cana-5309	60	5	hyperbolic	hyperbolic	ADJ
cana-5309	60	6	trigonometry	trigonometry	NOUN
cana-5309	60	7	scientists	scientist	NOUN
cana-5309	60	8	gained	gain	VERB
cana-5309	60	9	knowledge	knowledge	NOUN
cana-5309	60	10	needed	need	VERB
cana-5309	60	11	to	to	PART
cana-5309	60	12	create	create	VERB
cana-5309	60	13	formulas	formula	NOUN
cana-5309	60	14	and	and	CCONJ
cana-5309	60	15	metrics	metric	NOUN
cana-5309	60	16	that	that	PRON
cana-5309	60	17	excel	excel	VERB
cana-5309	60	18	in	in	ADP
cana-5309	60	19	negatively	negatively	ADV
cana-5309	60	20	curved	curved	ADJ
cana-5309	60	21	spaces	space	NOUN
cana-5309	60	22	.	.	PUNCT
cana-5309	61	1	mathematical	mathematical	ADJ
cana-5309	61	2	foundation	foundation	NOUN
cana-5309	61	3	that	that	PRON
cana-5309	61	4	was	be	AUX
cana-5309	61	5	developed	develop	VERB
cana-5309	61	6	allowed	allow	VERB
cana-5309	61	7	following	follow	VERB
cana-5309	61	8	generations	generation	NOUN
cana-5309	61	9	to	to	PART
cana-5309	61	10	create	create	VERB
cana-5309	61	11	new	new	ADJ
cana-5309	61	12	discoveries	discovery	NOUN
cana-5309	61	13	for	for	ADP
cana-5309	61	14	theoretical	theoretical	ADJ
cana-5309	61	15	and	and	CCONJ
cana-5309	61	16	applied	applied	ADJ
cana-5309	61	17	research	research	NOUN
cana-5309	61	18	.	.	PUNCT
cana-5309	62	1	scientists	scientist	NOUN
cana-5309	62	2	utilize	utilize	VERB
cana-5309	62	3	hyperbolic	hyperbolic	ADJ
cana-5309	62	4	geometry	geometry	NOUN
cana-5309	62	5	today	today	NOUN
cana-5309	62	6	to	to	AUX
cana-5309	62	7	model	model	NOUN
cana-5309	62	8	complex	complex	ADJ
cana-5309	62	9	networks	network	NOUN
cana-5309	62	10	through	through	ADP
cana-5309	62	11	which	which	PRON
cana-5309	62	12	hyperbolic	hyperbolic	ADJ
cana-5309	62	13	space	space	NOUN
cana-5309	62	14	enables	enable	VERB
cana-5309	62	15	efficient	efficient	ADJ
cana-5309	62	16	data	data	NOUN
cana-5309	62	17	structure	structure	NOUN
cana-5309	62	18	embedding	embed	VERB
cana-5309	62	19	despite	despite	SCONJ
cana-5309	62	20	its	its	PRON
cana-5309	62	21	hierarchical	hierarchical	ADJ
cana-5309	62	22	and	and	CCONJ
cana-5309	62	23	expansive	expansive	ADJ
cana-5309	62	24	aspect	aspect	NOUN
cana-5309	62	25	.	.	PUNCT
cana-5309	63	1	studies	study	NOUN
cana-5309	63	2	proved	prove	VERB
cana-5309	63	3	that	that	SCONJ
cana-5309	63	4	hyperbolic	hyperbolic	ADJ
cana-5309	63	5	models	model	NOUN
cana-5309	63	6	consume	consume	VERB
cana-5309	63	7	less	less	ADJ
cana-5309	63	8	space	space	NOUN
cana-5309	63	9	than	than	ADP
cana-5309	63	10	euclidean	euclidean	ADJ
cana-5309	63	11	embeddings	embedding	NOUN
cana-5309	63	12	when	when	SCONJ
cana-5309	63	13	representing	represent	VERB
cana-5309	63	14	internet	internet	NOUN
cana-5309	63	15	topology	topology	NOUN
cana-5309	63	16	social	social	ADJ
cana-5309	63	17	networks	network	NOUN
cana-5309	63	18	and	and	CCONJ
cana-5309	63	19	biological	biological	ADJ
cana-5309	63	20	systems	system	NOUN
cana-5309	63	21	because	because	SCONJ
cana-5309	63	22	they	they	PRON
cana-5309	63	23	effectively	effectively	ADV
cana-5309	63	24	represent	represent	VERB
cana-5309	63	25	relationships	relationship	NOUN
cana-5309	63	26	.	.	PUNCT
cana-5309	64	1	many	many	ADJ
cana-5309	64	2	visualization	visualization	NOUN
cana-5309	64	3	techniques	technique	NOUN
cana-5309	64	4	evolve	evolve	VERB
cana-5309	64	5	to	to	PART
cana-5309	64	6	better	well	ADV
cana-5309	64	7	show	show	VERB
cana-5309	64	8	hyperbolic	hyperbolic	ADJ
cana-5309	64	9	relationships	relationship	NOUN
cana-5309	64	10	and	and	CCONJ
cana-5309	64	11	computer	computer	NOUN
cana-5309	64	12	programs	program	NOUN
cana-5309	64	13	now	now	ADV
cana-5309	64	14	let	let	VERB
cana-5309	64	15	us	we	PRON
cana-5309	64	16	explore	explore	VERB
cana-5309	64	17	dynamic	dynamic	ADJ
cana-5309	64	18	views	view	NOUN
cana-5309	64	19	of	of	ADP
cana-5309	64	20	geodesics	geodesic	NOUN
cana-5309	64	21	and	and	CCONJ
cana-5309	64	22	growth	growth	NOUN
cana-5309	64	23	processes	process	NOUN
cana-5309	64	24	and	and	CCONJ
cana-5309	64	25	tilings	tiling	NOUN
cana-5309	64	26	within	within	ADP
cana-5309	64	27	hyperbolic	hyperbolic	ADJ
cana-5309	64	28	spaces	space	NOUN
cana-5309	64	29	.	.	PUNCT
cana-5309	65	1	in	in	ADP
cana-5309	65	2	2024	2024	NUM
cana-5309	65	3	j.	j.	PROPN
cana-5309	65	4	miller	miller	PROPN
cana-5309	65	5	et.al	et.al	PROPN
cana-5309	65	6	.	.	PROPN
cana-5309	65	7	,	,	PUNCT
cana-5309	65	8	d.	d.	PROPN
cana-5309	65	9	bhatia	bhatia	PROPN
cana-5309	65	10	et.al	et.al	PROPN
cana-5309	65	11	.	.	PROPN
cana-5309	65	12	,	,	PUNCT
cana-5309	65	13	and	and	CCONJ
cana-5309	65	14	s.	s.	PROPN
cana-5309	65	15	kobourov	kobourov	PROPN
cana-5309	65	16	et.al	et.al	PROPN
cana-5309	65	17	.	.	PROPN
cana-5309	65	18	,	,	PUNCT
cana-5309	65	19	[	[	X
cana-5309	65	20	5	5	NUM
cana-5309	65	21	]	]	PUNCT
cana-5309	65	22	proposed	propose	VERB
cana-5309	65	23	theoretical	theoretical	ADJ
cana-5309	65	24	physics	physics	NOUN
cana-5309	65	25	makes	make	VERB
cana-5309	65	26	regular	regular	ADJ
cana-5309	65	27	use	use	NOUN
cana-5309	65	28	of	of	ADP
cana-5309	65	29	hyperbolic	hyperbolic	ADJ
cana-5309	65	30	geometry	geometry	NOUN
cana-5309	65	31	mainly	mainly	ADV
cana-5309	65	32	when	when	SCONJ
cana-5309	65	33	studying	study	VERB
cana-5309	65	34	general	general	ADJ
cana-5309	65	35	relativity	relativity	NOUN
cana-5309	65	36	together	together	ADV
cana-5309	65	37	with	with	ADP
cana-5309	65	38	negative	negative	ADJ
cana-5309	65	39	-	-	PUNCT
cana-5309	65	40	curvature	curvature	NOUN
cana-5309	65	41	cosmological	cosmological	ADJ
cana-5309	65	42	systems	system	NOUN
cana-5309	65	43	.	.	PUNCT
cana-5309	66	1	this	this	DET
cana-5309	66	2	mathematical	mathematical	ADJ
cana-5309	66	3	concept	concept	NOUN
cana-5309	66	4	has	have	AUX
cana-5309	66	5	shaped	shape	VERB
cana-5309	66	6	new	new	ADJ
cana-5309	66	7	procedures	procedure	NOUN
cana-5309	66	8	in	in	ADP
cana-5309	66	9	optimization	optimization	NOUN
cana-5309	66	10	as	as	ADV
cana-5309	66	11	well	well	ADV
cana-5309	66	12	as	as	ADP
cana-5309	66	13	artistic	artistic	ADJ
cana-5309	66	14	developments	development	NOUN
cana-5309	66	15	of	of	ADP
cana-5309	66	16	infinite	infinite	ADJ
cana-5309	66	17	structures	structure	NOUN
cana-5309	66	18	through	through	ADP
cana-5309	66	19	hyperbolic	hyperbolic	ADJ
cana-5309	66	20	tessellations	tessellation	NOUN
cana-5309	66	21	.	.	PUNCT
cana-5309	67	1	current	current	ADJ
cana-5309	67	2	education	education	NOUN
cana-5309	67	3	systems	system	NOUN
cana-5309	67	4	face	face	VERB
cana-5309	67	5	obstacles	obstacle	NOUN
cana-5309	67	6	to	to	PART
cana-5309	67	7	teach	teach	VERB
cana-5309	67	8	hyperbolic	hyperbolic	ADJ
cana-5309	67	9	concepts	concept	NOUN
cana-5309	67	10	intuitively	intuitively	ADV
cana-5309	67	11	to	to	ADP
cana-5309	67	12	persons	person	NOUN
cana-5309	67	13	outside	outside	ADP
cana-5309	67	14	the	the	DET
cana-5309	67	15	pure	pure	ADJ
cana-5309	67	16	mathematics	mathematic	NOUN
cana-5309	67	17	field	field	NOUN
cana-5309	67	18	.	.	PUNCT
cana-5309	68	1	the	the	DET
cana-5309	68	2	relevance	relevance	NOUN
cana-5309	68	3	in	in	ADP
cana-5309	68	4	our	our	PRON
cana-5309	68	5	time	time	NOUN
cana-5309	68	6	requires	require	VERB
cana-5309	68	7	us	we	PRON
cana-5309	68	8	to	to	PART
cana-5309	68	9	establish	establish	VERB
cana-5309	68	10	new	new	ADJ
cana-5309	68	11	educational	educational	ADJ
cana-5309	68	12	tools	tool	NOUN
cana-5309	68	13	and	and	CCONJ
cana-5309	68	14	computational	computational	ADJ
cana-5309	68	15	frameworks	framework	NOUN
cana-5309	68	16	with	with	ADP
cana-5309	68	17	interactive	interactive	ADJ
cana-5309	68	18	visualizations	visualization	NOUN
cana-5309	68	19	to	to	PART
cana-5309	68	20	help	help	VERB
cana-5309	68	21	understand	understand	VERB
cana-5309	68	22	hyperbolic	hyperbolic	ADJ
cana-5309	68	23	concepts	concept	NOUN
cana-5309	68	24	better	well	ADJ
cana-5309	68	25	.	.	PUNCT
cana-5309	69	1	research	research	NOUN
cana-5309	69	2	on	on	ADP
cana-5309	69	3	hyperbolic	hyperbolic	ADJ
cana-5309	69	4	geometry	geometry	NOUN
cana-5309	69	5	demonstrates	demonstrate	VERB
cana-5309	69	6	its	its	PRON
cana-5309	69	7	fundamental	fundamental	ADJ
cana-5309	69	8	nature	nature	NOUN
cana-5309	69	9	as	as	ADP
cana-5309	69	10	theoretical	theoretical	ADJ
cana-5309	69	11	component	component	NOUN
cana-5309	69	12	and	and	CCONJ
cana-5309	69	13	useful	useful	ADJ
cana-5309	69	14	tool	tool	NOUN
cana-5309	69	15	through	through	ADP
cana-5309	69	16	ongoing	ongoing	ADJ
cana-5309	69	17	efforts	effort	NOUN
cana-5309	69	18	to	to	PART
cana-5309	69	19	increase	increase	VERB
cana-5309	69	20	knowledge	knowledge	NOUN
cana-5309	69	21	and	and	CCONJ
cana-5309	69	22	remove	remove	VERB
cana-5309	69	23	barriers	barrier	NOUN
cana-5309	69	24	for	for	ADP
cana-5309	69	25	different	different	ADJ
cana-5309	69	26	fields	field	NOUN
cana-5309	69	27	of	of	ADP
cana-5309	69	28	study	study	NOUN
cana-5309	69	29	.	.	PUNCT
cana-5309	70	1	iii	iii	X
cana-5309	70	2	.	.	PROPN
cana-5309	70	3	proposed	propose	VERB
cana-5309	70	4	methodology	methodology	NOUN
cana-5309	70	5	in	in	ADP
cana-5309	70	6	order	order	NOUN
cana-5309	70	7	to	to	PART
cana-5309	70	8	deeply	deeply	ADV
cana-5309	70	9	explore	explore	VERB
cana-5309	70	10	hyperbolic	hyperbolic	ADJ
cana-5309	70	11	geometry	geometry	NOUN
cana-5309	70	12	and	and	CCONJ
cana-5309	70	13	its	its	PRON
cana-5309	70	14	properties	property	NOUN
cana-5309	70	15	,	,	PUNCT
cana-5309	70	16	we	we	PRON
cana-5309	70	17	begin	begin	VERB
cana-5309	70	18	by	by	ADP
cana-5309	70	19	constructing	construct	VERB
cana-5309	70	20	a	a	DET
cana-5309	70	21	systematic	systematic	ADJ
cana-5309	70	22	mathematical	mathematical	ADJ
cana-5309	70	23	framework	framework	NOUN
cana-5309	70	24	that	that	PRON
cana-5309	70	25	uses	use	VERB
cana-5309	70	26	established	establish	VERB
cana-5309	70	27	models	model	NOUN
cana-5309	70	28	and	and	CCONJ
cana-5309	70	29	introduces	introduce	VERB
cana-5309	70	30	new	new	ADJ
cana-5309	70	31	ways	way	NOUN
cana-5309	70	32	to	to	PART
cana-5309	70	33	compute	compute	VERB
cana-5309	70	34	,	,	PUNCT
cana-5309	70	35	visualize	visualize	VERB
cana-5309	70	36	,	,	PUNCT
cana-5309	70	37	and	and	CCONJ
cana-5309	70	38	compare	compare	VERB
cana-5309	70	39	hyperbolic	hyperbolic	ADJ
cana-5309	70	40	structures	structure	NOUN
cana-5309	70	41	.	.	PUNCT
cana-5309	71	1	the	the	DET
cana-5309	71	2	methodology	methodology	NOUN
cana-5309	71	3	is	be	AUX
cana-5309	71	4	divided	divide	VERB
cana-5309	71	5	into	into	ADP
cana-5309	71	6	five	five	NUM
cana-5309	71	7	main	main	ADJ
cana-5309	71	8	phases	phase	NOUN
cana-5309	71	9	:	:	PUNCT
cana-5309	71	10	model	model	NOUN
cana-5309	71	11	selection	selection	NOUN
cana-5309	71	12	,	,	PUNCT
cana-5309	71	13	geometric	geometric	ADJ
cana-5309	71	14	definition	definition	NOUN
cana-5309	71	15	,	,	PUNCT
cana-5309	71	16	metric	metric	ADJ
cana-5309	71	17	derivation	derivation	NOUN
cana-5309	71	18	,	,	PUNCT
cana-5309	71	19	visualization	visualization	NOUN
cana-5309	71	20	,	,	PUNCT
cana-5309	71	21	and	and	CCONJ
cana-5309	71	22	application	application	NOUN
cana-5309	71	23	testing	testing	NOUN
cana-5309	71	24	[	[	X
cana-5309	71	25	8	8	NUM
cana-5309	71	26	]	]	PUNCT
cana-5309	71	27	.	.	PUNCT
cana-5309	72	1	a.	a.	NOUN
cana-5309	72	2	model	model	PROPN
cana-5309	72	3	selection	selection	PROPN
cana-5309	72	4	first	first	ADV
cana-5309	72	5	,	,	PUNCT
cana-5309	72	6	we	we	PRON
cana-5309	72	7	select	select	VERB
cana-5309	72	8	the	the	DET
cana-5309	72	9	poincaré	poincaré	PROPN
cana-5309	72	10	disk	disk	NOUN
cana-5309	72	11	model	model	NOUN
cana-5309	72	12	as	as	ADP
cana-5309	72	13	our	our	PRON
cana-5309	72	14	primary	primary	ADJ
cana-5309	72	15	working	working	NOUN
cana-5309	72	16	model	model	NOUN
cana-5309	72	17	due	due	ADP
cana-5309	72	18	to	to	ADP
cana-5309	72	19	its	its	PRON
cana-5309	72	20	conformality	conformality	NOUN
cana-5309	72	21	,	,	PUNCT
cana-5309	72	22	which	which	PRON
cana-5309	72	23	preserves	preserve	VERB
cana-5309	72	24	angles	angle	NOUN
cana-5309	72	25	and	and	CCONJ
cana-5309	72	26	makes	make	VERB
cana-5309	72	27	calculations	calculation	NOUN
cana-5309	72	28	involving	involve	VERB
cana-5309	72	29	angles	angle	NOUN
cana-5309	72	30	more	more	ADV
cana-5309	72	31	intuitive	intuitive	ADJ
cana-5309	72	32	.	.	PUNCT
cana-5309	73	1	in	in	ADP
cana-5309	73	2	the	the	DET
cana-5309	73	3	poincaré	poincaré	PROPN
cana-5309	73	4	disk	disk	NOUN
cana-5309	73	5	,	,	PUNCT
cana-5309	73	6	the	the	DET
cana-5309	73	7	set	set	NOUN
cana-5309	73	8	of	of	ADP
cana-5309	73	9	points	point	NOUN
cana-5309	73	10	(	(	PUNCT
cana-5309	73	11	𝑥	𝑥	NOUN
cana-5309	73	12	,	,	PUNCT
cana-5309	73	13	𝑦	𝑦	NOUN
cana-5309	73	14	)	)	PUNCT
cana-5309	73	15	satisfying	satisfy	VERB
cana-5309	73	16	𝑥2	𝑥2	NOUN
cana-5309	73	17	+	+	CCONJ
cana-5309	73	18	𝑦2	𝑦2	PROPN
cana-5309	73	19	<	<	X
cana-5309	73	20	1	1	NUM
cana-5309	73	21	defines	define	VERB
cana-5309	73	22	the	the	DET
cana-5309	73	23	space	space	NOUN
cana-5309	73	24	.	.	PUNCT
cana-5309	74	1	the	the	DET
cana-5309	74	2	metric	metric	NOUN
cana-5309	74	3	in	in	ADP
cana-5309	74	4	the	the	DET
cana-5309	74	5	disk	disk	NOUN
cana-5309	74	6	is	be	AUX
cana-5309	74	7	not	not	PART
cana-5309	74	8	euclidean	euclidean	ADJ
cana-5309	74	9	.	.	PUNCT
cana-5309	75	1	instead	instead	ADV
cana-5309	75	2	,	,	PUNCT
cana-5309	75	3	the	the	DET
cana-5309	75	4	infinitesimal	infinitesimal	ADJ
cana-5309	75	5	hyperbolic	hyperbolic	ADJ
cana-5309	75	6	distance	distance	NOUN
cana-5309	75	7	𝑑𝑠	𝑑𝑠	NOUN
cana-5309	75	8	is	be	AUX
cana-5309	75	9	given	give	VERB
cana-5309	75	10	by	by	ADP
cana-5309	75	11	:	:	PUNCT
cana-5309	75	12	communications	communication	NOUN
cana-5309	75	13	on	on	ADP
cana-5309	75	14	applied	apply	VERB
cana-5309	75	15	nonlinear	nonlinear	ADJ
cana-5309	75	16	analysis	analysis	NOUN
cana-5309	75	17	issn	issn	NOUN
cana-5309	75	18	:	:	PUNCT
cana-5309	75	19	1074	1074	NUM
cana-5309	75	20	-	-	PUNCT
cana-5309	75	21	133x	133x	NUM
cana-5309	75	22	vol	vol	NOUN
cana-5309	75	23	31	31	NUM
cana-5309	75	24	no	no	NOUN
cana-5309	75	25	.	.	PUNCT
cana-5309	76	1	4s	4s	NUM
cana-5309	76	2	(	(	PUNCT
cana-5309	76	3	2024	2024	NUM
cana-5309	76	4	)	)	PUNCT
cana-5309	76	5	676	676	NUM
cana-5309	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	76	7	𝑑𝑠	𝑑𝑠	NOUN
cana-5309	76	8	=	=	SYM
cana-5309	76	9	2√𝑑𝑥2	2√𝑑𝑥2	NUM
cana-5309	77	1	+	+	CCONJ
cana-5309	77	2	𝑑𝑦2	𝑑𝑦2	NOUN
cana-5309	77	3	1	1	NUM
cana-5309	77	4	−	−	NOUN
cana-5309	77	5	(	(	PUNCT
cana-5309	77	6	𝑥2	𝑥2	NOUN
cana-5309	77	7	+	+	CCONJ
cana-5309	77	8	𝑦2	𝑦2	NOUN
cana-5309	77	9	)	)	PUNCT
cana-5309	77	10	this	this	DET
cana-5309	77	11	equation	equation	NOUN
cana-5309	77	12	will	will	AUX
cana-5309	77	13	serve	serve	VERB
cana-5309	77	14	as	as	ADP
cana-5309	77	15	the	the	DET
cana-5309	77	16	foundation	foundation	NOUN
cana-5309	77	17	for	for	ADP
cana-5309	77	18	further	further	ADJ
cana-5309	77	19	geometric	geometric	ADJ
cana-5309	77	20	derivations	derivation	NOUN
cana-5309	77	21	.	.	PUNCT
cana-5309	78	1	b.	b.	PROPN
cana-5309	78	2	geometric	geometric	ADJ
cana-5309	78	3	definition	definition	NOUN
cana-5309	78	4	we	we	PRON
cana-5309	78	5	define	define	VERB
cana-5309	78	6	a	a	DET
cana-5309	78	7	hyperbolic	hyperbolic	ADJ
cana-5309	78	8	line	line	NOUN
cana-5309	78	9	in	in	ADP
cana-5309	78	10	the	the	DET
cana-5309	78	11	poincaré	poincaré	PROPN
cana-5309	78	12	disk	disk	NOUN
cana-5309	78	13	as	as	ADP
cana-5309	78	14	either	either	CCONJ
cana-5309	78	15	a	a	DET
cana-5309	78	16	euclidean	euclidean	ADJ
cana-5309	78	17	circle	circle	NOUN
cana-5309	78	18	orthogonal	orthogonal	NOUN
cana-5309	78	19	to	to	ADP
cana-5309	78	20	the	the	DET
cana-5309	78	21	boundary	boundary	NOUN
cana-5309	78	22	or	or	CCONJ
cana-5309	78	23	a	a	DET
cana-5309	78	24	diameter	diameter	NOUN
cana-5309	78	25	.	.	PUNCT
cana-5309	79	1	the	the	DET
cana-5309	79	2	general	general	ADJ
cana-5309	79	3	form	form	NOUN
cana-5309	79	4	of	of	ADP
cana-5309	79	5	a	a	DET
cana-5309	79	6	geodesic	geodesic	ADJ
cana-5309	79	7	(	(	PUNCT
cana-5309	79	8	line	line	NOUN
cana-5309	79	9	)	)	PUNCT
cana-5309	79	10	is	be	AUX
cana-5309	79	11	captured	capture	VERB
cana-5309	79	12	as	as	ADP
cana-5309	79	13	:	:	PUNCT
cana-5309	79	14	(	(	PUNCT
cana-5309	79	15	𝑥	𝑥	X
cana-5309	79	16	−	−	PROPN
cana-5309	79	17	𝑎)2	𝑎)2	NOUN
cana-5309	79	18	+	+	CCONJ
cana-5309	79	19	(	(	PUNCT
cana-5309	79	20	𝑦	𝑦	NOUN
cana-5309	79	21	−	−	PROPN
cana-5309	79	22	𝑏)2	𝑏)2	PROPN
cana-5309	79	23	=	=	NOUN
cana-5309	79	24	𝑟2	𝑟2	NOUN
cana-5309	79	25	where	where	SCONJ
cana-5309	79	26	𝑟2	𝑟2	NOUN
cana-5309	79	27	=	=	SYM
cana-5309	79	28	𝑎2	𝑎2	PROPN
cana-5309	79	29	+	+	PROPN
cana-5309	79	30	𝑏2	𝑏2	PROPN
cana-5309	79	31	−	−	PROPN
cana-5309	79	32	1	1	NUM
cana-5309	79	33	,	,	PUNCT
cana-5309	79	34	ensuring	ensure	VERB
cana-5309	79	35	orthogonality	orthogonality	NOUN
cana-5309	79	36	to	to	ADP
cana-5309	79	37	the	the	DET
cana-5309	79	38	unit	unit	NOUN
cana-5309	79	39	circle	circle	PROPN
cana-5309	79	40	boundary	boundary	PROPN
cana-5309	79	41	.	.	PUNCT
cana-5309	80	1	for	for	ADP
cana-5309	80	2	two	two	NUM
cana-5309	80	3	points	point	NOUN
cana-5309	80	4	𝑃	𝑃	NOUN
cana-5309	80	5	and	and	CCONJ
cana-5309	80	6	𝑄	𝑄	NOUN
cana-5309	80	7	inside	inside	ADP
cana-5309	80	8	the	the	DET
cana-5309	80	9	disk	disk	NOUN
cana-5309	80	10	,	,	PUNCT
cana-5309	80	11	the	the	DET
cana-5309	80	12	hyperbolic	hyperbolic	ADJ
cana-5309	80	13	distance	distance	NOUN
cana-5309	80	14	𝑑(𝑃	𝑑(𝑃	NOUN
cana-5309	80	15	,	,	PUNCT
cana-5309	80	16	𝑄	𝑄	PROPN
cana-5309	80	17	)	)	PUNCT
cana-5309	80	18	is	be	AUX
cana-5309	80	19	computed	compute	VERB
cana-5309	80	20	as	as	ADP
cana-5309	80	21	:	:	PUNCT
cana-5309	80	22	𝑑(𝑃	𝑑(𝑃	NOUN
cana-5309	80	23	,	,	PUNCT
cana-5309	80	24	𝑄	𝑄	PROPN
cana-5309	80	25	)	)	PUNCT
cana-5309	80	26	=	=	SYM
cana-5309	80	27	arccosh	arccosh	NOUN
cana-5309	80	28	(	(	PUNCT
cana-5309	80	29	1	1	NUM
cana-5309	80	30	+	+	NUM
cana-5309	80	31	2‖𝑃	2‖𝑃	NUM
cana-5309	80	32	−	−	PROPN
cana-5309	80	33	𝑄‖2	𝑄‖2	PROPN
cana-5309	80	34	(	(	PUNCT
cana-5309	80	35	1	1	NUM
cana-5309	80	36	−	−	PROPN
cana-5309	80	37	‖𝑃‖2)(1	‖𝑃‖2)(1	NOUN
cana-5309	80	38	−	−	PROPN
cana-5309	80	39	‖𝑄‖2	‖𝑄‖2	PROPN
cana-5309	80	40	)	)	PUNCT
cana-5309	80	41	)	)	PUNCT
cana-5309	80	42	where	where	SCONJ
cana-5309	80	43	||	||	NOUN
cana-5309	80	44	•	•	NOUN
cana-5309	80	45	||	||	NOUN
cana-5309	80	46	denotes	denote	VERB
cana-5309	80	47	the	the	DET
cana-5309	80	48	euclidean	euclidean	PROPN
cana-5309	80	49	norm	norm	NOUN
cana-5309	80	50	.	.	PUNCT
cana-5309	81	1	c.	c.	PROPN
cana-5309	81	2	metric	metric	ADJ
cana-5309	81	3	derivation	derivation	NOUN
cana-5309	81	4	a	a	DET
cana-5309	81	5	critical	critical	ADJ
cana-5309	81	6	aspect	aspect	NOUN
cana-5309	81	7	of	of	ADP
cana-5309	81	8	hyperbolic	hyperbolic	ADJ
cana-5309	81	9	geometry	geometry	NOUN
cana-5309	81	10	is	be	AUX
cana-5309	81	11	the	the	DET
cana-5309	81	12	calculation	calculation	NOUN
cana-5309	81	13	of	of	ADP
cana-5309	81	14	areas	area	NOUN
cana-5309	81	15	and	and	CCONJ
cana-5309	81	16	angles	angle	NOUN
cana-5309	81	17	.	.	PUNCT
cana-5309	82	1	in	in	ADP
cana-5309	82	2	hyperbolic	hyperbolic	ADJ
cana-5309	82	3	triangles	triangle	NOUN
cana-5309	82	4	,	,	PUNCT
cana-5309	82	5	the	the	DET
cana-5309	82	6	sum	sum	NOUN
cana-5309	82	7	of	of	ADP
cana-5309	82	8	angles	angle	NOUN
cana-5309	82	9	𝛼	𝛼	NOUN
cana-5309	82	10	+	+	NOUN
cana-5309	82	11	𝛽	𝛽	NOUN
cana-5309	82	12	+	+	NOUN
cana-5309	82	13	𝛾	𝛾	NOUN
cana-5309	82	14	satisfies	satisfie	NOUN
cana-5309	82	15	:	:	PUNCT
cana-5309	82	16	𝛼	𝛼	X
cana-5309	83	1	+	+	X
cana-5309	83	2	𝛽	𝛽	NOUN
cana-5309	83	3	+	+	NOUN
cana-5309	83	4	𝛾	𝛾	ADP
cana-5309	83	5	<	<	X
cana-5309	83	6	𝜋	𝜋	X
cana-5309	83	7	the	the	DET
cana-5309	83	8	area	area	NOUN
cana-5309	83	9	𝑨	𝑨	PROPN
cana-5309	83	10	of	of	ADP
cana-5309	83	11	a	a	DET
cana-5309	83	12	hyperbolic	hyperbolic	ADJ
cana-5309	83	13	triangle	triangle	NOUN
cana-5309	83	14	is	be	AUX
cana-5309	83	15	directly	directly	ADV
cana-5309	83	16	related	relate	VERB
cana-5309	83	17	to	to	ADP
cana-5309	83	18	its	its	PRON
cana-5309	83	19	angle	angle	NOUN
cana-5309	83	20	deficit	deficit	NOUN
cana-5309	83	21	:	:	PUNCT
cana-5309	83	22	𝐴	𝐴	PROPN
cana-5309	83	23	=	=	PROPN
cana-5309	83	24	𝜋	𝜋	PRON
cana-5309	83	25	−	−	PROPN
cana-5309	83	26	(	(	PUNCT
cana-5309	83	27	𝛼	𝛼	X
cana-5309	84	1	+	+	X
cana-5309	84	2	𝛽	𝛽	NOUN
cana-5309	84	3	+	+	CCONJ
cana-5309	84	4	𝛾	𝛾	NOUN
cana-5309	84	5	)	)	PUNCT
cana-5309	84	6	additionally	additionally	ADV
cana-5309	84	7	,	,	PUNCT
cana-5309	84	8	the	the	DET
cana-5309	84	9	hyperbolic	hyperbolic	ADJ
cana-5309	84	10	law	law	NOUN
cana-5309	84	11	of	of	ADP
cana-5309	84	12	cosines	cosines	PROPN
cana-5309	84	13	differs	differ	VERB
cana-5309	84	14	from	from	ADP
cana-5309	84	15	its	its	PRON
cana-5309	84	16	euclidean	euclidean	ADJ
cana-5309	84	17	counterpart	counterpart	NOUN
cana-5309	84	18	.	.	PUNCT
cana-5309	85	1	for	for	ADP
cana-5309	85	2	a	a	DET
cana-5309	85	3	triangle	triangle	NOUN
cana-5309	85	4	with	with	ADP
cana-5309	85	5	sides	side	NOUN
cana-5309	85	6	𝑎	𝑎	ADP
cana-5309	85	7	,	,	PUNCT
cana-5309	85	8	𝑏	𝑏	NOUN
cana-5309	85	9	,	,	PUNCT
cana-5309	85	10	𝑐	𝑐	NOUN
cana-5309	85	11	and	and	CCONJ
cana-5309	85	12	opposite	opposite	ADJ
cana-5309	85	13	angles	angle	NOUN
cana-5309	85	14	𝛼	𝛼	NOUN
cana-5309	85	15	,	,	PUNCT
cana-5309	85	16	𝛽	𝛽	NOUN
cana-5309	85	17	,	,	PUNCT
cana-5309	85	18	𝛾	𝛾	PROPN
cana-5309	85	19	,	,	PUNCT
cana-5309	85	20	it	it	PRON
cana-5309	85	21	is	be	AUX
cana-5309	85	22	:	:	PUNCT
cana-5309	85	23	cosh⁡(𝑐	cosh⁡(𝑐	NOUN
cana-5309	85	24	)	)	PUNCT
cana-5309	85	25	=	=	PUNCT
cana-5309	85	26	cosh⁡(𝑎)cosh⁡(𝑏	cosh⁡(𝑎)cosh⁡(𝑏	NOUN
cana-5309	85	27	)	)	PUNCT
cana-5309	86	1	−	−	NOUN
cana-5309	86	2	sinh⁡(𝑎)sinh⁡(𝑏)cos⁡(𝛾	sinh⁡(𝑎)sinh⁡(𝑏)cos⁡(𝛾	NOUN
cana-5309	86	3	)	)	PUNCT
cana-5309	86	4	where	where	SCONJ
cana-5309	86	5	cosh	cosh	NOUN
cana-5309	86	6	and	and	CCONJ
cana-5309	86	7	sinh	sinh	NOUN
cana-5309	86	8	are	be	AUX
cana-5309	86	9	hyperbolic	hyperbolic	ADJ
cana-5309	86	10	cosine	cosine	NOUN
cana-5309	86	11	and	and	CCONJ
cana-5309	86	12	sine	sine	ADJ
cana-5309	86	13	functions	function	NOUN
cana-5309	86	14	,	,	PUNCT
cana-5309	86	15	respectively	respectively	ADV
cana-5309	86	16	.	.	PUNCT
cana-5309	87	1	for	for	ADP
cana-5309	87	2	circles	circle	NOUN
cana-5309	87	3	in	in	ADP
cana-5309	87	4	hyperbolic	hyperbolic	ADJ
cana-5309	87	5	space	space	NOUN
cana-5309	87	6	,	,	PUNCT
cana-5309	87	7	the	the	DET
cana-5309	87	8	circumference	circumference	NOUN
cana-5309	87	9	𝐶	𝐶	PROPN
cana-5309	87	10	grows	grow	VERB
cana-5309	87	11	exponentially	exponentially	ADV
cana-5309	87	12	with	with	ADP
cana-5309	87	13	radius	radius	NOUN
cana-5309	87	14	𝑟	𝑟	NOUN
cana-5309	87	15	,	,	PUNCT
cana-5309	87	16	and	and	CCONJ
cana-5309	87	17	is	be	AUX
cana-5309	87	18	given	give	VERB
cana-5309	87	19	by	by	ADP
cana-5309	87	20	:	:	PUNCT
cana-5309	87	21	𝐶	𝐶	PROPN
cana-5309	87	22	=	=	PROPN
cana-5309	87	23	2𝜋sinh⁡(𝑟	2𝜋sinh⁡(𝑟	NUM
cana-5309	87	24	)	)	PUNCT
cana-5309	87	25	rather	rather	ADV
cana-5309	87	26	than	than	ADP
cana-5309	87	27	linearly	linearly	ADV
cana-5309	87	28	,	,	PUNCT
cana-5309	87	29	as	as	ADP
cana-5309	87	30	in	in	ADP
cana-5309	87	31	euclidean	euclidean	ADJ
cana-5309	87	32	geometry	geometry	NOUN
cana-5309	87	33	.	.	PUNCT
cana-5309	88	1	the	the	DET
cana-5309	88	2	area	area	NOUN
cana-5309	88	3	𝐴	𝐴	PROPN
cana-5309	88	4	enclosed	enclose	VERB
cana-5309	88	5	by	by	ADP
cana-5309	88	6	a	a	DET
cana-5309	88	7	hyperbolic	hyperbolic	ADJ
cana-5309	88	8	circle	circle	NOUN
cana-5309	88	9	of	of	ADP
cana-5309	88	10	radius	radius	NOUN
cana-5309	88	11	𝑟	𝑟	NOUN
cana-5309	88	12	is	be	AUX
cana-5309	88	13	:	:	PUNCT
cana-5309	88	14	communications	communication	NOUN
cana-5309	88	15	on	on	ADP
cana-5309	88	16	applied	apply	VERB
cana-5309	88	17	nonlinear	nonlinear	ADJ
cana-5309	88	18	analysis	analysis	NOUN
cana-5309	88	19	issn	issn	NOUN
cana-5309	88	20	:	:	PUNCT
cana-5309	88	21	1074	1074	NUM
cana-5309	88	22	-	-	PUNCT
cana-5309	88	23	133x	133x	NUM
cana-5309	88	24	vol	vol	NOUN
cana-5309	88	25	31	31	NUM
cana-5309	88	26	no	no	NOUN
cana-5309	88	27	.	.	PUNCT
cana-5309	89	1	4s	4s	NUM
cana-5309	89	2	(	(	PUNCT
cana-5309	89	3	2024	2024	NUM
cana-5309	89	4	)	)	PUNCT
cana-5309	89	5	677	677	NUM
cana-5309	89	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	89	7	𝐴	𝐴	PROPN
cana-5309	89	8	=	=	PUNCT
cana-5309	89	9	2𝜋(cosh⁡(𝑟	2𝜋(cosh⁡(𝑟	NUM
cana-5309	89	10	)	)	PUNCT
cana-5309	89	11	−	−	NOUN
cana-5309	89	12	1	1	X
cana-5309	89	13	)	)	PUNCT
cana-5309	89	14	thus	thus	ADV
cana-5309	89	15	demonstrating	demonstrate	VERB
cana-5309	89	16	a	a	DET
cana-5309	89	17	strong	strong	ADJ
cana-5309	89	18	departure	departure	NOUN
cana-5309	89	19	from	from	ADP
cana-5309	89	20	flat	flat	ADJ
cana-5309	89	21	-	-	PUNCT
cana-5309	89	22	space	space	NOUN
cana-5309	89	23	intuitions	intuition	NOUN
cana-5309	89	24	.	.	PUNCT
cana-5309	90	1	d.	d.	PROPN
cana-5309	90	2	visualization	visualization	NOUN
cana-5309	90	3	to	to	PART
cana-5309	90	4	aid	aid	VERB
cana-5309	90	5	understanding	understanding	NOUN
cana-5309	90	6	,	,	PUNCT
cana-5309	90	7	we	we	PRON
cana-5309	90	8	implement	implement	VERB
cana-5309	90	9	computational	computational	ADJ
cana-5309	90	10	simulations	simulation	NOUN
cana-5309	90	11	of	of	ADP
cana-5309	90	12	hyperbolic	hyperbolic	ADJ
cana-5309	90	13	structures	structure	NOUN
cana-5309	90	14	using	use	VERB
cana-5309	90	15	visualization	visualization	NOUN
cana-5309	90	16	libraries	library	NOUN
cana-5309	90	17	.	.	PUNCT
cana-5309	91	1	geodesics	geodesic	NOUN
cana-5309	91	2	are	be	AUX
cana-5309	91	3	drawn	draw	VERB
cana-5309	91	4	as	as	ADP
cana-5309	91	5	arcs	arc	NOUN
cana-5309	91	6	of	of	ADP
cana-5309	91	7	circles	circle	NOUN
cana-5309	91	8	,	,	PUNCT
cana-5309	91	9	and	and	CCONJ
cana-5309	91	10	hyperbolic	hyperbolic	ADJ
cana-5309	91	11	tessellations	tessellation	NOUN
cana-5309	91	12	are	be	AUX
cana-5309	91	13	created	create	VERB
cana-5309	91	14	using	use	VERB
cana-5309	91	15	recursive	recursive	ADJ
cana-5309	91	16	symmetry	symmetry	NOUN
cana-5309	91	17	operations	operation	NOUN
cana-5309	91	18	.	.	PUNCT
cana-5309	92	1	given	give	VERB
cana-5309	92	2	a	a	DET
cana-5309	92	3	transformation	transformation	NOUN
cana-5309	92	4	matrix	matrix	NOUN
cana-5309	92	5	𝑇	𝑇	PROPN
cana-5309	92	6	for	for	ADP
cana-5309	92	7	hyperbolic	hyperbolic	ADJ
cana-5309	92	8	motions	motion	NOUN
cana-5309	92	9	:	:	PUNCT
cana-5309	92	10	𝑇	𝑇	PROPN
cana-5309	92	11	=	=	SYM
cana-5309	92	12	[	[	PUNCT
cana-5309	92	13	cosh⁡(𝜃	cosh⁡(𝜃	NOUN
cana-5309	92	14	)	)	PUNCT
cana-5309	92	15	sinh⁡(𝜃	sinh⁡(𝜃	NOUN
cana-5309	92	16	)	)	PUNCT
cana-5309	92	17	sinh⁡(𝜃	sinh⁡(𝜃	NOUN
cana-5309	92	18	)	)	PUNCT
cana-5309	92	19	cosh⁡(𝜃	cosh⁡(𝜃	NOUN
cana-5309	92	20	)	)	PUNCT
cana-5309	92	21	]	]	PUNCT
cana-5309	93	1	we	we	PRON
cana-5309	93	2	simulate	simulate	VERB
cana-5309	93	3	translations	translation	NOUN
cana-5309	93	4	along	along	ADP
cana-5309	93	5	geodesics	geodesic	NOUN
cana-5309	93	6	.	.	PUNCT
cana-5309	94	1	the	the	DET
cana-5309	94	2	transformations	transformation	NOUN
cana-5309	94	3	are	be	AUX
cana-5309	94	4	essential	essential	ADJ
cana-5309	94	5	for	for	ADP
cana-5309	94	6	dynamically	dynamically	ADV
cana-5309	94	7	generating	generate	VERB
cana-5309	94	8	hyperbolic	hyperbolic	ADJ
cana-5309	94	9	tilings	tiling	NOUN
cana-5309	94	10	and	and	CCONJ
cana-5309	94	11	animations	animation	NOUN
cana-5309	94	12	.	.	PUNCT
cana-5309	95	1	reflections	reflection	NOUN
cana-5309	95	2	across	across	ADP
cana-5309	95	3	geodesics	geodesic	NOUN
cana-5309	95	4	are	be	AUX
cana-5309	95	5	another	another	DET
cana-5309	95	6	important	important	ADJ
cana-5309	95	7	operation	operation	NOUN
cana-5309	95	8	,	,	PUNCT
cana-5309	95	9	expressed	express	VERB
cana-5309	95	10	mathematically	mathematically	ADV
cana-5309	95	11	by	by	ADP
cana-5309	95	12	:	:	PUNCT
cana-5309	95	13	𝑃′	𝑃′	PROPN
cana-5309	95	14	=	=	SYM
cana-5309	95	15	𝑅(𝑃	𝑅(𝑃	PROPN
cana-5309	95	16	)	)	PUNCT
cana-5309	95	17	where	where	SCONJ
cana-5309	95	18	𝑅	𝑅	PROPN
cana-5309	95	19	is	be	AUX
cana-5309	95	20	the	the	DET
cana-5309	95	21	reflection	reflection	NOUN
cana-5309	95	22	operator	operator	NOUN
cana-5309	95	23	corresponding	correspond	VERB
cana-5309	95	24	to	to	ADP
cana-5309	95	25	a	a	DET
cana-5309	95	26	given	give	VERB
cana-5309	95	27	geodesic	geodesic	NOUN
cana-5309	95	28	.	.	PUNCT
cana-5309	96	1	e.	e.	PROPN
cana-5309	96	2	application	application	PROPN
cana-5309	96	3	testing	testing	NOUN
cana-5309	96	4	finally	finally	ADV
cana-5309	96	5	,	,	PUNCT
cana-5309	96	6	we	we	PRON
cana-5309	96	7	apply	apply	VERB
cana-5309	96	8	the	the	DET
cana-5309	96	9	methodology	methodology	NOUN
cana-5309	96	10	to	to	PART
cana-5309	96	11	simulate	simulate	VERB
cana-5309	96	12	network	network	NOUN
cana-5309	96	13	embeddings	embedding	NOUN
cana-5309	96	14	into	into	ADP
cana-5309	96	15	hyperbolic	hyperbolic	ADJ
cana-5309	96	16	space	space	NOUN
cana-5309	96	17	.	.	PUNCT
cana-5309	97	1	by	by	ADP
cana-5309	97	2	minimizing	minimize	VERB
cana-5309	97	3	the	the	DET
cana-5309	97	4	loss	loss	NOUN
cana-5309	97	5	function	function	NOUN
cana-5309	97	6	:	:	PUNCT
cana-5309	97	7	ℒ	ℒ	PROPN
cana-5309	97	8	=	=	SYM
cana-5309	97	9	∑	∑	PART
cana-5309	97	10	  	  	SPACE
cana-5309	97	11	𝑖,𝑗	𝑖,𝑗	PROPN
cana-5309	97	12	𝑤𝑖𝑗(𝑑𝐻(𝑥𝑖	𝑤𝑖𝑗(𝑑𝐻(𝑥𝑖	PROPN
cana-5309	97	13	,	,	PUNCT
cana-5309	97	14	𝑥𝑗	𝑥𝑗	PROPN
cana-5309	97	15	)	)	PUNCT
cana-5309	97	16	−	−	PROPN
cana-5309	98	1	𝛿𝑖𝑗	𝛿𝑖𝑗	NOUN
cana-5309	98	2	)	)	PUNCT
cana-5309	98	3	2	2	NUM
cana-5309	98	4	where	where	SCONJ
cana-5309	98	5	𝑤𝑖𝑗	𝑤𝑖𝑗	PROPN
cana-5309	98	6	are	be	AUX
cana-5309	98	7	weights	weight	NOUN
cana-5309	98	8	,	,	PUNCT
cana-5309	98	9	𝑑𝐻	𝑑𝐻	ADJ
cana-5309	98	10	denotes	denote	NOUN
cana-5309	98	11	hyperbolic	hyperbolic	ADJ
cana-5309	98	12	distance	distance	NOUN
cana-5309	98	13	,	,	PUNCT
cana-5309	98	14	and	and	CCONJ
cana-5309	98	15	𝛿𝑖𝑗	𝛿𝑖𝑗	NOUN
cana-5309	98	16	is	be	AUX
cana-5309	98	17	the	the	DET
cana-5309	98	18	target	target	NOUN
cana-5309	98	19	distance	distance	NOUN
cana-5309	99	1	,	,	PUNCT
cana-5309	99	2	we	we	PRON
cana-5309	99	3	optimize	optimize	VERB
cana-5309	99	4	embeddings	embedding	NOUN
cana-5309	99	5	to	to	PART
cana-5309	99	6	preserve	preserve	VERB
cana-5309	99	7	structural	structural	ADJ
cana-5309	99	8	relationships	relationship	NOUN
cana-5309	99	9	.	.	PUNCT
cana-5309	100	1	for	for	ADP
cana-5309	100	2	optimization	optimization	NOUN
cana-5309	100	3	,	,	PUNCT
cana-5309	100	4	we	we	PRON
cana-5309	100	5	rely	rely	VERB
cana-5309	100	6	on	on	ADP
cana-5309	100	7	hyperbolic	hyperbolic	ADJ
cana-5309	100	8	gradient	gradient	NOUN
cana-5309	100	9	descent	descent	NOUN
cana-5309	100	10	,	,	PUNCT
cana-5309	100	11	where	where	SCONJ
cana-5309	100	12	the	the	DET
cana-5309	100	13	update	update	NOUN
cana-5309	100	14	rule	rule	NOUN
cana-5309	100	15	for	for	ADP
cana-5309	100	16	a	a	DET
cana-5309	100	17	point	point	NOUN
cana-5309	100	18	𝑥	𝑥	NOUN
cana-5309	100	19	is	be	AUX
cana-5309	100	20	:	:	PUNCT
cana-5309	100	21	𝑥′	𝑥′	PROPN
cana-5309	100	22	=	=	SYM
cana-5309	100	23	exp𝑥⁡(−𝜂∇ℒ(𝑥	exp𝑥⁡(−𝜂∇ℒ(𝑥	X
cana-5309	100	24	)	)	PUNCT
cana-5309	100	25	)	)	PUNCT
cana-5309	100	26	with	with	ADP
cana-5309	100	27	exp𝑥	exp𝑥	NOUN
cana-5309	100	28	being	be	AUX
cana-5309	100	29	the	the	DET
cana-5309	100	30	exponential	exponential	ADJ
cana-5309	100	31	map	map	NOUN
cana-5309	100	32	at	at	ADP
cana-5309	100	33	𝑥	𝑥	PROPN
cana-5309	100	34	,	,	PUNCT
cana-5309	100	35	𝜂	𝜂	DET
cana-5309	100	36	the	the	DET
cana-5309	100	37	learning	learning	NOUN
cana-5309	100	38	rate	rate	NOUN
cana-5309	100	39	,	,	PUNCT
cana-5309	100	40	and	and	CCONJ
cana-5309	100	41	∇ℒ	∇ℒ	PROPN
cana-5309	100	42	the	the	DET
cana-5309	100	43	riemannian	riemannian	ADJ
cana-5309	100	44	gradient	gradient	NOUN
cana-5309	100	45	.	.	PUNCT
cana-5309	101	1	throughout	throughout	ADP
cana-5309	101	2	the	the	DET
cana-5309	101	3	process	process	NOUN
cana-5309	101	4	,	,	PUNCT
cana-5309	101	5	continuous	continuous	ADJ
cana-5309	101	6	monitoring	monitoring	NOUN
cana-5309	101	7	of	of	ADP
cana-5309	101	8	error	error	NOUN
cana-5309	101	9	metrics	metric	NOUN
cana-5309	101	10	such	such	ADJ
cana-5309	101	11	as	as	ADP
cana-5309	101	12	mean	mean	NOUN
cana-5309	101	13	squared	square	VERB
cana-5309	101	14	error	error	NOUN
cana-5309	101	15	(	(	PUNCT
cana-5309	101	16	mse	mse	NOUN
cana-5309	101	17	)	)	PUNCT
cana-5309	101	18	is	be	AUX
cana-5309	101	19	maintained	maintain	VERB
cana-5309	101	20	,	,	PUNCT
cana-5309	101	21	defined	define	VERB
cana-5309	101	22	as	as	ADP
cana-5309	101	23	:	:	PUNCT
cana-5309	101	24	mse	mse	NOUN
cana-5309	101	25	=	=	SYM
cana-5309	101	26	1	1	NUM
cana-5309	101	27	𝑛	𝑛	PRON
cana-5309	101	28	∑	∑	DET
cana-5309	101	29	  	  	SPACE
cana-5309	101	30	𝑛	𝑛	PRON
cana-5309	101	31	𝑖=1	𝑖=1	PROPN
cana-5309	101	32	(	(	PUNCT
cana-5309	101	33	𝑑𝐻(𝑥𝑖	𝑑𝐻(𝑥𝑖	PROPN
cana-5309	101	34	,	,	PUNCT
cana-5309	101	35	𝑦𝑖	𝑦𝑖	PROPN
cana-5309	101	36	)	)	PUNCT
cana-5309	101	37	−	−	PROPN
cana-5309	101	38	𝛿𝑖	𝛿𝑖	NOUN
cana-5309	101	39	)	)	PUNCT
cana-5309	101	40	2	2	NUM
cana-5309	101	41	communications	communication	NOUN
cana-5309	101	42	on	on	ADP
cana-5309	101	43	applied	apply	VERB
cana-5309	101	44	nonlinear	nonlinear	ADJ
cana-5309	101	45	analysis	analysis	NOUN
cana-5309	101	46	issn	issn	NOUN
cana-5309	101	47	:	:	PUNCT
cana-5309	101	48	1074	1074	NUM
cana-5309	101	49	-	-	PUNCT
cana-5309	101	50	133x	133x	NUM
cana-5309	101	51	vol	vol	NOUN
cana-5309	101	52	31	31	NUM
cana-5309	101	53	no	no	NOUN
cana-5309	101	54	.	.	PUNCT
cana-5309	102	1	4s	4s	NUM
cana-5309	102	2	(	(	PUNCT
cana-5309	102	3	2024	2024	NUM
cana-5309	102	4	)	)	PUNCT
cana-5309	102	5	678	678	NUM
cana-5309	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	102	7	to	to	PART
cana-5309	102	8	ensure	ensure	VERB
cana-5309	102	9	the	the	DET
cana-5309	102	10	fidelity	fidelity	NOUN
cana-5309	102	11	of	of	ADP
cana-5309	102	12	the	the	DET
cana-5309	102	13	hyperbolic	hyperbolic	ADJ
cana-5309	102	14	embedding	embed	VERB
cana-5309	102	15	’s	’s	NOUN
cana-5309	102	16	.	.	PUNCT
cana-5309	103	1	figure	figure	NOUN
cana-5309	103	2	1	1	NUM
cana-5309	103	3	:	:	PUNCT
cana-5309	103	4	proposed	propose	VERB
cana-5309	103	5	methodology	methodology	NOUN
cana-5309	103	6	for	for	ADP
cana-5309	103	7	hyperbolic	hyperbolic	ADJ
cana-5309	103	8	geometry	geometry	NOUN
cana-5309	103	9	study	study	NOUN
cana-5309	103	10	iv	iv	NUM
cana-5309	103	11	.	.	PUNCT
cana-5309	103	12	result	result	PROPN
cana-5309	103	13	&	&	CCONJ
cana-5309	103	14	discussions	discussion	NOUN
cana-5309	103	15	the	the	DET
cana-5309	103	16	deployment	deployment	NOUN
cana-5309	103	17	of	of	ADP
cana-5309	103	18	this	this	DET
cana-5309	103	19	proposed	propose	VERB
cana-5309	103	20	method	method	NOUN
cana-5309	103	21	produced	produce	VERB
cana-5309	103	22	many	many	ADJ
cana-5309	103	23	valuable	valuable	ADJ
cana-5309	103	24	observations	observation	NOUN
cana-5309	103	25	about	about	ADP
cana-5309	103	26	hyperbolic	hyperbolic	ADJ
cana-5309	103	27	spaces	space	NOUN
cana-5309	103	28	.	.	PUNCT
cana-5309	104	1	observing	observe	VERB
cana-5309	104	2	geodesics	geodesic	NOUN
cana-5309	104	3	together	together	ADV
cana-5309	104	4	with	with	ADP
cana-5309	104	5	tessellations	tessellation	NOUN
cana-5309	104	6	under	under	ADP
cana-5309	104	7	the	the	DET
cana-5309	104	8	poincaré	poincaré	PROPN
cana-5309	104	9	disk	disk	NOUN
cana-5309	104	10	model	model	NOUN
cana-5309	104	11	helped	helped	AUX
cana-5309	104	12	demonstrate	demonstrate	VERB
cana-5309	104	13	mathematically	mathematically	ADV
cana-5309	104	14	as	as	ADV
cana-5309	104	15	well	well	ADV
cana-5309	104	16	as	as	ADP
cana-5309	104	17	visually	visually	ADV
cana-5309	104	18	how	how	SCONJ
cana-5309	104	19	angular	angular	ADJ
cana-5309	104	20	measures	measure	NOUN
cana-5309	104	21	stayed	stay	VERB
cana-5309	104	22	intact	intact	ADJ
cana-5309	104	23	while	while	SCONJ
cana-5309	104	24	distance	distance	NOUN
cana-5309	104	25	measurements	measurement	NOUN
cana-5309	104	26	were	be	AUX
cana-5309	104	27	altered	alter	VERB
cana-5309	104	28	.	.	PUNCT
cana-5309	105	1	spatial	spatial	ADJ
cana-5309	105	2	intuition	intuition	NOUN
cana-5309	105	3	works	work	VERB
cana-5309	105	4	substantially	substantially	ADV
cana-5309	105	5	different	different	ADJ
cana-5309	105	6	in	in	ADP
cana-5309	105	7	hyperbolic	hyperbolic	ADJ
cana-5309	105	8	space	space	NOUN
cana-5309	105	9	because	because	SCONJ
cana-5309	105	10	distances	distance	NOUN
cana-5309	105	11	become	become	VERB
cana-5309	105	12	dramatically	dramatically	ADV
cana-5309	105	13	greater	great	ADJ
cana-5309	105	14	near	near	ADP
cana-5309	105	15	the	the	DET
cana-5309	105	16	boundary	boundary	NOUN
cana-5309	105	17	.	.	PUNCT
cana-5309	106	1	calculation	calculation	NOUN
cana-5309	106	2	results	result	NOUN
cana-5309	106	3	showed	show	VERB
cana-5309	106	4	that	that	SCONJ
cana-5309	106	5	randomly	randomly	ADV
cana-5309	106	6	picked	pick	VERB
cana-5309	106	7	hyperbolic	hyperbolic	ADJ
cana-5309	106	8	distance	distance	NOUN
cana-5309	106	9	measurements	measurement	NOUN
cana-5309	106	10	exceeded	exceed	VERB
cana-5309	106	11	their	their	PRON
cana-5309	106	12	equivalent	equivalent	ADJ
cana-5309	106	13	euclidean	euclidean	ADJ
cana-5309	106	14	values	value	NOUN
cana-5309	106	15	in	in	ADP
cana-5309	106	16	all	all	DET
cana-5309	106	17	tests	test	NOUN
cana-5309	106	18	thereby	thereby	ADV
cana-5309	106	19	validating	validate	VERB
cana-5309	106	20	theoretical	theoretical	ADJ
cana-5309	106	21	hyperbolic	hyperbolic	ADJ
cana-5309	106	22	expansion	expansion	NOUN
cana-5309	106	23	principles	principle	NOUN
cana-5309	106	24	[	[	X
cana-5309	106	25	10	10	NUM
cana-5309	106	26	]	]	PUNCT
cana-5309	106	27	.	.	PUNCT
cana-5309	107	1	communications	communication	NOUN
cana-5309	107	2	on	on	ADP
cana-5309	107	3	applied	apply	VERB
cana-5309	107	4	nonlinear	nonlinear	ADJ
cana-5309	107	5	analysis	analysis	NOUN
cana-5309	107	6	issn	issn	NOUN
cana-5309	107	7	:	:	PUNCT
cana-5309	107	8	1074	1074	NUM
cana-5309	107	9	-	-	PUNCT
cana-5309	107	10	133x	133x	NUM
cana-5309	107	11	vol	vol	NOUN
cana-5309	107	12	31	31	NUM
cana-5309	107	13	no	no	NOUN
cana-5309	107	14	.	.	PUNCT
cana-5309	108	1	4s	4s	NUM
cana-5309	108	2	(	(	PUNCT
cana-5309	108	3	2024	2024	NUM
cana-5309	108	4	)	)	PUNCT
cana-5309	108	5	679	679	NUM
cana-5309	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	108	7	area	area	NOUN
cana-5309	108	8	expansion	expansion	NOUN
cana-5309	108	9	between	between	ADP
cana-5309	108	10	euclidean	euclidean	ADJ
cana-5309	108	11	and	and	CCONJ
cana-5309	108	12	hyperbolic	hyperbolic	ADJ
cana-5309	108	13	circles	circle	NOUN
cana-5309	108	14	was	be	AUX
cana-5309	108	15	evaluated	evaluate	VERB
cana-5309	108	16	as	as	ADP
cana-5309	108	17	one	one	NUM
cana-5309	108	18	method	method	NOUN
cana-5309	108	19	to	to	PART
cana-5309	108	20	analyze	analyze	VERB
cana-5309	108	21	structural	structural	ADJ
cana-5309	108	22	properties	property	NOUN
cana-5309	108	23	.	.	PUNCT
cana-5309	109	1	the	the	DET
cana-5309	109	2	hyperbolic	hyperbolic	ADJ
cana-5309	109	3	space	space	NOUN
cana-5309	109	4	shows	show	VERB
cana-5309	109	5	faster	fast	ADJ
cana-5309	109	6	growth	growth	NOUN
cana-5309	109	7	of	of	ADP
cana-5309	109	8	area	area	NOUN
cana-5309	109	9	compared	compare	VERB
cana-5309	109	10	to	to	ADP
cana-5309	109	11	euclidean	euclidean	ADJ
cana-5309	109	12	space	space	NOUN
cana-5309	109	13	when	when	SCONJ
cana-5309	109	14	both	both	DET
cana-5309	109	15	spaces	space	NOUN
cana-5309	109	16	experience	experience	VERB
cana-5309	109	17	a	a	DET
cana-5309	109	18	similar	similar	ADJ
cana-5309	109	19	increase	increase	NOUN
cana-5309	109	20	in	in	ADP
cana-5309	109	21	radius	radius	NOUN
cana-5309	109	22	(	(	PUNCT
cana-5309	109	23	figure	figure	NOUN
cana-5309	109	24	2	2	NUM
cana-5309	109	25	illustrates	illustrate	VERB
cana-5309	109	26	the	the	DET
cana-5309	109	27	results	result	NOUN
cana-5309	109	28	)	)	PUNCT
cana-5309	109	29	.	.	PUNCT
cana-5309	110	1	the	the	DET
cana-5309	110	2	non	non	ADJ
cana-5309	110	3	-	-	ADJ
cana-5309	110	4	linear	linear	ADJ
cana-5309	110	5	area	area	NOUN
cana-5309	110	6	growth	growth	NOUN
cana-5309	110	7	demonstrates	demonstrate	VERB
cana-5309	110	8	both	both	CCONJ
cana-5309	110	9	the	the	DET
cana-5309	110	10	fundamental	fundamental	ADJ
cana-5309	110	11	curvature	curvature	NOUN
cana-5309	110	12	principles	principle	NOUN
cana-5309	110	13	while	while	SCONJ
cana-5309	110	14	showing	show	VERB
cana-5309	110	15	that	that	SCONJ
cana-5309	110	16	basic	basic	ADJ
cana-5309	110	17	geometric	geometric	ADJ
cana-5309	110	18	structures	structure	NOUN
cana-5309	110	19	have	have	VERB
cana-5309	110	20	totally	totally	ADV
cana-5309	110	21	different	different	ADJ
cana-5309	110	22	behaviors	behavior	NOUN
cana-5309	110	23	.	.	PUNCT
cana-5309	111	1	boundary	boundary	ADJ
cana-5309	111	2	limitations	limitation	NOUN
cana-5309	111	3	enable	enable	VERB
cana-5309	111	4	hyperbolic	hyperbolic	ADJ
cana-5309	111	5	geometries	geometry	NOUN
cana-5309	111	6	to	to	PART
cana-5309	111	7	contain	contain	VERB
cana-5309	111	8	substantially	substantially	ADV
cana-5309	111	9	more	more	ADJ
cana-5309	111	10	space	space	NOUN
cana-5309	111	11	than	than	ADP
cana-5309	111	12	flat	flat	ADJ
cana-5309	111	13	geometries	geometry	NOUN
cana-5309	111	14	because	because	SCONJ
cana-5309	111	15	of	of	ADP
cana-5309	111	16	their	their	PRON
cana-5309	111	17	distinctive	distinctive	ADJ
cana-5309	111	18	circle	circle	NOUN
cana-5309	111	19	growth	growth	NOUN
cana-5309	111	20	patterns	pattern	NOUN
cana-5309	111	21	.	.	PUNCT
cana-5309	112	1	figure	figure	NOUN
cana-5309	112	2	2	2	NUM
cana-5309	112	3	:	:	PUNCT
cana-5309	112	4	area	area	NOUN
cana-5309	112	5	growth	growth	NOUN
cana-5309	112	6	comparison	comparison	NOUN
cana-5309	112	7	between	between	ADP
cana-5309	112	8	euclidean	euclidean	ADJ
cana-5309	112	9	and	and	CCONJ
cana-5309	112	10	hyperbolic	hyperbolic	ADJ
cana-5309	112	11	circles	circle	NOUN
cana-5309	112	12	the	the	DET
cana-5309	112	13	research	research	NOUN
cana-5309	112	14	also	also	ADV
cana-5309	112	15	focused	focus	VERB
cana-5309	112	16	on	on	ADP
cana-5309	112	17	developing	develop	VERB
cana-5309	112	18	tessellation	tessellation	NOUN
cana-5309	112	19	examples	example	NOUN
cana-5309	112	20	through	through	ADP
cana-5309	112	21	regular	regular	ADJ
cana-5309	112	22	polygons	polygon	NOUN
cana-5309	112	23	in	in	ADP
cana-5309	112	24	hyperbolic	hyperbolic	ADJ
cana-5309	112	25	space	space	NOUN
cana-5309	112	26	.	.	PUNCT
cana-5309	113	1	the	the	DET
cana-5309	113	2	analysis	analysis	NOUN
cana-5309	113	3	examined	examine	VERB
cana-5309	113	4	how	how	SCONJ
cana-5309	113	5	triangles	triangle	NOUN
cana-5309	113	6	and	and	CCONJ
cana-5309	113	7	pentagons	pentagon	NOUN
cana-5309	113	8	tessellate	tessellate	VERB
cana-5309	113	9	side	side	NOUN
cana-5309	113	10	by	by	ADP
cana-5309	113	11	side	side	NOUN
cana-5309	113	12	to	to	PART
cana-5309	113	13	bring	bring	VERB
cana-5309	113	14	out	out	ADP
cana-5309	113	15	angular	angular	ADJ
cana-5309	113	16	deficits	deficit	NOUN
cana-5309	113	17	that	that	PRON
cana-5309	113	18	create	create	VERB
cana-5309	113	19	unique	unique	ADJ
cana-5309	113	20	patterns	pattern	NOUN
cana-5309	113	21	that	that	PRON
cana-5309	113	22	can	can	AUX
cana-5309	113	23	not	not	PART
cana-5309	113	24	occur	occur	VERB
cana-5309	113	25	in	in	ADP
cana-5309	113	26	euclidean	euclidean	ADJ
cana-5309	113	27	geometry	geometry	NOUN
cana-5309	113	28	.	.	PUNCT
cana-5309	114	1	the	the	DET
cana-5309	114	2	hyperbolic	hyperbolic	ADJ
cana-5309	114	3	geometry	geometry	NOUN
cana-5309	114	4	showed	show	VERB
cana-5309	114	5	a	a	DET
cana-5309	114	6	greater	great	ADJ
cana-5309	114	7	number	number	NOUN
cana-5309	114	8	of	of	ADP
cana-5309	114	9	connecting	connect	VERB
cana-5309	114	10	polygons	polygon	NOUN
cana-5309	114	11	at	at	ADP
cana-5309	114	12	each	each	DET
cana-5309	114	13	singular	singular	ADJ
cana-5309	114	14	point	point	NOUN
cana-5309	114	15	than	than	SCONJ
cana-5309	114	16	occurs	occur	VERB
cana-5309	114	17	in	in	ADP
cana-5309	114	18	euclidean	euclidean	ADJ
cana-5309	114	19	tilings	tiling	NOUN
cana-5309	114	20	according	accord	VERB
cana-5309	114	21	to	to	ADP
cana-5309	114	22	table	table	NOUN
cana-5309	114	23	1	1	NUM
cana-5309	114	24	.	.	PUNCT
cana-5309	114	25	table	table	NOUN
cana-5309	114	26	1	1	NUM
cana-5309	114	27	:	:	PUNCT
cana-5309	114	28	number	number	NOUN
cana-5309	114	29	of	of	ADP
cana-5309	114	30	polygons	polygon	NOUN
cana-5309	114	31	at	at	ADP
cana-5309	114	32	a	a	DET
cana-5309	114	33	single	single	ADJ
cana-5309	114	34	vertex	vertex	NOUN
cana-5309	114	35	in	in	ADP
cana-5309	114	36	different	different	ADJ
cana-5309	114	37	geometries	geometry	NOUN
cana-5309	114	38	geometry	geometry	NOUN
cana-5309	114	39	type	type	NOUN
cana-5309	114	40	polygon	polygon	NOUN
cana-5309	114	41	type	type	NOUN
cana-5309	114	42	number	number	NOUN
cana-5309	114	43	of	of	ADP
cana-5309	114	44	polygons	polygon	NOUN
cana-5309	114	45	at	at	ADP
cana-5309	114	46	vertex	vertex	NOUN
cana-5309	114	47	euclidean	euclidean	NOUN
cana-5309	114	48	triangle	triangle	NOUN
cana-5309	114	49	6	6	NUM
cana-5309	114	50	hyperbolic	hyperbolic	ADJ
cana-5309	114	51	triangle	triangle	NOUN
cana-5309	114	52	7	7	NUM
cana-5309	114	53	or	or	CCONJ
cana-5309	114	54	more	more	ADJ
cana-5309	114	55	euclidean	euclidean	ADJ
cana-5309	114	56	pentagon	pentagon	PROPN
cana-5309	114	57	impossible	impossible	ADJ
cana-5309	114	58	hyperbolic	hyperbolic	ADJ
cana-5309	114	59	pentagon	pentagon	NOUN
cana-5309	114	60	4	4	NUM
cana-5309	114	61	or	or	CCONJ
cana-5309	114	62	5	5	NUM
cana-5309	114	63	3.14	3.14	NUM
cana-5309	114	64	12.57	12.57	NUM
cana-5309	114	65	28.27	28.27	NUM
cana-5309	114	66	3.88	3.88	NUM
cana-5309	114	67	20.31	20.31	NUM
cana-5309	114	68	59.52	59.52	NUM
cana-5309	114	69	0	0	NUM
cana-5309	114	70	10	10	NUM
cana-5309	114	71	20	20	NUM
cana-5309	114	72	30	30	NUM
cana-5309	114	73	40	40	NUM
cana-5309	114	74	50	50	NUM
cana-5309	114	75	60	60	NUM
cana-5309	114	76	70	70	NUM
cana-5309	114	77	1	1	NUM
cana-5309	114	78	2	2	NUM
cana-5309	114	79	3	3	NUM
cana-5309	114	80	area	area	NOUN
cana-5309	114	81	growth	growth	NOUN
cana-5309	114	82	comparison	comparison	NOUN
cana-5309	114	83	between	between	ADP
cana-5309	114	84	euclidean	euclidean	ADJ
cana-5309	114	85	and	and	CCONJ
cana-5309	114	86	hyperbolic	hyperbolic	ADJ
cana-5309	114	87	circles	circle	NOUN
cana-5309	114	88	euclidean	euclidean	ADJ
cana-5309	114	89	area	area	NOUN
cana-5309	114	90	(	(	PUNCT
cana-5309	114	91	πr²	πr²	NOUN
cana-5309	114	92	)	)	PUNCT
cana-5309	114	93	hyperbolic	hyperbolic	ADJ
cana-5309	114	94	area	area	NOUN
cana-5309	114	95	(	(	PUNCT
cana-5309	114	96	approx	approx	PROPN
cana-5309	114	97	.	.	PUNCT
cana-5309	114	98	)	)	PUNCT
cana-5309	115	1	communications	communication	NOUN
cana-5309	115	2	on	on	ADP
cana-5309	115	3	applied	apply	VERB
cana-5309	115	4	nonlinear	nonlinear	ADJ
cana-5309	115	5	analysis	analysis	NOUN
cana-5309	115	6	issn	issn	NOUN
cana-5309	115	7	:	:	PUNCT
cana-5309	115	8	1074	1074	NUM
cana-5309	115	9	-	-	PUNCT
cana-5309	115	10	133x	133x	NUM
cana-5309	115	11	vol	vol	NOUN
cana-5309	115	12	31	31	NUM
cana-5309	115	13	no	no	NOUN
cana-5309	115	14	.	.	PUNCT
cana-5309	116	1	4s	4s	NUM
cana-5309	116	2	(	(	PUNCT
cana-5309	116	3	2024	2024	NUM
cana-5309	116	4	)	)	PUNCT
cana-5309	117	1	680	680	NUM
cana-5309	117	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	117	3	the	the	DET
cana-5309	117	4	evaluation	evaluation	NOUN
cana-5309	117	5	of	of	ADP
cana-5309	117	6	large	large	ADJ
cana-5309	117	7	datasets	dataset	NOUN
cana-5309	117	8	through	through	ADP
cana-5309	117	9	hyperbolic	hyperbolic	ADJ
cana-5309	117	10	embedding	embed	VERB
cana-5309	117	11	required	require	VERB
cana-5309	117	12	simulated	simulated	ADJ
cana-5309	117	13	network	network	NOUN
cana-5309	117	14	graphs	graph	NOUN
cana-5309	117	15	to	to	PART
cana-5309	117	16	be	be	AUX
cana-5309	117	17	embedded	embed	VERB
cana-5309	117	18	into	into	ADP
cana-5309	117	19	euclidean	euclidean	ADJ
cana-5309	117	20	and	and	CCONJ
cana-5309	117	21	hyperbolic	hyperbolic	ADJ
cana-5309	117	22	spaces	space	NOUN
cana-5309	117	23	.	.	PUNCT
cana-5309	118	1	table	table	NOUN
cana-5309	118	2	2	2	NUM
cana-5309	118	3	presents	present	VERB
cana-5309	118	4	the	the	DET
cana-5309	118	5	summary	summary	NOUN
cana-5309	118	6	of	of	ADP
cana-5309	118	7	performance	performance	NOUN
cana-5309	118	8	metrics	metric	NOUN
cana-5309	118	9	regarding	regard	VERB
cana-5309	118	10	average	average	ADJ
cana-5309	118	11	distortion	distortion	NOUN
cana-5309	118	12	together	together	ADV
cana-5309	118	13	with	with	ADP
cana-5309	118	14	path	path	NOUN
cana-5309	118	15	preservation	preservation	NOUN
cana-5309	118	16	results	result	NOUN
cana-5309	118	17	(	(	PUNCT
cana-5309	118	18	see	see	VERB
cana-5309	118	19	table	table	NOUN
cana-5309	118	20	2	2	NUM
cana-5309	118	21	)	)	PUNCT
cana-5309	118	22	.	.	PUNCT
cana-5309	119	1	table	table	NOUN
cana-5309	119	2	2	2	NUM
cana-5309	119	3	:	:	PUNCT
cana-5309	119	4	performance	performance	NOUN
cana-5309	119	5	metrics	metric	NOUN
cana-5309	119	6	for	for	ADP
cana-5309	119	7	network	network	NOUN
cana-5309	119	8	embeddings	embedding	NOUN
cana-5309	119	9	in	in	ADP
cana-5309	119	10	euclidean	euclidean	NOUN
cana-5309	119	11	vs	vs	ADP
cana-5309	119	12	hyperbolic	hyperbolic	ADJ
cana-5309	119	13	space	space	NOUN
cana-5309	119	14	metric	metric	ADJ
cana-5309	119	15	euclidean	euclidean	NOUN
cana-5309	119	16	embedding	embed	VERB
cana-5309	119	17	hyperbolic	hyperbolic	ADJ
cana-5309	119	18	embedding	embed	VERB
cana-5309	119	19	average	average	ADJ
cana-5309	119	20	distortion	distortion	NOUN
cana-5309	119	21	0.42	0.42	NUM
cana-5309	119	22	0.15	0.15	NUM
cana-5309	119	23	shortest	short	ADJ
cana-5309	119	24	path	path	NOUN
cana-5309	119	25	preservation	preservation	NOUN
cana-5309	119	26	73	73	NUM
cana-5309	119	27	%	%	NOUN
cana-5309	119	28	89	89	NUM
cana-5309	119	29	%	%	NOUN
cana-5309	119	30	average	average	NOUN
cana-5309	119	31	embedding	embed	VERB
cana-5309	119	32	error	error	NOUN
cana-5309	119	33	0.39	0.39	NUM
cana-5309	119	34	0.21	0.21	NUM
cana-5309	119	35	the	the	DET
cana-5309	119	36	visual	visual	ADJ
cana-5309	119	37	representation	representation	NOUN
cana-5309	119	38	of	of	ADP
cana-5309	119	39	embedding	embed	VERB
cana-5309	119	40	’s	’	VERB
cana-5309	119	41	became	become	VERB
cana-5309	119	42	vital	vital	ADJ
cana-5309	119	43	to	to	PART
cana-5309	119	44	observe	observe	VERB
cana-5309	119	45	the	the	DET
cana-5309	119	46	spatial	spatial	ADJ
cana-5309	119	47	structures	structure	NOUN
cana-5309	119	48	within	within	ADP
cana-5309	119	49	the	the	DET
cana-5309	119	50	data	datum	NOUN
cana-5309	119	51	.	.	PUNCT
cana-5309	120	1	the	the	DET
cana-5309	120	2	networks	network	NOUN
cana-5309	120	3	in	in	ADP
cana-5309	120	4	euclidean	euclidean	ADJ
cana-5309	120	5	space	space	NOUN
cana-5309	120	6	experienced	experience	VERB
cana-5309	120	7	stretching	stretching	NOUN
cana-5309	120	8	while	while	SCONJ
cana-5309	120	9	needing	need	VERB
cana-5309	120	10	substantial	substantial	ADJ
cana-5309	120	11	amounts	amount	NOUN
cana-5309	120	12	of	of	ADP
cana-5309	120	13	space	space	NOUN
cana-5309	120	14	to	to	PART
cana-5309	120	15	preserve	preserve	VERB
cana-5309	120	16	their	their	PRON
cana-5309	120	17	network	network	NOUN
cana-5309	120	18	integrity	integrity	NOUN
cana-5309	120	19	.	.	PUNCT
cana-5309	121	1	the	the	DET
cana-5309	121	2	natural	natural	ADJ
cana-5309	121	3	structure	structure	NOUN
cana-5309	121	4	of	of	ADP
cana-5309	121	5	hyperbolic	hyperbolic	ADJ
cana-5309	121	6	embedding	embedding	NOUN
cana-5309	121	7	’s	’s	PART
cana-5309	121	8	could	could	AUX
cana-5309	121	9	compact	compact	VERB
cana-5309	121	10	networks	network	NOUN
cana-5309	121	11	effectively	effectively	ADV
cana-5309	121	12	since	since	SCONJ
cana-5309	121	13	they	they	PRON
cana-5309	121	14	preserved	preserve	VERB
cana-5309	121	15	proximity	proximity	NOUN
cana-5309	121	16	and	and	CCONJ
cana-5309	121	17	hierarchy	hierarchy	NOUN
cana-5309	121	18	seamlessly	seamlessly	ADV
cana-5309	121	19	.	.	PUNCT
cana-5309	122	1	the	the	DET
cana-5309	122	2	two	two	NUM
cana-5309	122	3	embedding	embed	VERB
cana-5309	122	4	’s	’s	PART
cana-5309	122	5	possess	possess	VERB
cana-5309	122	6	different	different	ADJ
cana-5309	122	7	structural	structural	ADJ
cana-5309	122	8	characteristics	characteristic	NOUN
cana-5309	122	9	which	which	PRON
cana-5309	122	10	show	show	VERB
cana-5309	122	11	up	up	ADP
cana-5309	122	12	in	in	ADP
cana-5309	122	13	the	the	DET
cana-5309	122	14	network	network	NOUN
cana-5309	122	15	graph	graph	NOUN
cana-5309	122	16	visualizations	visualization	NOUN
cana-5309	122	17	(	(	PUNCT
cana-5309	122	18	see	see	VERB
cana-5309	122	19	figure	figure	NOUN
cana-5309	122	20	3	3	NUM
cana-5309	122	21	)	)	PUNCT
cana-5309	122	22	.	.	PUNCT
cana-5309	123	1	figure	figure	VERB
cana-5309	123	2	3	3	NUM
cana-5309	123	3	:	:	PUNCT
cana-5309	123	4	network	network	NOUN
cana-5309	123	5	embedding	embed	VERB
cana-5309	123	6	comparison	comparison	NOUN
cana-5309	123	7	in	in	ADP
cana-5309	123	8	euclidean	euclidean	NOUN
cana-5309	123	9	vs	vs	ADP
cana-5309	123	10	hyperbolic	hyperbolic	ADJ
cana-5309	123	11	spaces	space	NOUN
cana-5309	123	12	the	the	DET
cana-5309	123	13	tests	test	NOUN
cana-5309	123	14	demonstrated	demonstrate	VERB
cana-5309	123	15	that	that	SCONJ
cana-5309	123	16	connections	connection	NOUN
cana-5309	123	17	between	between	ADP
cana-5309	123	18	nodes	node	NOUN
cana-5309	123	19	in	in	ADP
cana-5309	123	20	hyperbolic	hyperbolic	ADJ
cana-5309	123	21	geometry	geometry	NOUN
cana-5309	123	22	featured	feature	VERB
cana-5309	123	23	easier	easy	ADJ
cana-5309	123	24	routes	route	NOUN
cana-5309	123	25	which	which	PRON
cana-5309	123	26	required	require	VERB
cana-5309	123	27	less	less	ADJ
cana-5309	123	28	complex	complex	ADJ
cana-5309	123	29	paths	path	NOUN
cana-5309	123	30	than	than	ADP
cana-5309	123	31	euclidean	euclidean	ADJ
cana-5309	123	32	geometry	geometry	NOUN
cana-5309	123	33	would	would	AUX
cana-5309	123	34	have	have	AUX
cana-5309	123	35	needed	need	VERB
cana-5309	123	36	.	.	PUNCT
cana-5309	124	1	experimental	experimental	ADJ
cana-5309	124	2	results	result	NOUN
cana-5309	124	3	from	from	ADP
cana-5309	124	4	all	all	DET
cana-5309	124	5	investigations	investigation	NOUN
cana-5309	124	6	prove	prove	VERB
cana-5309	124	7	that	that	SCONJ
cana-5309	124	8	natural	natural	ADJ
cana-5309	124	9	hyperbolic	hyperbolic	ADJ
cana-5309	124	10	structures	structure	NOUN
cana-5309	124	11	offer	offer	VERB
cana-5309	124	12	superior	superior	ADJ
cana-5309	124	13	data	datum	NOUN
cana-5309	124	14	representation	representation	NOUN
cana-5309	124	15	capabilities	capability	NOUN
cana-5309	124	16	for	for	ADP
cana-5309	124	17	extensive	extensive	ADJ
cana-5309	124	18	hierarchical	hierarchical	ADJ
cana-5309	124	19	information	information	NOUN
cana-5309	124	20	systems	system	NOUN
cana-5309	124	21	.	.	PUNCT
cana-5309	125	1	5.2	5.2	NUM
cana-5309	125	2	7.8	7.8	NUM
cana-5309	125	3	6.4	6.4	NUM
cana-5309	125	4	3.1	3.1	NUM
cana-5309	125	5	4.5	4.5	NUM
cana-5309	125	6	3.8	3.8	NUM
cana-5309	125	7	0	0	NUM
cana-5309	125	8	2	2	NUM
cana-5309	125	9	4	4	NUM
cana-5309	125	10	6	6	NUM
cana-5309	125	11	8	8	NUM
cana-5309	125	12	10	10	NUM
cana-5309	125	13	a	a	DET
cana-5309	125	14	b	b	PROPN
cana-5309	125	15	b	b	PROPN
cana-5309	125	16	c	c	NOUN
cana-5309	125	17	c	c	NOUN
cana-5309	125	18	d	d	NOUN
cana-5309	125	19	network	network	NOUN
cana-5309	125	20	embedding	embed	VERB
cana-5309	125	21	comparison	comparison	NOUN
cana-5309	125	22	in	in	ADP
cana-5309	125	23	euclidean	euclidean	NOUN
cana-5309	125	24	vs	vs	ADP
cana-5309	125	25	hyperbolic	hyperbolic	ADJ
cana-5309	125	26	spaces	space	NOUN
cana-5309	125	27	euclidean	euclidean	ADJ
cana-5309	125	28	distance	distance	NOUN
cana-5309	125	29	(	(	PUNCT
cana-5309	125	30	units	unit	NOUN
cana-5309	125	31	)	)	PUNCT
cana-5309	125	32	hyperbolic	hyperbolic	ADJ
cana-5309	125	33	distance	distance	NOUN
cana-5309	125	34	(	(	PUNCT
cana-5309	125	35	units	unit	NOUN
cana-5309	125	36	)	)	PUNCT
cana-5309	125	37	communications	communication	NOUN
cana-5309	125	38	on	on	ADP
cana-5309	125	39	applied	apply	VERB
cana-5309	125	40	nonlinear	nonlinear	ADJ
cana-5309	125	41	analysis	analysis	NOUN
cana-5309	125	42	issn	issn	NOUN
cana-5309	125	43	:	:	PUNCT
cana-5309	125	44	1074	1074	NUM
cana-5309	125	45	-	-	PUNCT
cana-5309	125	46	133x	133x	NUM
cana-5309	125	47	vol	vol	NOUN
cana-5309	125	48	31	31	NUM
cana-5309	125	49	no	no	NOUN
cana-5309	125	50	.	.	PUNCT
cana-5309	126	1	4s	4s	NUM
cana-5309	126	2	(	(	PUNCT
cana-5309	126	3	2024	2024	NUM
cana-5309	126	4	)	)	PUNCT
cana-5309	126	5	681	681	NUM
cana-5309	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	126	7	test	test	NOUN
cana-5309	126	8	participants	participant	NOUN
cana-5309	126	9	needed	need	VERB
cana-5309	126	10	time	time	NOUN
cana-5309	126	11	to	to	PART
cana-5309	126	12	adjust	adjust	VERB
cana-5309	126	13	their	their	PRON
cana-5309	126	14	understanding	understanding	NOUN
cana-5309	126	15	of	of	ADP
cana-5309	126	16	hyperbolic	hyperbolic	ADJ
cana-5309	126	17	visualization	visualization	NOUN
cana-5309	126	18	rules	rule	NOUN
cana-5309	126	19	but	but	CCONJ
cana-5309	126	20	eventually	eventually	ADV
cana-5309	126	21	showed	show	VERB
cana-5309	126	22	quick	quick	ADJ
cana-5309	126	23	adoption	adoption	NOUN
cana-5309	126	24	of	of	ADP
cana-5309	126	25	these	these	DET
cana-5309	126	26	rules	rule	NOUN
cana-5309	126	27	during	during	ADP
cana-5309	126	28	their	their	PRON
cana-5309	126	29	interactions	interaction	NOUN
cana-5309	126	30	.	.	PUNCT
cana-5309	127	1	the	the	DET
cana-5309	127	2	general	general	ADJ
cana-5309	127	3	usefulness	usefulness	NOUN
cana-5309	127	4	of	of	ADP
cana-5309	127	5	hyperbolic	hyperbolic	ADJ
cana-5309	127	6	space	space	NOUN
cana-5309	127	7	visualizations	visualization	NOUN
cana-5309	127	8	becomes	become	VERB
cana-5309	127	9	possible	possible	ADJ
cana-5309	127	10	in	in	ADP
cana-5309	127	11	real	real	ADJ
cana-5309	127	12	applications	application	NOUN
cana-5309	127	13	involving	involve	VERB
cana-5309	127	14	virtual	virtual	ADJ
cana-5309	127	15	reality	reality	NOUN
cana-5309	127	16	environments	environment	NOUN
cana-5309	127	17	and	and	CCONJ
cana-5309	127	18	educational	educational	ADJ
cana-5309	127	19	software	software	NOUN
cana-5309	127	20	and	and	CCONJ
cana-5309	127	21	data	datum	NOUN
cana-5309	127	22	science	science	NOUN
cana-5309	127	23	tools	tool	NOUN
cana-5309	127	24	because	because	SCONJ
cana-5309	127	25	of	of	ADP
cana-5309	127	26	their	their	PRON
cana-5309	127	27	spatial	spatial	ADJ
cana-5309	127	28	efficiency	efficiency	NOUN
cana-5309	127	29	and	and	CCONJ
cana-5309	127	30	visual	visual	ADJ
cana-5309	127	31	hierarchy	hierarchy	NOUN
cana-5309	127	32	capabilities	capability	NOUN
cana-5309	127	33	.	.	PUNCT
cana-5309	128	1	the	the	DET
cana-5309	128	2	research	research	NOUN
cana-5309	128	3	verified	verify	VERB
cana-5309	128	4	theory	theory	NOUN
cana-5309	128	5	-	-	PUNCT
cana-5309	128	6	based	base	VERB
cana-5309	128	7	predictions	prediction	NOUN
cana-5309	128	8	in	in	ADP
cana-5309	128	9	addition	addition	NOUN
cana-5309	128	10	to	to	ADP
cana-5309	128	11	creating	create	VERB
cana-5309	128	12	practical	practical	ADJ
cana-5309	128	13	potential	potential	NOUN
cana-5309	128	14	for	for	ADP
cana-5309	128	15	hyperbolic	hyperbolic	ADJ
cana-5309	128	16	implementation	implementation	NOUN
cana-5309	128	17	across	across	ADP
cana-5309	128	18	modern	modern	ADJ
cana-5309	128	19	technology	technology	NOUN
cana-5309	128	20	fields	field	NOUN
cana-5309	128	21	.	.	PUNCT
cana-5309	129	1	the	the	DET
cana-5309	129	2	combination	combination	NOUN
cana-5309	129	3	of	of	ADP
cana-5309	129	4	visual	visual	ADJ
cana-5309	129	5	and	and	CCONJ
cana-5309	129	6	computational	computational	ADJ
cana-5309	129	7	and	and	CCONJ
cana-5309	129	8	comparative	comparative	ADJ
cana-5309	129	9	evidence	evidence	NOUN
cana-5309	129	10	demonstrates	demonstrate	VERB
cana-5309	129	11	that	that	SCONJ
cana-5309	129	12	hyperbolic	hyperbolic	ADJ
cana-5309	129	13	geometry	geometry	NOUN
cana-5309	129	14	serves	serve	VERB
cana-5309	129	15	as	as	ADP
cana-5309	129	16	an	an	DET
cana-5309	129	17	advanced	advanced	ADJ
cana-5309	129	18	tool	tool	NOUN
cana-5309	129	19	beyond	beyond	ADP
cana-5309	129	20	academic	academic	ADJ
cana-5309	129	21	theories	theory	NOUN
cana-5309	129	22	because	because	SCONJ
cana-5309	129	23	it	it	PRON
cana-5309	129	24	delivers	deliver	VERB
cana-5309	129	25	effective	effective	ADJ
cana-5309	129	26	solutions	solution	NOUN
cana-5309	129	27	for	for	ADP
cana-5309	129	28	current	current	ADJ
cana-5309	129	29	science	science	NOUN
cana-5309	129	30	and	and	CCONJ
cana-5309	129	31	engineering	engineering	NOUN
cana-5309	129	32	challenges	challenge	NOUN
cana-5309	129	33	.	.	PUNCT
cana-5309	130	1	v.	v.	ADP
cana-5309	130	2	conclusion	conclusion	NOUN
cana-5309	130	3	the	the	DET
cana-5309	130	4	alternative	alternative	ADJ
cana-5309	130	5	system	system	NOUN
cana-5309	130	6	of	of	ADP
cana-5309	130	7	hyperbolic	hyperbolic	ADJ
cana-5309	130	8	geometry	geometry	NOUN
cana-5309	130	9	substitutes	substitute	VERB
cana-5309	130	10	the	the	DET
cana-5309	130	11	traditional	traditional	ADJ
cana-5309	130	12	concepts	concept	NOUN
cana-5309	130	13	of	of	ADP
cana-5309	130	14	euclidean	euclidean	ADJ
cana-5309	130	15	intuitions	intuition	NOUN
cana-5309	130	16	to	to	PART
cana-5309	130	17	create	create	VERB
cana-5309	130	18	new	new	ADJ
cana-5309	130	19	frameworks	framework	NOUN
cana-5309	130	20	of	of	ADP
cana-5309	130	21	understanding	understand	VERB
cana-5309	130	22	geometric	geometric	ADJ
cana-5309	130	23	space	space	NOUN
cana-5309	130	24	and	and	CCONJ
cana-5309	130	25	spatial	spatial	ADJ
cana-5309	130	26	relationships	relationship	NOUN
cana-5309	130	27	.	.	PUNCT
cana-5309	131	1	theoretical	theoretical	ADJ
cana-5309	131	2	along	along	ADP
cana-5309	131	3	with	with	ADP
cana-5309	131	4	computational	computational	ADJ
cana-5309	131	5	studies	study	NOUN
cana-5309	131	6	brought	bring	VERB
cana-5309	131	7	forward	forward	ADV
cana-5309	131	8	diverse	diverse	ADJ
cana-5309	131	9	behavioral	behavioral	ADJ
cana-5309	131	10	patterns	pattern	NOUN
cana-5309	131	11	which	which	PRON
cana-5309	131	12	exist	exist	VERB
cana-5309	131	13	within	within	ADP
cana-5309	131	14	hyperbolic	hyperbolic	ADJ
cana-5309	131	15	spaces	space	NOUN
cana-5309	131	16	and	and	CCONJ
cana-5309	131	17	their	their	PRON
cana-5309	131	18	corresponding	corresponding	ADJ
cana-5309	131	19	mathematical	mathematical	ADJ
cana-5309	131	20	models	model	NOUN
cana-5309	131	21	.	.	PUNCT
cana-5309	132	1	modern	modern	ADJ
cana-5309	132	2	scientific	scientific	ADJ
cana-5309	132	3	interrogation	interrogation	NOUN
cana-5309	132	4	relies	rely	VERB
cana-5309	132	5	on	on	ADP
cana-5309	132	6	hyperbolic	hyperbolic	ADJ
cana-5309	132	7	geometry	geometry	NOUN
cana-5309	132	8	for	for	ADP
cana-5309	132	9	extensive	extensive	ADJ
cana-5309	132	10	physical	physical	ADJ
cana-5309	132	11	applications	application	NOUN
cana-5309	132	12	and	and	CCONJ
cana-5309	132	13	computing	computing	NOUN
cana-5309	132	14	applications	application	NOUN
cana-5309	132	15	together	together	ADV
cana-5309	132	16	with	with	ADP
cana-5309	132	17	artistic	artistic	ADJ
cana-5309	132	18	applications	application	NOUN
cana-5309	132	19	.	.	PUNCT
cana-5309	133	1	references	reference	NOUN
cana-5309	133	2	[	[	X
cana-5309	133	3	1	1	NUM
cana-5309	133	4	]	]	X
cana-5309	133	5	n.	n.	NOUN
cana-5309	133	6	abrosimov	abrosimov	PROPN
cana-5309	133	7	and	and	CCONJ
cana-5309	133	8	a.	a.	NOUN
cana-5309	133	9	mednykh	mednykh	NOUN
cana-5309	133	10	,	,	PUNCT
cana-5309	133	11	“	"	PUNCT
cana-5309	133	12	area	area	NOUN
cana-5309	133	13	and	and	CCONJ
cana-5309	133	14	volume	volume	NOUN
cana-5309	133	15	in	in	ADP
cana-5309	133	16	non	non	ADJ
cana-5309	133	17	-	-	ADJ
cana-5309	133	18	euclidean	euclidean	ADJ
cana-5309	133	19	geometry	geometry	NOUN
cana-5309	133	20	,	,	PUNCT
cana-5309	133	21	”	"	PUNCT
cana-5309	133	22	in	in	ADP
cana-5309	133	23	ems	ems	PROPN
cana-5309	133	24	press	press	NOUN
cana-5309	133	25	ebooks	ebook	NOUN
cana-5309	133	26	,	,	PUNCT
cana-5309	133	27	2019	2019	NUM
cana-5309	133	28	,	,	PUNCT
cana-5309	133	29	pp	pp	ADJ
cana-5309	133	30	.	.	PUNCT
cana-5309	134	1	151–189	151–189	NUM
cana-5309	134	2	.	.	X
cana-5309	135	1	doi	doi	NOUN
cana-5309	135	2	:	:	PUNCT
cana-5309	135	3	10.4171/196	10.4171/196	NUM
cana-5309	135	4	-	-	SYM
cana-5309	135	5	1/11	1/11	NUM
cana-5309	135	6	.	.	PUNCT
cana-5309	136	1	[	[	X
cana-5309	136	2	2	2	NUM
cana-5309	136	3	]	]	PUNCT
cana-5309	136	4	d.	d.	PROPN
cana-5309	136	5	ferrarello	ferrarello	PROPN
cana-5309	136	6	,	,	PUNCT
cana-5309	136	7	m.	m.	PROPN
cana-5309	136	8	f.	f.	PROPN
cana-5309	136	9	mammana	mammana	PROPN
cana-5309	136	10	,	,	PUNCT
cana-5309	136	11	and	and	CCONJ
cana-5309	136	12	e.	e.	PROPN
cana-5309	136	13	taranto	taranto	PROPN
cana-5309	136	14	,	,	PUNCT
cana-5309	136	15	“	"	PUNCT
cana-5309	136	16	non	non	ADJ
cana-5309	136	17	-	-	ADJ
cana-5309	136	18	euclidean	euclidean	ADJ
cana-5309	136	19	geometry	geometry	NOUN
cana-5309	136	20	with	with	ADP
cana-5309	136	21	art	art	NOUN
cana-5309	136	22	by	by	ADP
cana-5309	136	23	means	mean	NOUN
cana-5309	136	24	of	of	ADP
cana-5309	136	25	geogebra	geogebra	NOUN
cana-5309	136	26	,	,	PUNCT
cana-5309	136	27	”	"	PUNCT
cana-5309	136	28	international	international	ADJ
cana-5309	136	29	journal	journal	NOUN
cana-5309	136	30	for	for	ADP
cana-5309	136	31	technology	technology	NOUN
cana-5309	136	32	in	in	ADP
cana-5309	136	33	mathematics	mathematics	PROPN
cana-5309	136	34	education	education	NOUN
cana-5309	136	35	,	,	PUNCT
cana-5309	136	36	vol	vol	NOUN
cana-5309	136	37	.	.	PROPN
cana-5309	136	38	26	26	NUM
cana-5309	136	39	,	,	PUNCT
cana-5309	136	40	no	no	INTJ
cana-5309	136	41	.	.	NOUN
cana-5309	136	42	3	3	NUM
cana-5309	136	43	,	,	PUNCT
cana-5309	136	44	pp	pp	ADJ
cana-5309	136	45	.	.	PUNCT
cana-5309	137	1	113–119	113–119	NUM
cana-5309	137	2	,	,	PUNCT
cana-5309	137	3	sep	sep	PROPN
cana-5309	137	4	.	.	PROPN
cana-5309	137	5	2019	2019	NUM
cana-5309	137	6	,	,	PUNCT
cana-5309	137	7	doi	doi	NOUN
cana-5309	137	8	:	:	PUNCT
cana-5309	137	9	10.1564	10.1564	NUM
cana-5309	137	10	/	/	SYM
cana-5309	137	11	tme_v26.3.02	tme_v26.3.02	NOUN
cana-5309	137	12	.	.	PUNCT
cana-5309	138	1	[	[	X
cana-5309	138	2	3	3	NUM
cana-5309	138	3	]	]	X
cana-5309	138	4	i.	i.	PROPN
cana-5309	138	5	stevenson	stevenson	PROPN
cana-5309	138	6	,	,	PUNCT
cana-5309	138	7	“	"	PUNCT
cana-5309	138	8	hyperbolic	hyperbolic	ADJ
cana-5309	138	9	space	space	NOUN
cana-5309	138	10	:	:	PUNCT
cana-5309	138	11	designing	design	VERB
cana-5309	138	12	a	a	DET
cana-5309	138	13	computational	computational	ADJ
cana-5309	138	14	context	context	NOUN
cana-5309	138	15	for	for	ADP
cana-5309	138	16	learning	learn	VERB
cana-5309	138	17	non	non	ADJ
cana-5309	138	18	-	-	ADJ
cana-5309	138	19	euclidean	euclidean	ADJ
cana-5309	138	20	geometry	geometry	NOUN
cana-5309	138	21	,	,	PUNCT
cana-5309	138	22	”	"	PUNCT
cana-5309	138	23	international	international	ADJ
cana-5309	138	24	journal	journal	NOUN
cana-5309	138	25	of	of	ADP
cana-5309	138	26	computers	computer	NOUN
cana-5309	138	27	for	for	ADP
cana-5309	138	28	mathematical	mathematical	ADJ
cana-5309	138	29	learning	learning	NOUN
cana-5309	138	30	,	,	PUNCT
cana-5309	138	31	vol	vol	NOUN
cana-5309	138	32	.	.	PROPN
cana-5309	138	33	5	5	NUM
cana-5309	138	34	,	,	PUNCT
cana-5309	138	35	no	no	INTJ
cana-5309	138	36	.	.	NOUN
cana-5309	138	37	2	2	NUM
cana-5309	138	38	,	,	PUNCT
cana-5309	138	39	pp	pp	ADJ
cana-5309	138	40	.	.	PUNCT
cana-5309	139	1	143–167	143–167	NUM
cana-5309	139	2	,	,	PUNCT
cana-5309	139	3	jan	jan	PROPN
cana-5309	139	4	.	.	PROPN
cana-5309	139	5	2000	2000	NUM
cana-5309	139	6	,	,	PUNCT
cana-5309	139	7	doi	doi	NOUN
cana-5309	139	8	:	:	PUNCT
cana-5309	139	9	10.1023	10.1023	NUM
cana-5309	139	10	/	/	SYM
cana-5309	139	11	a:1009802004645	a:1009802004645	NOUN
cana-5309	139	12	.	.	PUNCT
cana-5309	140	1	[	[	X
cana-5309	140	2	4	4	X
cana-5309	140	3	]	]	X
cana-5309	140	4	n.	n.	NOUN
cana-5309	140	5	a’campo	a’campo	PROPN
cana-5309	140	6	and	and	CCONJ
cana-5309	140	7	a.	a.	NOUN
cana-5309	140	8	papadopoulos	papadopoulos	PROPN
cana-5309	140	9	,	,	PUNCT
cana-5309	140	10	“	"	PUNCT
cana-5309	140	11	on	on	ADP
cana-5309	140	12	klein	klein	PROPN
cana-5309	140	13	’s	’s	PART
cana-5309	140	14	so	so	ADV
cana-5309	140	15	-	-	PUNCT
cana-5309	140	16	called	call	VERB
cana-5309	140	17	non	non	ADJ
cana-5309	140	18	-	-	ADJ
cana-5309	140	19	euclidean	euclidean	ADJ
cana-5309	140	20	geometry	geometry	NOUN
cana-5309	140	21	,	,	PUNCT
cana-5309	140	22	”	"	PUNCT
cana-5309	140	23	in	in	ADP
cana-5309	140	24	ems	ems	PROPN
cana-5309	140	25	press	press	NOUN
cana-5309	140	26	ebooks	ebook	NOUN
cana-5309	140	27	,	,	PUNCT
cana-5309	140	28	2015	2015	NUM
cana-5309	140	29	,	,	PUNCT
cana-5309	140	30	pp	pp	ADJ
cana-5309	140	31	.	.	PUNCT
cana-5309	141	1	91–136	91–136	NUM
cana-5309	141	2	.	.	PUNCT
cana-5309	142	1	doi	doi	NOUN
cana-5309	142	2	:	:	PUNCT
cana-5309	142	3	10.4171/148	10.4171/148	NUM
cana-5309	142	4	-	-	SYM
cana-5309	142	5	1/5	1/5	NUM
cana-5309	142	6	.	.	PUNCT
cana-5309	143	1	[	[	X
cana-5309	143	2	5	5	X
cana-5309	143	3	]	]	PUNCT
cana-5309	143	4	j.	j.	PROPN
cana-5309	143	5	miller	miller	PROPN
cana-5309	143	6	,	,	PUNCT
cana-5309	143	7	d.	d.	PROPN
cana-5309	143	8	bhatia	bhatia	PROPN
cana-5309	143	9	,	,	PUNCT
cana-5309	143	10	and	and	CCONJ
cana-5309	143	11	s.	s.	PROPN
cana-5309	143	12	kobourov	kobourov	PROPN
cana-5309	143	13	,	,	PUNCT
cana-5309	143	14	“	"	PUNCT
cana-5309	143	15	state	state	NOUN
cana-5309	143	16	of	of	ADP
cana-5309	143	17	the	the	DET
cana-5309	143	18	art	art	NOUN
cana-5309	143	19	of	of	ADP
cana-5309	143	20	graph	graph	NOUN
cana-5309	143	21	visualization	visualization	NOUN
cana-5309	143	22	in	in	ADP
cana-5309	143	23	non‐	non‐	ADJ
cana-5309	143	24	euclidean	euclidean	ADJ
cana-5309	143	25	spaces	space	NOUN
cana-5309	143	26	,	,	PUNCT
cana-5309	143	27	”	"	PUNCT
cana-5309	143	28	computer	computer	NOUN
cana-5309	143	29	graphics	graphic	NOUN
cana-5309	143	30	forum	forum	PROPN
cana-5309	143	31	,	,	PUNCT
cana-5309	143	32	vol	vol	NOUN
cana-5309	143	33	.	.	PROPN
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cana-5309	143	35	,	,	PUNCT
cana-5309	143	36	no	no	INTJ
cana-5309	143	37	.	.	NOUN
cana-5309	143	38	3	3	NUM
cana-5309	143	39	,	,	PUNCT
cana-5309	143	40	jun	jun	PROPN
cana-5309	143	41	.	.	PROPN
cana-5309	143	42	2024	2024	NUM
cana-5309	143	43	,	,	PUNCT
cana-5309	143	44	doi	doi	NOUN
cana-5309	143	45	:	:	PUNCT
cana-5309	143	46	10.1111	10.1111	NUM
cana-5309	143	47	/	/	SYM
cana-5309	143	48	cgf.15113	cgf.15113	PROPN
cana-5309	143	49	.	.	PUNCT
cana-5309	144	1	[	[	X
cana-5309	144	2	6	6	NUM
cana-5309	144	3	]	]	PUNCT
cana-5309	144	4	c.	c.	PROPN
cana-5309	144	5	saxena	saxena	PROPN
cana-5309	144	6	,	,	PUNCT
cana-5309	144	7	t.	t.	PROPN
cana-5309	144	8	liu	liu	PROPN
cana-5309	144	9	,	,	PUNCT
cana-5309	144	10	and	and	CCONJ
cana-5309	144	11	i.	i.	PROPN
cana-5309	144	12	king	king	PROPN
cana-5309	144	13	,	,	PUNCT
cana-5309	144	14	“	"	PUNCT
cana-5309	144	15	a	a	DET
cana-5309	144	16	survey	survey	NOUN
cana-5309	144	17	of	of	ADP
cana-5309	144	18	graph	graph	NOUN
cana-5309	144	19	curvature	curvature	NOUN
cana-5309	144	20	and	and	CCONJ
cana-5309	144	21	embedding	embed	VERB
cana-5309	144	22	in	in	ADP
cana-5309	144	23	noneuclidean	noneuclidean	ADJ
cana-5309	144	24	spaces	space	NOUN
cana-5309	144	25	,	,	PUNCT
cana-5309	144	26	”	"	PUNCT
cana-5309	144	27	in	in	ADP
cana-5309	144	28	lecture	lecture	NOUN
cana-5309	144	29	notes	note	NOUN
cana-5309	144	30	in	in	ADP
cana-5309	144	31	computer	computer	NOUN
cana-5309	144	32	science	science	NOUN
cana-5309	144	33	,	,	PUNCT
cana-5309	144	34	2020	2020	NUM
cana-5309	144	35	,	,	PUNCT
cana-5309	144	36	pp	pp	ADP
cana-5309	144	37	.	.	PUNCT
cana-5309	145	1	127–139	127–139	NUM
cana-5309	145	2	.	.	PUNCT
cana-5309	146	1	doi	doi	NOUN
cana-5309	146	2	:	:	PUNCT
cana-5309	146	3	10.1007/978	10.1007/978	NUM
cana-5309	146	4	-	-	SYM
cana-5309	146	5	3	3	NUM
cana-5309	146	6	-	-	PUNCT
cana-5309	146	7	030	030	NUM
cana-5309	146	8	-	-	PUNCT
cana-5309	146	9	63833	63833	NUM
cana-5309	146	10	-	-	SYM
cana-5309	146	11	7_11	7_11	NUM
cana-5309	146	12	.	.	PUNCT
cana-5309	147	1	[	[	X
cana-5309	147	2	7	7	X
cana-5309	147	3	]	]	X
cana-5309	147	4	l.	l.	PROPN
cana-5309	147	5	moreno	moreno	PROPN
cana-5309	147	6	-	-	PUNCT
cana-5309	147	7	armella	armella	PROPN
cana-5309	147	8	,	,	PUNCT
cana-5309	147	9	c.	c.	PROPN
cana-5309	147	10	brady	brady	PROPN
cana-5309	147	11	,	,	PUNCT
cana-5309	147	12	and	and	CCONJ
cana-5309	147	13	r.	r.	PROPN
cana-5309	147	14	elizondo	elizondo	PROPN
cana-5309	147	15	-	-	PUNCT
cana-5309	147	16	ramirez	ramirez	PROPN
cana-5309	147	17	,	,	PUNCT
cana-5309	147	18	“	"	PUNCT
cana-5309	147	19	dynamic	dynamic	ADJ
cana-5309	147	20	hyperbolic	hyperbolic	ADJ
cana-5309	147	21	geometry	geometry	NOUN
cana-5309	147	22	:	:	PUNCT
cana-5309	147	23	building	building	NOUN
cana-5309	147	24	intuition	intuition	NOUN
cana-5309	147	25	and	and	CCONJ
cana-5309	147	26	understanding	understanding	NOUN
cana-5309	147	27	mediated	mediate	VERB
cana-5309	147	28	by	by	ADP
cana-5309	147	29	a	a	DET
cana-5309	147	30	euclidean	euclidean	ADJ
cana-5309	147	31	model	model	NOUN
cana-5309	147	32	,	,	PUNCT
cana-5309	147	33	”	"	PUNCT
cana-5309	147	34	international	international	ADJ
cana-5309	147	35	journal	journal	NOUN
cana-5309	147	36	of	of	ADP
cana-5309	147	37	mathematical	mathematical	ADJ
cana-5309	147	38	education	education	NOUN
cana-5309	147	39	in	in	ADP
cana-5309	147	40	science	science	NOUN
cana-5309	147	41	and	and	CCONJ
cana-5309	147	42	technology	technology	NOUN
cana-5309	147	43	,	,	PUNCT
cana-5309	147	44	vol	vol	NOUN
cana-5309	147	45	.	.	PROPN
cana-5309	147	46	49	49	NUM
cana-5309	147	47	,	,	PUNCT
cana-5309	147	48	no	no	INTJ
cana-5309	147	49	.	.	NOUN
cana-5309	147	50	4	4	NUM
cana-5309	147	51	,	,	PUNCT
cana-5309	147	52	pp	pp	ADJ
cana-5309	147	53	.	.	PUNCT
cana-5309	148	1	594–612	594–612	NUM
cana-5309	148	2	,	,	PUNCT
cana-5309	148	3	jan	jan	PROPN
cana-5309	148	4	.	.	PROPN
cana-5309	148	5	2018	2018	NUM
cana-5309	148	6	,	,	PUNCT
cana-5309	148	7	doi	doi	NOUN
cana-5309	148	8	:	:	PUNCT
cana-5309	148	9	10.1080/0020739x.2017.1418915	10.1080/0020739x.2017.1418915	NOUN
cana-5309	148	10	.	.	PUNCT
cana-5309	149	1	communications	communication	NOUN
cana-5309	149	2	on	on	ADP
cana-5309	149	3	applied	apply	VERB
cana-5309	149	4	nonlinear	nonlinear	ADJ
cana-5309	149	5	analysis	analysis	NOUN
cana-5309	149	6	issn	issn	NOUN
cana-5309	149	7	:	:	PUNCT
cana-5309	149	8	1074	1074	NUM
cana-5309	149	9	-	-	PUNCT
cana-5309	149	10	133x	133x	NUM
cana-5309	149	11	vol	vol	NOUN
cana-5309	149	12	31	31	NUM
cana-5309	149	13	no	no	NOUN
cana-5309	149	14	.	.	PUNCT
cana-5309	150	1	4s	4s	NUM
cana-5309	150	2	(	(	PUNCT
cana-5309	150	3	2024	2024	NUM
cana-5309	150	4	)	)	PUNCT
cana-5309	150	5	682	682	NUM
cana-5309	150	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5309	151	1	[	[	X
cana-5309	151	2	8	8	X
cana-5309	151	3	]	]	PUNCT
cana-5309	151	4	s.	s.	PROPN
cana-5309	151	5	d.	d.	PROPN
cana-5309	151	6	casey	casey	PROPN
cana-5309	151	7	,	,	PUNCT
cana-5309	151	8	“	"	PUNCT
cana-5309	151	9	harmonic	harmonic	VERB
cana-5309	151	10	analysis	analysis	NOUN
cana-5309	151	11	in	in	ADP
cana-5309	151	12	non	non	ADJ
cana-5309	151	13	-	-	ADJ
cana-5309	151	14	euclidean	euclidean	ADJ
cana-5309	151	15	spaces	space	NOUN
cana-5309	151	16	:	:	PUNCT
cana-5309	151	17	theory	theory	NOUN
cana-5309	151	18	and	and	CCONJ
cana-5309	151	19	application	application	NOUN
cana-5309	151	20	,	,	PUNCT
cana-5309	151	21	”	"	PUNCT
cana-5309	151	22	in	in	ADP
cana-5309	151	23	applied	applied	ADJ
cana-5309	151	24	and	and	CCONJ
cana-5309	151	25	numerical	numerical	ADJ
cana-5309	151	26	harmonic	harmonic	ADJ
cana-5309	151	27	analysis	analysis	NOUN
cana-5309	151	28	,	,	PUNCT
cana-5309	151	29	2017	2017	NUM
cana-5309	151	30	,	,	PUNCT
cana-5309	151	31	pp	pp	ADJ
cana-5309	151	32	.	.	PUNCT
cana-5309	152	1	565–601	565–601	NUM
cana-5309	152	2	.	.	PUNCT
cana-5309	152	3	doi	doi	NOUN
cana-5309	152	4	:	:	PUNCT
cana-5309	152	5	10.1007/978	10.1007/978	NUM
cana-5309	152	6	-	-	SYM
cana-5309	152	7	3319	3319	NUM
cana-5309	152	8	-	-	PUNCT
cana-5309	152	9	55556	55556	NUM
cana-5309	152	10	-	-	PUNCT
cana-5309	152	11	0_6	0_6	NUM
cana-5309	152	12	.	.	PUNCT
cana-5309	153	1	[	[	X
cana-5309	153	2	9	9	NUM
cana-5309	153	3	]	]	PUNCT
cana-5309	153	4	t.	t.	PROPN
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cana-5309	153	6	,	,	PUNCT
cana-5309	153	7	v.	v.	PROPN
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cana-5309	153	11	and	and	CCONJ
cana-5309	153	12	l.	l.	PROPN
cana-5309	153	13	velho	velho	PROPN
cana-5309	153	14	,	,	PUNCT
cana-5309	153	15	“	"	PUNCT
cana-5309	153	16	global	global	ADJ
cana-5309	153	17	illumination	illumination	NOUN
cana-5309	153	18	of	of	ADP
cana-5309	153	19	non	non	ADJ
cana-5309	153	20	-	-	ADJ
cana-5309	153	21	euclidean	euclidean	ADJ
cana-5309	153	22	spaces	space	NOUN
cana-5309	153	23	,	,	PUNCT
cana-5309	153	24	”	"	PUNCT
cana-5309	153	25	computers	computer	NOUN
cana-5309	153	26	&	&	CCONJ
cana-5309	153	27	graphics	graphic	NOUN
cana-5309	153	28	,	,	PUNCT
cana-5309	153	29	vol	vol	NOUN
cana-5309	153	30	.	.	PROPN
cana-5309	153	31	93	93	NUM
cana-5309	153	32	,	,	PUNCT
cana-5309	153	33	pp	pp	ADJ
cana-5309	153	34	.	.	PUNCT
cana-5309	154	1	61–70	61–70	NOUN
cana-5309	154	2	,	,	PUNCT
cana-5309	154	3	oct	oct	PROPN
cana-5309	154	4	.	.	PROPN
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cana-5309	154	6	,	,	PUNCT
cana-5309	154	7	doi	doi	NOUN
cana-5309	154	8	:	:	PUNCT
cana-5309	154	9	10.1016	10.1016	NUM
cana-5309	154	10	/	/	SYM
cana-5309	154	11	j.cag.2020.09.014	j.cag.2020.09.014	PROPN
cana-5309	154	12	.	.	PUNCT
cana-5309	155	1	[	[	X
cana-5309	155	2	10	10	NUM
cana-5309	155	3	]	]	X
cana-5309	155	4	s.	s.	PROPN
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cana-5309	155	9	,	,	PUNCT
cana-5309	155	10	“	"	PUNCT
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cana-5309	155	12	euclid	euclid	PROPN
cana-5309	155	13	:	:	PUNCT
cana-5309	155	14	an	an	DET
cana-5309	155	15	illustrated	illustrated	ADJ
cana-5309	155	16	guide	guide	NOUN
cana-5309	155	17	to	to	ADP
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cana-5309	155	19	machine	machine	NOUN
cana-5309	155	20	learning	learn	VERB
cana-5309	155	21	with	with	ADP
cana-5309	155	22	geometric	geometric	ADJ
cana-5309	155	23	,	,	PUNCT
cana-5309	155	24	topological	topological	ADJ
cana-5309	155	25	,	,	PUNCT
cana-5309	155	26	and	and	CCONJ
cana-5309	155	27	algebraic	algebraic	ADJ
cana-5309	155	28	structures	structure	NOUN
cana-5309	155	29	,	,	PUNCT
cana-5309	155	30	”	"	PUNCT
cana-5309	155	31	arxiv	arxiv	PROPN
cana-5309	155	32	(	(	PUNCT
cana-5309	155	33	cornell	cornell	PROPN
cana-5309	155	34	university	university	PROPN
cana-5309	155	35	)	)	PUNCT
cana-5309	155	36	,	,	PUNCT
cana-5309	155	37	jul	jul	PROPN
cana-5309	155	38	.	.	PROPN
cana-5309	155	39	2024	2024	NUM
cana-5309	155	40	,	,	PUNCT
cana-5309	155	41	doi	doi	NOUN
cana-5309	155	42	:	:	PUNCT
cana-5309	155	43	10.48550	10.48550	NUM
cana-5309	155	44	/	/	SYM
cana-5309	155	45	arxiv.2407.09468	arxiv.2407.09468	NOUN
cana-5309	155	46	.	.	PUNCT
cana-5309	156	1	[	[	X
cana-5309	156	2	11	11	NUM
cana-5309	156	3	]	]	PUNCT
cana-5309	156	4	j.	j.	PROPN
cana-5309	156	5	batle	batle	PROPN
cana-5309	156	6	,	,	PUNCT
cana-5309	156	7	a.	a.	NOUN
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cana-5309	156	9	,	,	PUNCT
cana-5309	156	10	m.	m.	NOUN
cana-5309	156	11	abdel	abdel	PROPN
cana-5309	156	12	-	-	PUNCT
cana-5309	156	13	aty	aty	PROPN
cana-5309	156	14	,	,	PUNCT
cana-5309	156	15	and	and	CCONJ
cana-5309	156	16	s.	s.	PROPN
cana-5309	156	17	abdalla	abdalla	PROPN
cana-5309	156	18	,	,	PUNCT
cana-5309	156	19	“	"	PUNCT
cana-5309	156	20	generalized	generalized	ADJ
cana-5309	156	21	thomson	thomson	NOUN
cana-5309	156	22	problem	problem	NOUN
cana-5309	156	23	in	in	ADP
cana-5309	156	24	arbitrary	arbitrary	ADJ
cana-5309	156	25	dimensions	dimension	NOUN
cana-5309	156	26	and	and	CCONJ
cana-5309	156	27	non	non	ADJ
cana-5309	156	28	-	-	ADJ
cana-5309	156	29	euclidean	euclidean	ADJ
cana-5309	156	30	geometries	geometry	NOUN
cana-5309	156	31	,	,	PUNCT
cana-5309	156	32	”	"	PUNCT
cana-5309	156	33	physica	physica	VERB
cana-5309	156	34	a	a	DET
cana-5309	156	35	statistical	statistical	ADJ
cana-5309	156	36	mechanics	mechanic	NOUN
cana-5309	156	37	and	and	CCONJ
cana-5309	156	38	its	its	PRON
cana-5309	156	39	applications	application	NOUN
cana-5309	156	40	,	,	PUNCT
cana-5309	156	41	vol	vol	NOUN
cana-5309	156	42	.	.	PROPN
cana-5309	157	1	451	451	NUM
cana-5309	157	2	,	,	PUNCT
cana-5309	157	3	pp	pp	ADJ
cana-5309	157	4	.	.	PUNCT
cana-5309	158	1	237–250	237–250	NUM
cana-5309	158	2	,	,	PUNCT
cana-5309	158	3	feb	feb	PROPN
cana-5309	158	4	.	.	PROPN
cana-5309	158	5	2016	2016	NUM
cana-5309	158	6	,	,	PUNCT
cana-5309	158	7	doi	doi	NOUN
cana-5309	158	8	:	:	PUNCT
cana-5309	158	9	10.1016	10.1016	NUM
cana-5309	158	10	/	/	SYM
cana-5309	158	11	j.physa.2016.01.069	j.physa.2016.01.069	PROPN
cana-5309	158	12	.	.	PUNCT
cana-5309	159	1	[	[	X
cana-5309	159	2	12	12	NUM
cana-5309	159	3	]	]	X
cana-5309	159	4	s.	s.	PROPN
cana-5309	159	5	huckemann	huckemann	PROPN
cana-5309	159	6	and	and	CCONJ
cana-5309	159	7	b.	b.	PROPN
cana-5309	159	8	eltzner	eltzner	PROPN
cana-5309	159	9	,	,	PUNCT
cana-5309	159	10	“	"	PUNCT
cana-5309	159	11	statistical	statistical	ADJ
cana-5309	159	12	methods	method	NOUN
cana-5309	159	13	generalizing	generalize	VERB
cana-5309	159	14	principal	principal	ADJ
cana-5309	159	15	component	component	NOUN
cana-5309	159	16	analysis	analysis	NOUN
cana-5309	159	17	to	to	ADP
cana-5309	159	18	non	non	ADJ
cana-5309	159	19	-	-	ADJ
cana-5309	159	20	euclidean	euclidean	ADJ
cana-5309	159	21	spaces	space	NOUN
cana-5309	159	22	,	,	PUNCT
cana-5309	159	23	”	"	PUNCT
cana-5309	159	24	in	in	ADP
cana-5309	159	25	springer	springer	NOUN
cana-5309	159	26	ebooks	ebook	NOUN
cana-5309	159	27	,	,	PUNCT
cana-5309	159	28	2020	2020	NUM
cana-5309	159	29	,	,	PUNCT
cana-5309	159	30	pp	pp	ADV
cana-5309	159	31	.	.	PUNCT
cana-5309	160	1	317–338	317–338	NUM
cana-5309	160	2	.	.	PUNCT
cana-5309	160	3	doi	doi	NOUN
cana-5309	160	4	:	:	PUNCT
cana-5309	160	5	10.1007/978	10.1007/978	NUM
cana-5309	160	6	-	-	SYM
cana-5309	160	7	3	3	NUM
cana-5309	160	8	-	-	PUNCT
cana-5309	160	9	030	030	NUM
cana-5309	160	10	-	-	PUNCT
cana-5309	160	11	31351	31351	NUM
cana-5309	160	12	-	-	PUNCT
cana-5309	160	13	7_10	7_10	NOUN
cana-5309	160	14	.	.	PUNCT
cana-5309	161	1	[	[	X
cana-5309	161	2	13	13	NUM
cana-5309	161	3	]	]	PUNCT
cana-5309	161	4	w.	w.	PROPN
cana-5309	161	5	peng	peng	PROPN
cana-5309	161	6	,	,	PUNCT
cana-5309	161	7	t.	t.	PROPN
cana-5309	161	8	varanka	varanka	PROPN
cana-5309	161	9	,	,	PUNCT
cana-5309	161	10	a.	a.	PROPN
cana-5309	161	11	mostafa	mostafa	PROPN
cana-5309	161	12	,	,	PUNCT
cana-5309	161	13	h.	h.	PROPN
cana-5309	161	14	shi	shi	PROPN
cana-5309	161	15	,	,	PUNCT
cana-5309	161	16	and	and	CCONJ
cana-5309	161	17	g.	g.	PROPN
cana-5309	161	18	zhao	zhao	PROPN
cana-5309	161	19	,	,	PUNCT
cana-5309	161	20	“	"	PUNCT
cana-5309	161	21	hyperbolic	hyperbolic	ADJ
cana-5309	161	22	deep	deep	ADJ
cana-5309	161	23	neural	neural	ADJ
cana-5309	161	24	networks	network	NOUN
cana-5309	161	25	:	:	PUNCT
cana-5309	161	26	a	a	DET
cana-5309	161	27	survey	survey	NOUN
cana-5309	161	28	,	,	PUNCT
cana-5309	161	29	”	"	PUNCT
cana-5309	161	30	ieee	ieee	NOUN
cana-5309	161	31	transactions	transaction	NOUN
cana-5309	161	32	on	on	ADP
cana-5309	161	33	pattern	pattern	NOUN
cana-5309	161	34	analysis	analysis	NOUN
cana-5309	161	35	and	and	CCONJ
cana-5309	161	36	machine	machine	NOUN
cana-5309	161	37	intelligence	intelligence	NOUN
cana-5309	161	38	,	,	PUNCT
cana-5309	161	39	vol	vol	NOUN
cana-5309	161	40	.	.	PROPN
cana-5309	161	41	44	44	NUM
cana-5309	161	42	,	,	PUNCT
cana-5309	161	43	no	no	INTJ
cana-5309	161	44	.	.	NOUN
cana-5309	161	45	12	12	NUM
cana-5309	161	46	,	,	PUNCT
cana-5309	161	47	pp	pp	ADJ
cana-5309	161	48	.	.	PUNCT
cana-5309	162	1	10023–10044	10023–10044	NUM
cana-5309	162	2	,	,	PUNCT
cana-5309	162	3	dec	dec	PROPN
cana-5309	162	4	.	.	PROPN
cana-5309	162	5	2021	2021	NUM
cana-5309	162	6	,	,	PUNCT
cana-5309	162	7	doi	doi	NOUN
cana-5309	162	8	:	:	PUNCT
cana-5309	162	9	10.1109	10.1109	NUM
cana-5309	162	10	/	/	SYM
cana-5309	162	11	tpami.2021.3136921	tpami.2021.3136921	PROPN
cana-5309	162	12	.	.	PUNCT
cana-5309	163	1	[	[	X
cana-5309	163	2	14	14	NUM
cana-5309	163	3	]	]	X
cana-5309	163	4	y.	y.	PROPN
cana-5309	163	5	l.	l.	PROPN
cana-5309	163	6	sukestiyarno	sukestiyarno	PROPN
cana-5309	163	7	,	,	PUNCT
cana-5309	163	8	k.	k.	PROPN
cana-5309	163	9	u.	u.	PROPN
cana-5309	163	10	z.	z.	PROPN
cana-5309	163	11	nugroho	nugroho	PROPN
cana-5309	163	12	,	,	PUNCT
cana-5309	163	13	s.	s.	PROPN
cana-5309	163	14	sugiman	sugiman	NOUN
cana-5309	163	15	,	,	PUNCT
cana-5309	163	16	and	and	CCONJ
cana-5309	163	17	b.	b.	PROPN
cana-5309	163	18	waluya	waluya	PROPN
cana-5309	163	19	,	,	PUNCT
cana-5309	163	20	“	"	PUNCT
cana-5309	163	21	learning	learn	VERB
cana-5309	163	22	trajectory	trajectory	NOUN
cana-5309	163	23	of	of	ADP
cana-5309	163	24	non	non	ADJ
cana-5309	163	25	-	-	ADJ
cana-5309	163	26	euclidean	euclidean	ADJ
cana-5309	163	27	geometry	geometry	NOUN
cana-5309	163	28	through	through	ADP
cana-5309	163	29	ethnomathematics	ethnomathematic	NOUN
cana-5309	163	30	learning	learn	VERB
cana-5309	163	31	approaches	approach	NOUN
cana-5309	163	32	to	to	PART
cana-5309	163	33	improve	improve	VERB
cana-5309	163	34	spatial	spatial	ADJ
cana-5309	163	35	ability	ability	NOUN
cana-5309	163	36	,	,	PUNCT
cana-5309	163	37	”	"	PUNCT
cana-5309	163	38	eurasia	eurasia	PROPN
cana-5309	163	39	journal	journal	PROPN
cana-5309	163	40	of	of	ADP
cana-5309	163	41	mathematics	mathematics	PROPN
cana-5309	163	42	science	science	NOUN
cana-5309	163	43	and	and	CCONJ
cana-5309	163	44	technology	technology	NOUN
cana-5309	163	45	education	education	NOUN
cana-5309	163	46	,	,	PUNCT
cana-5309	163	47	vol	vol	NOUN
cana-5309	163	48	.	.	PROPN
cana-5309	163	49	19	19	NUM
cana-5309	163	50	,	,	PUNCT
cana-5309	163	51	no	no	INTJ
cana-5309	163	52	.	.	NOUN
cana-5309	163	53	6	6	NUM
cana-5309	163	54	,	,	PUNCT
cana-5309	163	55	p.	p.	NOUN
cana-5309	163	56	em2285	em2285	PROPN
cana-5309	163	57	,	,	PUNCT
cana-5309	163	58	may	may	AUX
cana-5309	163	59	2023	2023	NUM
cana-5309	163	60	,	,	PUNCT
cana-5309	163	61	doi	doi	NOUN
cana-5309	163	62	:	:	PUNCT
cana-5309	163	63	10.29333	10.29333	NUM
cana-5309	163	64	/	/	SYM
cana-5309	163	65	ejmste/13269	ejmste/13269	NOUN
cana-5309	163	66	.	.	PUNCT
cana-5309	164	1	[	[	X
cana-5309	164	2	15	15	NUM
cana-5309	164	3	]	]	X
cana-5309	164	4	a.	a.	NOUN
cana-5309	164	5	a.	a.	NOUN
cana-5309	164	6	ungar	ungar	NOUN
cana-5309	164	7	,	,	PUNCT
cana-5309	164	8	“	"	PUNCT
cana-5309	164	9	gyrations	gyration	NOUN
cana-5309	164	10	:	:	PUNCT
cana-5309	164	11	the	the	DET
cana-5309	164	12	missing	miss	VERB
cana-5309	164	13	link	link	NOUN
cana-5309	164	14	between	between	ADP
cana-5309	164	15	classical	classical	ADJ
cana-5309	164	16	mechanics	mechanic	NOUN
cana-5309	164	17	with	with	ADP
cana-5309	164	18	its	its	PRON
cana-5309	164	19	underlying	underlie	VERB
cana-5309	164	20	euclidean	euclidean	ADJ
cana-5309	164	21	geometry	geometry	NOUN
cana-5309	164	22	and	and	CCONJ
cana-5309	164	23	relativistic	relativistic	ADJ
cana-5309	164	24	mechanics	mechanic	NOUN
cana-5309	164	25	with	with	ADP
cana-5309	164	26	its	its	PRON
cana-5309	164	27	underlying	underlying	ADJ
cana-5309	164	28	hyperbolic	hyperbolic	ADJ
cana-5309	164	29	geometry	geometry	NOUN
cana-5309	164	30	,	,	PUNCT
cana-5309	164	31	”	"	PUNCT
cana-5309	164	32	in	in	ADP
cana-5309	164	33	springer	springer	NOUN
cana-5309	164	34	ebooks	ebook	NOUN
cana-5309	164	35	,	,	PUNCT
cana-5309	164	36	2012	2012	NUM
cana-5309	164	37	,	,	PUNCT
cana-5309	164	38	pp	pp	ADJ
cana-5309	164	39	.	.	PUNCT
cana-5309	165	1	463–504	463–504	NUM
cana-5309	165	2	.	.	PUNCT
cana-5309	166	1	doi	doi	NOUN
cana-5309	166	2	:	:	PUNCT
cana-5309	166	3	10.1007/978	10.1007/978	NUM
cana-5309	166	4	-	-	SYM
cana-5309	166	5	3642	3642	NUM
cana-5309	166	6	-	-	PUNCT
cana-5309	166	7	28821	28821	NUM
cana-5309	166	8	-	-	PUNCT
cana-5309	166	9	0_18	0_18	NOUN
cana-5309	166	10	.	.	PUNCT