ICE Topology and parametric equations

[Daniel Brassard]

The three little pigs!

[Daniel Brassard]

The Cylinder!

Our next object will deform the grid into a tube or open cylinder.

The parametric equations are:

r = 1 (this is the radius of the cylinder)
x = f(u,v) = r * Cos(u)
y = g(u,v) = v (this will control the length of the cylinder)
z = h(u,v) = r * Sin(u)

The domain will be

u_start = 0
u_end = 2*Pi
v_start = 0
v_end = 1

Open the Parametric_Sandbox compound and change the inside as follows. Rename to "Cylinder" and export.

[Daniel Brassard]

The disk!

The disk also use polar coordinates. The v parameter controls the radius of the circles (v_start control the inside circle with zero at the center, v_end the outside circle).

The equations are as follows:

x = f(u,v) = v * Sin (u)
y = g(u,v) = 0 (we are centering the disk at the origin)
z = h(u,v) = v * Cos (u)

u_start = 0, u_end = 2* PI, v_start = 0, v_end = 1

Open the Parametric_Sandbox and modify it as follows. Rename "Disk" and export. Play with the sliders to see the effects!

[Daniel Brassard]

Our last object for tonight is the sphere!

The sphere also use polar coordinates. The radius "r" control the size of the sphere. The equations are as follow:

x = f(u,v) = r * Sin(u) * Cos (v)
y = g(u,v) = r * Cos (u)
z = h(u,v) = r * Sin(u) * Sin (v)

u_start = 0
u_end = PI
v_start = 0
v_end = 2*PI
r = 1

Modify the Parametric_Sandbox as follows. Rename to "Sphere" and export the compound.

That's it for tonight. Cheers and happy experiment!

[Hirazi Blue]

Let me say it one more time: Wow! ;)

[guillaume]

It looks like you've got a lot of fun with parametric equations Daniel ! Nice compounds, well done !

The Create Topo node (the little guy responsible of converting arrays to a true polygon mesh) is one of the simplest topo nodes, but I was convinced that it would be one of the most useful. Nice to see good applications of it like in your tools or in Implosia FX too !

[Daniel Brassard]

Thanks Guillaume,

I like the topology node, it is a very useful node. The most simpliest thing are often the most useful. Thanks to you and the other Dev behind Softimage, you rock!

[Daniel Brassard]

The Sphere revisited (second mapping)

Due to the property of the sine and cosine, the sphere has two parametric version.

The Sine can be seen as a cosine phased out (moved) by 90 degrees (half-pi) as illustrated below.

The second version of the sphere is:

x = f(u,v) = r * Cos(u) * Cos (v)
y = g(u,v) = r * Sin(v)
z = h(u,v) = r * Sin(u) * Cos (v)

u_start = 0
u_end = Pi
v_start = 0
v_end = 2 * Pi
r = 1

This version will create a sphere by rotating a full circle 180 degrees.

Below is the ICE tree and the result. I did not do a full 180 degree to show how the sphere is created.

[Daniel Brassard]

Experimentation with the sphere equations

If you remenber the original equation of the sphere, we have:

x = f(u,v) = r * Sin(u) * Cos (v)
y = g(u,v) = r * Cos (u)
z = h(u,v) = r * Sin(u) * Sin (v)

u_start = 0
u_end = PI
v_start = 0
v_end = 2*PI
r = 1

but what if r was different for x, y and z like so?

x = f(u,v) = rx * Sin(u) * Cos (v)
y = g(u,v) = ry * Cos (u)
z = h(u,v) = rz * Sin(u) * Sin (v)

u_start = 0
u_end = PI
v_start = 0
v_end = 2*PI
rx = 1
ry = .5
rz = .75

We Get the Ellipsoid (squashed sphere). Here is the ICE Tree and result.

[Daniel Brassard]

Football

To create a Football shape, we need to modify our original sphere equations as follows:

x = f(u,v) = r * Sin(u) * Cos (v)
y = g(u,v) = r * Cos (u) - b * u (note the small modification)
z = h(u,v) = r * Sin(u) * Sin (v)

u_start = 0
u_end = PI
v_start = 0
v_end = 2*PI
r = 1
b = 1

The ICE tree and result is provided below.

[Daniel Brassard]

The Twisted Sphere

The Twisted Sphere is a variation of the second mapping of the sphere. The equations are:

x = f(u,v) = r * Cos(u) * Cos (v)
y = g(u,v) = r * Sin(v) + b * u (note the modification)
z = h(u,v) = r * Sin(u) * Cos (v)

This is similar to the one we have seen before but on the second version of the sphere equations.

u_start = 0
u_end = Pi
v_start = 0
v_end = 2 * Pi
r = 1
b = 1

The ICE tree and result is as below.

[Daniel Brassard]

The Top

The Top surface is a variation of the shpere where the radius is fed by a cosine function. The equations are:

x = f(u,v) = r * Sin(u) * Cos (v)
y = g(u,v) = r * Cos (u)
z = h(u,v) = r * Sin(u) * Sin (v)

r = Cos(2*u) (this is the only modification to the sphere equations)

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

The ICE tree and the result is illustrated below.

[Daniel Brassard]

The Cone

The cone is a variation of the disk. We use v to modify the shape and scale the circles. The equations are:

x = f(u,v) = v * Sin (u)
y = g(u,v) = v
z = h(u,v) = v * Cos(u)

u_start = 0
u_end = 2*PI
v_start = 0 (to invert the cone, start at -1 and end at 0)
v_end = 1

The ICE tree and result is as below.

[Daniel Brassard]

Cone Variation

In this variation of the cone, i use a scalar to control the height and base width. You can also modify this to have both control in x and z independently if desire.

The equations are

x = f(u,v) = r * v * Sin (u)
y = g(u,v) = h * v
z = h(u,v) = r * v * Cos(u)

u_start = 0
u_end = 2*PI
v_start = -1 (show the cone inverted)
v_end = 0

r = .126
h = 1

The ICE tree and result is as below.

[Daniel Brassard]

That is it for tonight ... next time The Torus!

[Daniel Brassard]

The Torus (hum ... Donuts!)

The torus is formed by rotating a circle that is offset from the origin. The torus is controled by two radius: a large radius that control the size of the offset and a small radius that control the thickness of the ring.

The torus equations are:

x = f(u,v) = Cos(u) * ( R0 + (R1 * Cos(v)))
y = g(u,v) = R1 * Sin(v)
z = h(u,v) = Sin(u) * (R0 + ( R1 * Cos(v)))

where

R0 is the large radius = 1
R1 is the small radius = 0.25
U_start = 0
U_End = 2*PI
V_Start = 0
V_End - 2*PI

The ICE Tree and result is as below.