A list of some of the preset shapes that Parplot variation produces. There’s more now, but this should give you a flavour of what to expect.

Preset ID	Preset	Example Image	Creator	Notes
0	3D lissajous		Andreas Maschke	x=cos(u)*(4+cos(v)) y=sin(u)*(4+cos(v)) z=4*sin(2*u)+sin(v)*(1.2-sin(v)) umin = 0.0, umax = 2 * pi vmin = 0.0, vmax = 2 * pi
1	cone		Andreas Maschke	color_modes 0-3 x=cos(v)*sin(2*u) y=sin(v)*sin(2*u) z=sin(2*v)*sqr((cos(u))) umin = 0.0, umax = pi vmin = 0.0. vmax = pi
2	shell		Andreas Maschke	color_mode=2; the center line is from the gradient. x=cos(u)*(exp(u/10)-1)*(cos(v)+0.8) y=sin(u)*(exp(u/10)-1)*(cos(v)+0.8) zformula=exp(u/10)-1)*sin(v) umin = 0.0, umax = 5 * pi vmin = 0.0, vmax = 2 * pi
3	torus		Andreas Maschke	xformula = cos(v)*(2+sin(u+v/3)) yformula = sin(v)*(2+sin(u+v/3)) zformula = cos(u+v/3) umin = 0.0, umax = 2 * pi vmin = 0.0, vmax = 2 * pi
4	helix		Andreas Maschke	color_mode set as indicated xformula = cos(u)*(2+cos(v)) yformula = sin(u)*(2+cos(v)) zformula = (u-2*pi)+sin(v) umin = 0.0; umax = 4.0 * pi vmin = 0.0, vmax = 2.0 * pi
5	wavy disc		Andreas Maschke	x = u*cos(v) y = u*sin(v) z = sqr(cos(4.0*u))*exp(0-u) umin = 0.0, umax = pi vmin = 0.0, vmax = 2.0 * pi
6	klein bottle		Andreas Maschke	x = x=cos(u)*(2.0+sqr(cos(u/2.0))*sin(v)) y = sin(u)*(2.0+sqr(cos(u/2.0))*sin(v)) z = sqr(cos(u/2.0))*cos(v) umin = 0.0 – pi, umax = pi vmin = 0.0 – pi, vmax = 2.0 * pi
7	Case 7		Andreas Maschke	x = cos(u)*(4+cos(v)) y = sin(u)*(4+cos(v)) z = 3*sin(u)+(sin(3*v)*(1.2+sin(3*v))) umin = 0.0, umax = 2.0 * pi vmin = 0.0, vmax = 2.0 * pi
8	Case 9		Andreas Maschke	x= = u*cos(v) y = v*cos(u) z = u*v*sin(u)*sin(v) umin = 0.0 – pi, umax = pi vmin = 0.0 – pi, vmax = pi
9	Hershey Kiss		Andreas Maschke	x = cos(u)*sin(v*v*v/(pi*pi)) y = sin(u)*sin(v) z = cos(v) umin = 0.0, umax = 2.0 * pi vmin = 0.0, vmax = pi
10	Case 10		Andreas Maschke	x`= cos(u)*((cos(3*u)+2)*sin(v)+0.5) y` = sin(u)*((cos(3*u)+2)*sin(v)+0.5) z` = (cos(3*u)+2)*cos(v) umin = 0.0, umax = 2.0 * pi vmin = 0.0, vmax=2.0 * pi
11	Varigated sphere		Andreas Maschke	solid render source:https://reference.wolfram.com/language/tutorial/Paramet ricPlots.html x= sin(u)*sin(v)+0.05*cos(20*v) y = cos(u)*sin(v)+0.05*cos(20*u) z = cos(v) umin = -pi, umax = pi vmin = -pi, vmax = pi
12	Shell		Dimitri Augusto Rocha	https://renklisheyler.wordpress.com/2012/04/27/algebraicsurfaces/ x=2.0*(1.0-exp(u/(6.0*pi)))*cos(u)*sqr(cos(v/2.0)) y = 2.0*(-1.0+exp(u/(6.0*pi)))*sin(u)*sqr(cos(v/2.0)) z =1.0-exp(u/(3.0*pi))-sin(v)+exp(u/(6.0*pi))*sin(v) umin = 0,umax = 6 * pi vmin = 0, vmax = 2* pi
13	Slinky		Don Town	http://mathworld.wolfram.com/Slinky.html color_modes 0-3 x=(6+2*cos(u*v))*cos(u) ---- R=6, a=2, h=2 y=(6+2*cos(u*v))*sin(u) z=(2*u+2*sin(u*v)) umin= 0.0, umax= 6 * pi vmin= 0.0, vmax= 6 * pi
14	Real slinky		Frank Baumann	x=(1.0+0.25*cos(75.0*u))*cos(u) y=(1.0+0.25*cos(75.0*u))*sin(u) z=u+sin(75.0*u) umin= -2 * pi, umax= 2 * pi vmin= -2 * pi, vmax= 2 * pi
15	Spherical spiral		Frank Baumann	x= 7.83*cos((v-pi)/2)*(cos(16.4*v)) y= 7.83*cos((v-pi)/2)*(sin(16.4*v)) z= 7.83*sin((v-pi)/2) umin= 0.0, umax= 2 * pi vmin= 0.0. vmax= 2 * pi
16	Spherical rose		Frank Baumann	xformula= (2 + sin(7*u + 5*v))*cos(u)*sin(v) y= (2 + sin(7*u + 5*v))*sin(u)*sin(v) z= (2 + sin(7*u + 5*v))*cos(v) umin= -pi, umax= pi vmin= -pi, vmax= pi
17	Folded box		Frank Baumann	Preset_id=17 x= sin(u)*sin(v) y= cos(v)*cos(u) z= sin(sin(u)+cos(v)) umin= -pi, umax= pi vmin= -pi, vmax= pi
18	Wavy heart		Frank Baumann	x= (2*v*cos(u)) y= 2.0*v*(sin(u))+v*fabs(cos(u)) z= cos(3*v)*sin(3*v)2 umin= 0.0, umax= 63 vmin= 0.0, vmax= 6 1Turn your heart right side up with the Affine transform-Vertical flip button in the XY plane. 2Change the amplitude of waves by changing zformula value as seen in this equation: a*cos(f*v)*a*sin(f*v) Change the frequency of waves by changing value f. a=amplitude; f=frequency. 3To remove the pierced look, set umin to below -0.3.
19	Interesting Shape		Frank Baumann	color_modes 0-3 x=v*sin(fabs(u)) y= u*sin(fabs(v)) z= u+fabs(sin(v*u)) umin= -pi, umax= pi vmin= -pi, vmax= pi
20	Bubble Gum		Frank Baumann	xformula= cos(u)*(6-(5/4+sin(3-v))*sin(v-3-u)) y= (6-(5/4+sin(3*v))*sin(v-3*u))*sin(u) z= -cos(v-3*u)*(5/4+sin(3*v)) umin= -pi, umax= pi vmin= -pi, vmax= pi
21	Twisted Torus		Frank Baumann	xformula= (4+(sin(4*(v+2*u))+1.25)*cos(v))*cos(u) y= (4+(sin(4*(v+2*u))+1.25)*cos(v))*sin(u) z= ((sin(4*(v+2*u))+1.25)*sin(v)) umin= -pi, umax= pi vmin= -pi, vmax= pi
22	Vase		Frank Baumann	xformula= u y= sin(v)*(u*u*u+2.0*u*u-2.0*u+2.0)/5.0 z= cos(v)*(u*u*u+2.0*u*u-2.0*u+2.0)/5.0 umin= -2.3, umax= 1.3 vmin= 0.0, vmax= 2 * pi
23	Breather		Frank Baumann	color_modes 0-3 xformula=-0.8*u+(2*0.75*cosh(0.5*u)*sinh(0.5*u))/ (0.5*((sqrt(0.75)*sqr(cosh(0.5*u))) +sqr(0.5*sin(sqrt(0.75)*v)))) y=(2.0*sqrt(0.75)*cosh(0.5*u)*(- (sqrt(0.75)*cos(v)*cos(sqrt(0.75)*v))- sin(v)*sin(sqrt(0.75)*v)))/(0.5*sqr((sqrt(0.75)*cosh(0.5*u)) +sqr(0.5*sin(sqrt(0.75)*v)))) z=(2.0*sqrt(0.75)*cosh(0.5*u)*((sqrt(0.75)*sin(v)*cos(sqrt (0.75)*v))+cos(v)*sin(sqrt(0.75)*v)))/(0.5*sqr((sqrt(0.75)* cosh(0.5*u)) +sqr(0.5*sin(sqrt(0.75)*v)))) umin= -15, umax= 15 vmin= -24.55, vmax= 22 1 This one can be slow to render, depending on your computer's capabilities)
24	Lissajous		Frank Baumann	x= cos(1*u+0)+0.06*sin(1*v) --- Change the 0 value for phase shift and the u multiplier for frequency y= cos(15*u+0)-0.6*cos(1*v) --- Change the 0 value for phase shift and the u multiplier for frequency z= sin(12*u+0)+0.06*sin(1*v) --- Change the 0 value for phase shift and the u multiplier for frequency umin= 0.0, umax= 2 * pi vmin= -0.5, vmax= 0.5 1 These figures are as written, with no changes to constants (0 values) or coefficients (u multipliers).
25	Double mushroom		Frank Baumann	(set pitch close to 90 degrees to see mushroom) x= (cos(2*u))/(sqrt(2)+sin(2*v)) y= sin(2*u)/(sqrt(2)+sin(2*v)) z= v/(sqrt(5)+cos(2*v)) umin= -pi, umax= pi vmin= -3, vmax= 8
26	Triangluoid trefoil		Frank Baumann	color_modes 0-3 x= 2*sin(3*u)/(2+cos(v)) y= 2*(sin(u)+2*sin(2*u))/(2+cos(v+2*pi/3)) z= (cos(u)-2*cos(2*u))*(2+cos(v))*(2+cos(v+2*pi/3))/4 umin= -pi, umax= pi vmin= -pi, vmax= pi
27	Shell#1		Frank Baumann	color_modes 0-3 x= pow(1.2,u)*(1+cos(v))*cos(u) y= pow(1.2,u)*(1+cos(v))*sin(u) z= pow(1.2,u)*sin(v)-1.5*pow(1.2,u) umin= -12, umax= 6 vmin= -pi, vmax= pi
28	Shell#2		Frank Baumann	color_modes 0-3 x= u*cos(u)*(cos(v)+1) y= u*sin(u)*(cos(v)+1) z= u*sin(v)-((u+3)/8*pi)*u/3 umin= 0.0, umax= 20 vmin= -pi, vmax= pi
29	Trefoil knot		Frank Baumann	x= cos(u)*cos(v)+3*cos(u)*(1.5+sin(u*5/3)/2) y= sin(u)*cos(v)+3*sin(u)*(1.5+sin(u*5/3)/2) z= sin(v)+2*cos(u*5/3) umin= 0.0, umax= 20 vmin= -pi. vmax= pi
30	Penne Rigate pasta		Frank Baumann	color_modes 0-3 x= 0.1*cos(u) y= -0.1*sin(u) z= v+0.1*sin(u) umin= 0.0, umax= 2 * pi vmin= -0.5, vmax= 0.5
31	Conchiglie Rigate		Frank Baumann	x= (u/(pi+pi))*(1.0-2.0*v*v)*cos(u) y= (u/(pi+pi))*(1.0-2.0*v*v)*sin(u) z= v umin= 0.5235988, umax= 6.8067841 vmin= -0.5, vmax= 0.5
32	Cavatappi (corkscrew)		Frank Baumann	x= (3+2*cos(v))*cos(u) y= (3+2*cos(v))*sin(u) z= 1*u+2*sin(v) umin= -12.5663706, umax= 2 * pi vmin= 0.0, vmax= 2 * pi
33	Farfalle(Bow-tie)		Frank Baumann	x= u+(1/10)*sin(10*v) y= ((2.0*v)/3.0)*(1.2-(1.0/(1.0+u*u))) z= sin(pi*v)/(2*pi*v) umin= -3, umax= 3 vmin= -pi, vmax= pi
34	Fusilli		Frank Baumann	color_modes 1-3 x= (v/3)*cos(u-(2*1*pi)/3) y= (v/3)*sin(u-(2*1*pi)/3) z= u/10+(v^2)/2 umin= -2 * pi, umax= 2 * pi vmin= 0.0, vmax= 0.5
35	Under the sea		Don Town	color_mode=2 x=u*cos(v) y=u*sin(v) z=exp(-u*u)*(sin(param_a*pi*(u))-u*cos(param_b*v))1 umin=0, umax=2 vmin=0,vmax=2 pi 1Set these values: param_a,param_b, to make changes A, front and B, back: param_a=0 and param_b=0. C: param_a=1 and param_b=1. D: param_a=0 and param_b=1. E: param_a=1 and param_b=0.
36	Parallelogram		Don Town	x=u*cos(v) y=u*sin(v) z=exp(-u*u)*(sin(param_a*pi*(u))-u*cos(param_b*v)) umin=0, umax=2 vmin=0,vmax=2 pi A, B and D: param_a=2 and param_b=5. C: param_a=-5 and param_b=5.
37	Sine wave surface
38	EllipticCone
39	Ellipsoid (Change (0,0) to create ovoid
40	Crossbar Twist
41	Rippled Ribbon
42	Channel Surface
43	Wavy surface
44	Rippled surface
45	Furled surface
46	Accordian surface

See Also …

• Crackle Presets
• Isoplot 3D Presets
• LSystem 3D Presets
• LSystem Presets