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!

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.

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!

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.

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.

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.

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.

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.

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.

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.