ICE Topology and parametric equations

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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 03:01

Before we go further, it's now time to share the sandbox.

Attached is the Parametric_Sandbox compound. It includes all the nodes seen above in a tight little compound ready for experiment. Download and install the compound in your users or workgroup directory.

We will be modifying this compound a lot. Make sure to rename the compound each time you make a new object before exporting and save often!

After you installed it, the compound will appear under topology. To use it, create an empty polygon mesh and create an ice tree. Connect the compound to the ICE tree. You should see a grid appears in the viewport. Explore the nodes and play with the sliders.

Let's move on!
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TwinSnakes007
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Re: ICE Topology and parametric equations (Warning math here!)

Post by TwinSnakes007 » 03 Dec 2011, 03:03

Dude...I freaking LOVE this stuff. Keep it coming!!!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 03:33

Good to know I am getting some fan! Let's keep it coming then.

Our next object will be the elliptic paraboloid.

The equation is

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

with u_start = -2, u_end = 2, v_start = -2, v_end = 2

Open the Parametric_Sandbox compound and modify it using two multiply nodes and an add nodes as follows.

Check the viewport! Not bad, we have deformed our grid like a cloth pulled by its corners!

Rename the compound "Elliptic_Paraboloid" and export the compound in your user or workgroup. You now have a new object!

It is that simple!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 03:50

The Hyperbolic paraboloid!

Next object is a variation of the previous object. It is also called the "saddle"

The parametric equations are as follows:

x = f(u,v) = u
y = g(u,v) = u*u - v*v (notice the difference from the previous object definition!!)
z = h(u,v) = v

the domain (bounding box) will be:

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

Open the elliptic paraboloid node and replace the add node with a substract node.

Rename the compound "Hyperbolic_Paraboloid" or "Saddle" and export it.

Voila! New object!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 03:59

For the next object, we will be using polar coordinates. The link below explain what polar coordinates are.

http://mathworld.wolfram.com/PolarCoordinates.html

Briefly, polar coordinates express x,y,and z using radius and angle.

Here is an illustration from Wikipedia for polar coordinates in two dimension.
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 04:07

Before we go further, note that the Cos, Sin and Tan factory ICE nodes accept degrees as input angles. Our formulas are providing numbers in radians. We will need to convert the numbers from radians to degrees before inputting to Cos, Sin or Tan nodes. To simplify the process, here three little compounds that do just that, accept radians as input.

Install them in your users or workgroup directory.
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Last edited by Daniel Brassard on 05 Dec 2011, 19:12, edited 3 times in total.
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 04:08

The three little pigs!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 04:29

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.
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 04:44

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!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 03 Dec 2011, 05:04

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!
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Last edited by Daniel Brassard on 05 Dec 2011, 19:18, edited 1 time in total.
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Hirazi Blue
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Hirazi Blue » 03 Dec 2011, 11:16

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

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guillaume
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Re: ICE Topology and parametric equations (Warning math here!)

Post by guillaume » 04 Dec 2011, 18:54

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 !

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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 05 Dec 2011, 14:52

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!
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 10 Dec 2011, 03:44

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.
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 10 Dec 2011, 03:58

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.
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Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)

Post by Daniel Brassard » 10 Dec 2011, 04:26

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.
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