ICE Topology and parametric equations

[Daniel Brassard]

The Stereographic Sphere (Riemann Sphere)

Here is an interesting parametric surface. This sphere parameters are based on the stereographic projection, which map the points of the sphere on the complex plane. The gap in the sphere cannot be closed without the u and v parameters extending to a very large number (infinity). If you want to close it, you will have to do it manually.

This surface is useful in constructing baskets, necklaces and domes.

The parameter are:

x = 2 * v / (1 + u*u + v*v)
y = (u*u + v*v - 1) / (1 + u*u + v*v)
z = 2 * u / (1 + u*u + v*v)

with

u_start = -10
u_end = 10
v_start = -10
v_end = 10

The ICE Tree and parametric surface is as follow. The last figure (Sphere_Stereographis3) shows you the necklace form with the parameters to achieve it. Enjoy!

[Memag]

Wow!
Insanely awesome for (below)average 3d user like me.
I just wish these could be put in "get primitive" menu for quick use.
Please continue.

[Daniel Brassard]

Thanks Memag.

The more I am getting into this, the more I am thinking of packaging all the surfaces into a Topo pack. Maybe I'll do that after I finish with this thread!

Sort of thank you and support to si-community!

[ActionArt]

That would be a VERY good idea. I think it would be quite useful and give a good starting point for those of us that are a little math challenged to build onto. Would be much appreciated if you decide to do that.

In production sometimes there isn't as much time to experiment as I'd like so a starting point is very helpful. I think it would be very popular.

[Daniel Brassard]

Egg Carton

This is a simple variation of the grid using sines.

The parameters are:

x = u
y = A * Sin(u) * Sin (v)
z = v

with

u_start = -10
u_end = 10
v_start = -10
v_end = 10
A = 2 (amplitude of the bump on the grid surface)

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Folded Strip

This next surface uses sine and cosine functions to fold the grid. The surface is useful for making bow ties and candies.

The parameters are:

x = Sin(u)
y = Cos(u+v)
z = v

with

u_start = 0 ... (if you make this -2*PI you get the bow tie)
u_end = 2*PI
v_start = -PI
v_end = PI

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Folded Strip Variation

This next version of the folded strip uses the absolute function to fold the strip on itself along the z axis creating an interesting shape.

The parameters are:

x = Sin(u)
y = Cos(u+v)
z = abs(v) / 2

with

u_start = 0
u_end = 2*PI
v_start = -20
v_end = 20

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Grid Ripple

This variation of the grid uses the sine function to create ripples on the surface. The amount of ripples is controlled by a scalar.

The parameters are:

x = u
y = Sin(u*v/Ripple)
z = v

with

u_start = -10
u_end = 10
v_start = -10
v_end = 10
Ripple = 5 (can be any amount except zero to avoid a "division by zero" error)

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Conical Spiral Disk (Cochlea)

Here is a variation of the spiral disk using u to deform the disk in a conical shape. The height of the disk on the y axis is controlled by a scalar.

The parameters are:

x = u * v * Sin(u)
y = Height * u
z = u * v * Cos(u)

with

u_start = -8*PI
u_end = 8*PI
v_start = 0
v_end = 1
Height = 1

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Torus Spiral (Coil)

A variation of the spiral helical torus with a constant width and no height. Useful to create clock springs, hoses and ornements.

The parameters are:

x = Cos(u) * (R0 + (R1 * Cos(v)))
y = R1 * Sin(v)
z = Sin(u) * (R0 + (R1 * Cos(v)))

with

u_start = 0
u_end = 8*PI
v_start = 0
v_end = 2*PI
R0 = Large radius = .5
R1 = Small radius = .5

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Whitney's Umbrella

An interesting surface with intersecting polygons. The parameters are so simple.

They are:

x = u
y = v * v
z = u * v

with

u_start = -1
u_end = 1
v_start = -1
v_end = 1

The ICE Tree and parametric surface is as follow.

[Daniel Brassard]

Little Update, I am working on the next set of ice surface posts. Here is a preview of the braided torus for your enjoyment!

And no, this is not the torus wireframe, this is the actual surface!

[dwigfor]

Great stuff. I hope to get this working in the near future after I finish a couple of projects I'm currently working on. Haven't had the chance to really take a look at ICE Modelling yet.

I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html

Great stuff - keep up the hard work!
-Dave

[Daniel Brassard]

Hi Dave,

Thank you for your kind encouragement.

I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html

For that, you will need some sort of ray marcher / marching cube / voxelizer method ....
maybe emPolygoniser can help!

Maybe something to propose in the community project thread! That would be an interesting project!

[Daniel Brassard]

Little update!

Just created a serie of supershapes based on the super torus.

Here is a rendered preview.

The SuperCone, SuperSphere, SuperCylinder, SuperDisk and SuperTorus!

Each shape can adopt many variations by adjusting its parameters! As an example, the rounded cube in the middle is actually the SuperSphere!

[ActionArt]

Very nice indeed! I've fallen behind in trying it myself (have to get some work done) but really admire what you've achieved here. Looking forward to more!