ICE Topology and parametric equations

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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! ;)
Stay safe, sane & healthy!

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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!
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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.
Attachments
Sin_Cos.jpg
Sin_Cos.jpg (16.38 KiB) Viewed 9535 times
Sphere3.jpg
Sphere3.jpg (70.25 KiB) Viewed 9535 times
Sphere4.jpg
Sphere4.jpg (76.8 KiB) Viewed 9535 times
Last edited by Daniel Brassard on 10 Dec 2011, 04:01, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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.
Attachments
Ellipsoid.jpg
Ellipsoid.jpg (72.84 KiB) Viewed 9532 times
Ellipsoid2.jpg
Ellipsoid2.jpg (60.84 KiB) Viewed 9532 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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.
Attachments
Football.jpg
Football.jpg (77.61 KiB) Viewed 9528 times
Football2.jpg
Football2.jpg (62.51 KiB) Viewed 9528 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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, 04:44

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.
Attachments
Sphere_Twisted.jpg
Sphere_Twisted.jpg (71.75 KiB) Viewed 9495 times
Sphere_Twisted2.jpg
Sphere_Twisted2.jpg (69.22 KiB) Viewed 9495 times
Last edited by Daniel Brassard on 10 Dec 2011, 05:01, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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, 05:00

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.
Attachments
Top.jpg
Top.jpg (73.54 KiB) Viewed 9493 times
Top2.jpg
Top2.jpg (70.91 KiB) Viewed 9493 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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, 05:56

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.
Attachments
Cone.jpg
Cone.jpg (59.21 KiB) Viewed 9491 times
Cone2.jpg
Cone2.jpg (78.27 KiB) Viewed 9491 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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, 06:13

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.
Attachments
Cone3.jpg
Cone3.jpg (56.44 KiB) Viewed 9490 times
Cone4.jpg
Cone4.jpg (68.97 KiB) Viewed 9490 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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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, 06:21

That is it for tonight ... next time The Torus!
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Re: ICE Topology and parametric equations (Warning math inside!)

Post by Daniel Brassard » 11 Dec 2011, 04:17

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.
Attachments
Torus.jpg
Torus.jpg (72.73 KiB) Viewed 2962 times
Torus2.jpg
Torus2.jpg (82.19 KiB) Viewed 2962 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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

Post by Daniel Brassard » 11 Dec 2011, 04:21

The Torus second mapping.

Like the sphere, the torus has a second mapping as follows:

x = f(u,v) = Cos(u) * ( R0 + (R1 * Sin(v)))
y = g(u,v) = R1 * Cos(v)
z = h(u,v) = Sin(u) * (R0 + ( R1 * Sin(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.
Attachments
Torus3.jpg
Torus3.jpg (71.08 KiB) Viewed 3515 times
Torus4.jpg
Torus4.jpg (86.42 KiB) Viewed 3515 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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

Post by Daniel Brassard » 11 Dec 2011, 04:25

Torus variations

Also like the sphere, we can modify the equations to control the height, width or depth by multiplying each equations with a scalar.

The torus equations are now:

x = f(u,v) = a * (Cos(u) * ( R0 + (R1 * Cos(v))))
y = g(u,v) = b * (R1 * Sin(v))
z = h(u,v) = c * (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
a, b and c are scalars to control the width, height and depth of the torus.


Below is a modification of the ICE tree in x,y and z with a stretch in height of the torus as an example.
Attachments
Torus5.jpg
Torus5.jpg (77.33 KiB) Viewed 3514 times
Torus6.jpg
Torus6.jpg (84.49 KiB) Viewed 3514 times
Last edited by Daniel Brassard on 11 Dec 2011, 04:47, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

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

Post by Daniel Brassard » 11 Dec 2011, 04:31

The Helical Torus

By adding a component to the y axis equation, we can change the torus to a spring type surface.

The equations are:

x = f(u,v) = Cos(u) * ( R0 + (R1 * Cos(v)))
y = g(u,v) = Stretch * u + (R1 * Sin(v)) (note the modification)
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 = 4*PI
V_Start = 0
V_End - 2*PI
Stretch = 1

The ICE Tree and result is as below.
Attachments
Helical-Torus.jpg
Helical-Torus.jpg (72.62 KiB) Viewed 3513 times
Helical-Torus2.jpg
Helical-Torus2.jpg (79.66 KiB) Viewed 3513 times
Last edited by Daniel Brassard on 11 Dec 2011, 04:44, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

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

Post by Daniel Brassard » 11 Dec 2011, 04:43

The Spiral Helical Torus (Shell)

An other of the Torus variation, this time we also control the size of the ring as its spiral around the y axis creating a Shell type surface.

The equations are:

x = f(u,v) = Cos(u) * ( R0 + (R1 * Cos(v)))
y = g(u,v) = u* (R1 * Sin(v)) - Stretch * u (note the modification)
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 = 4*PI
V_Start = 0
V_End - 2*PI
Stretch = 1.5

The ICE Tree and result is as below.
Attachments
Torus_Shell.jpg
Torus_Shell.jpg (76.68 KiB) Viewed 3511 times
Torus_Shell2.jpg
Torus_Shell2.jpg (82.86 KiB) Viewed 3511 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

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