ICE Topology and parametric equations

Here you can find ICE tutorials and learning material
User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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

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

The three little pigs!
Attachments
Three-little-pigs.jpg
Three-little-pigs.jpg (10.29 KiB) Viewed 2728 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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.
Attachments
Cylinder.jpg
Cylinder.jpg (81.46 KiB) Viewed 2728 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:13, edited 3 times in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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!
Attachments
Disk.jpg
Disk.jpg (75.07 KiB) Viewed 2727 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:17, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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!
Attachments
Sphere.jpg
Sphere.jpg (62.72 KiB) Viewed 2727 times
Sphere2.jpg
Sphere2.jpg (58.41 KiB) Viewed 2727 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:18, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Hirazi Blue
Administrator
Posts: 5107
Joined: 04 Jun 2009, 12:15

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!

User avatar
guillaume
Posts: 41
Joined: 19 Jul 2009, 00:01
Contact:

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 !

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9537 times
Sphere3.jpg
Sphere3.jpg (70.25 KiB) Viewed 9537 times
Sphere4.jpg
Sphere4.jpg (76.8 KiB) Viewed 9537 times
Last edited by Daniel Brassard on 10 Dec 2011, 04:01, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9534 times
Ellipsoid2.jpg
Ellipsoid2.jpg (60.84 KiB) Viewed 9534 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9530 times
Football2.jpg
Football2.jpg (62.51 KiB) Viewed 9530 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9496 times
Sphere_Twisted2.jpg
Sphere_Twisted2.jpg (69.22 KiB) Viewed 9496 times
Last edited by Daniel Brassard on 10 Dec 2011, 05:01, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9494 times
Top2.jpg
Top2.jpg (70.91 KiB) Viewed 9494 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9492 times
Cone2.jpg
Cone2.jpg (78.27 KiB) Viewed 9492 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 9491 times
Cone4.jpg
Cone4.jpg (68.97 KiB) Viewed 9491 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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

User avatar
Daniel Brassard
Posts: 878
Joined: 18 Mar 2010, 23:38
Location: St. Thomas, Ontario
Contact:

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 2963 times
Torus2.jpg
Torus2.jpg (82.19 KiB) Viewed 2963 times
$ifndef "Softimage"
set "Softimage" "true"
$endif

Post Reply

Who is online

Users browsing this forum: No registered users and 4 guests