Implicitization

From ErgaWiki

Contents 1 Implicitization experiments on curves and surfaces 1.1 Curves 1.2 Surfaces 2 Remarks

Implicitization experiments on curves and surfaces

Curves

No	curve [1]	equation	supports	# mixed subdivisions	Enum by reverse search (sec) [2]	TOPCOM point2alltriang (sec) [3]	TOPCOM point2triang(sec) [4]	# mixed cell configurations	# N(R) vertices	N(R) vertices	# all lattice points of N(R)
1.	astroid	a\cos(t)^3,a\sin(t)^3	supports	289	193.62	0.048	0.452	289	35	N(R)	454
2.	cardioid	$a(2\cos(t)-\cos(2t)),a(2\sin(t)-\sin(2t))$	supports	37	6.52	0.005	0.024	37	10	N(R)	33
3.	circle	$ \cos(t),\sin(t)$	supports	5	0.004	0.016	0.004	5	3	N(R)	4
4.	conchoid	$a+\cos(t),a\tan(t)+\sin(t)$	supports	12	0.84	0.003	0.008	12	4	N(R)	6
5.	ellipse	$a\cos(t),b\sin(t)$	supports	5	0.15	0.001	0.004	5	3	N(R)	4
6.	folium of Descartes	$3at/(1+t^3),3at^2/(1+t^3)$	supports	14	0.94	0.004	0.008	14	6	N(R)	10
7.	involute of a circle	$a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$	supports	14	1.00	0.001	0.007	14	6	N(R)	7
8.	nephroid	$a(3\cos(t)-\cos(3t)),a(3\sin(t)-\sin(3t))$	supports	289	195.27	0.004	0.240	289	35	N(R)	454
9a.	Plateau curve	$a\sin(3t)/\sin(t),2a\sin(2t)$	supports	94	33.02	0.012	0.064	94	15	N(R)	55
9b.	plateau curve	$a\sin(6t)/ \sin(2t), 2a\sin(4t)$	supports	42168	halt	25.934	85.597	42168	495	N(R)	not computed
10.	talbot's curve	$(a^2 + c^2 \sin( t)^2) \cos( t)/a, (a^2 - 2c^2 + c^2\sin(t)^2)\sin(t)/b $	supports	1944	3948.80	0.416	2.356	1944	84	N(R)	1600
11.	tricuspoid	$a(2\cos(t)+\cos(2t)),a(2\sin(t)-\sin(2t))$	supports	37	6.20	0.008	0.024	37	10	N(R)	33
12.	witch of Agnesi	$at,a/(1 + t^2)$	supports	2	0.03	0.007	0.004	2	2	N(R)	2
13.	circle (3 systems)	$(-t^2 +1)/s, 2t/s, t^2 -s +1$	supports	26	6.00	0.020	0.052	26	6	N(R)	7

Surfaces

No	surface	equation	supports	# mixed subdivisions	Enum by reverse search (sec)	TOPCOM point2alltriang (sec)	TOPCOM point2triang(sec)	# mixed cell configurations	# N(R) vertices	N(R) vertices	# all lattice points of N(R)
1.	cylinder	$\cos(t),\sin(t),s$	supports	5	0.24	0.003	0.006	5	3	N(R)	4
2.	cone	$s\cos(t),s\sin(t),s$	supports	122	73.45	0.192	0.288	98	8	N(R)	14
3.	paraboloid	$s\cos(t),s\sin(t),s^2$	supports	122	71.60	0.192	0.296	98	8	N(R)	37
4.	surface of revolution	$s\cos(t),s\sin(t),\cos(s)$	supports	122	71.80	0.193	0.288	98	8	N(R)	37
5.	sphere	$\sin(t)\cos(s),\sin(t)\sin(s),\cos(t)$	supports	104148	halt	19496.602	714.161	43018	21	N(R)	186
6.	sphere2	$\cos(t)\cos(s),\sin(t)\cos(s),\sin(s)$	supports	76280	halt	4492.977	397.157	32076	95	N(R)	776
7.	stereographic shpere	$2t/(1 t^2 s^2),2s/(1 t^2 s^2),(t^2 s^2-1)/(1 t^2 s^2)$	supports	3540	7112.54	25.402	11.025	3126	22	N(R)	283
8.	twisted shpere	$a(\cos(t) t(\sin(t)),a(\sin(t)-t\cos(t))$	supports	>1812221	not computed	not computed	not computed	not computed	not computed		not computed

Remarks

1. ↑ Many thanks to Tatjana Kalinka for providing this list of curves and surfaces.
2. ↑ This is the computation time of enumeration of regular triangulations algorithm using reverse search. I would like to thank very much Fumihiko TAKEUCHI for running the experiments and providing this results. Experiments were done on a Blade 100, 550Mhz, 2GB memory with SunOS 5.9.
3. ↑ This is the computation time of points2alltriangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.
4. ↑ This is the computation time of points2triangs client of TOPCOM package. Experiments were done on a Intel(R) Pentium(R) 4 CPU 3.20GHz, 1.5GB memory with x86_64 Debian GNU/Linux.