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The three little pigs!

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.

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!

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!

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

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 !

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!

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.

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.

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.

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.

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.

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.

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.

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

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.