ICE Topology and parametric equations
- Daniel Brassard
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Re: ICE Topology and parametric equations (Warning math here!)
The three little pigs!
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- Three-little-pigs.jpg (10.29 KiB) Viewed 2850 times
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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.
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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- Cylinder.jpg (81.46 KiB) Viewed 2850 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:13, edited 3 times in total.
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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!
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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- Disk.jpg (75.07 KiB) Viewed 2849 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:17, edited 1 time in total.
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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!
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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- Sphere.jpg (62.72 KiB) Viewed 2849 times
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- Sphere2.jpg (58.41 KiB) Viewed 2849 times
Last edited by Daniel Brassard on 05 Dec 2011, 19:18, edited 1 time in total.
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- Hirazi Blue
- Administrator
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- Joined: 04 Jun 2009, 12:15
Re: ICE Topology and parametric equations (Warning math here!)
Let me say it one more time: Wow! ;)
Stay safe, sane & healthy!
Re: ICE Topology and parametric equations (Warning math here!)
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 !
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
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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!
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"
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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.
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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- Sin_Cos.jpg (16.38 KiB) Viewed 9669 times
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- Sphere3.jpg (70.25 KiB) Viewed 9669 times
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- Sphere4.jpg (76.8 KiB) Viewed 9669 times
Last edited by Daniel Brassard on 10 Dec 2011, 04:01, edited 1 time in total.
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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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- Ellipsoid.jpg (72.84 KiB) Viewed 9666 times
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- Ellipsoid2.jpg (60.84 KiB) Viewed 9666 times
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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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- Football.jpg (77.61 KiB) Viewed 9662 times
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- Football2.jpg (62.51 KiB) Viewed 9662 times
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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 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
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- Sphere_Twisted.jpg (71.75 KiB) Viewed 9616 times
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- Sphere_Twisted2.jpg (69.22 KiB) Viewed 9616 times
Last edited by Daniel Brassard on 10 Dec 2011, 05:01, edited 1 time in total.
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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 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.
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- Top.jpg (73.54 KiB) Viewed 9614 times
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- Top2.jpg (70.91 KiB) Viewed 9614 times
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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
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- Cone.jpg (59.21 KiB) Viewed 9612 times
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- Cone2.jpg (78.27 KiB) Viewed 9612 times
$ifndef "Softimage"
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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
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- Cone3.jpg (56.44 KiB) Viewed 9611 times
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- Cone4.jpg (68.97 KiB) Viewed 9611 times
$ifndef "Softimage"
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
That is it for tonight ... next time The Torus!
$ifndef "Softimage"
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- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math inside!)
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.
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
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- Torus.jpg (72.73 KiB) Viewed 3092 times
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- Torus2.jpg (82.19 KiB) Viewed 3092 times
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