ICE Topology and parametric equations
- Hirazi Blue
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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
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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!
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$endif
set "Softimage" "true"
$endif
- Daniel Brassard
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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.
- Attachments
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- Sin_Cos.jpg (16.38 KiB) Viewed 9668 times
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- Sphere3.jpg (70.25 KiB) Viewed 9668 times
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- Sphere4.jpg (76.8 KiB) Viewed 9668 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
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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.
- Attachments
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- Ellipsoid.jpg (72.84 KiB) Viewed 9665 times
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- Ellipsoid2.jpg (60.84 KiB) Viewed 9665 times
$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!)
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.
- Attachments
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- Football.jpg (77.61 KiB) Viewed 9661 times
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- Football2.jpg (62.51 KiB) Viewed 9661 times
$ifndef "Softimage"
set "Softimage" "true"
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- Daniel Brassard
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- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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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 9615 times
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- Sphere_Twisted2.jpg (69.22 KiB) Viewed 9615 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
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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.
- Attachments
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- Top.jpg (73.54 KiB) Viewed 9613 times
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- Top2.jpg (70.91 KiB) Viewed 9613 times
$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 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
-
- Cone.jpg (59.21 KiB) Viewed 9611 times
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- Cone2.jpg (78.27 KiB) Viewed 9611 times
$ifndef "Softimage"
set "Softimage" "true"
$endif
set "Softimage" "true"
$endif
- 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!)
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
-
- Cone3.jpg (56.44 KiB) Viewed 9610 times
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- Cone4.jpg (68.97 KiB) Viewed 9610 times
$ifndef "Softimage"
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$endif
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$endif
- 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!
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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
-
- Torus.jpg (72.73 KiB) Viewed 3091 times
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- Torus2.jpg (82.19 KiB) Viewed 3091 times
$ifndef "Softimage"
set "Softimage" "true"
$endif
set "Softimage" "true"
$endif
- 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 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.
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 (71.08 KiB) Viewed 3602 times
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- Torus4.jpg (86.42 KiB) Viewed 3602 times
$ifndef "Softimage"
set "Softimage" "true"
$endif
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$endif
- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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 (77.33 KiB) Viewed 3601 times
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- Torus6.jpg (84.49 KiB) Viewed 3601 times
Last edited by Daniel Brassard on 11 Dec 2011, 04:47, edited 1 time in total.
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set "Softimage" "true"
$endif
set "Softimage" "true"
$endif
- 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 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.
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 (72.62 KiB) Viewed 3600 times
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- Helical-Torus2.jpg (79.66 KiB) Viewed 3600 times
Last edited by Daniel Brassard on 11 Dec 2011, 04:44, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
$endif
set "Softimage" "true"
$endif
- 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 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.
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 (76.68 KiB) Viewed 3598 times
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- Torus_Shell2.jpg (82.86 KiB) Viewed 3598 times
$ifndef "Softimage"
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