ICE Topology and parametric equations
 Daniel Brassard
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 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations (Warning math here!)
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!
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!
 Attachments

 Parametric_Sandbox.xsicompound
 (50.73 KiB) Downloaded 157 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:03, edited 3 times in total.
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 TwinSnakes007
 Posts: 316
 Joined: 06 Jun 2011, 16:00
Re: ICE Topology and parametric equations (Warning math here!)
Dude...I freaking LOVE this stuff. Keep it coming!!!
Last edited by TwinSnakes007 on 03 Dec 2011, 05:14, edited 1 time in total.
 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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!
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!
 Attachments

 Elliptic_Paraboloid.jpg (60.67 KiB) Viewed 2157 times

 Elliptic_Paraboloid2.jpg (92.78 KiB) Viewed 2157 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:08, edited 3 times in total.
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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!
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!
 Attachments

 Hyperbolic_Paraboloid.jpg (75.39 KiB) Viewed 2155 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:07, edited 2 times in total.
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
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.
 Attachments

 Polarcoordinate.jpg (4.56 KiB) Viewed 2153 times
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
Re: ICE Topology and parametric equations (Warning math here!)
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.
Install them in your users or workgroup directory.
 Attachments

 Tan_rad.xsicompound
 (5 KiB) Downloaded 110 times

 Sin_rad.xsicompound
 (5 KiB) Downloaded 111 times

 Cos_rad.xsicompound
 (5 KiB) Downloaded 110 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:12, edited 3 times in total.
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
Re: ICE Topology and parametric equations (Warning math here!)
The three little pigs!
 Attachments

 Threelittlepigs.jpg (10.29 KiB) Viewed 2148 times
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
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.
 Attachments

 Cylinder.jpg (81.46 KiB) Viewed 2148 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:13, edited 3 times in total.
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
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!
 Attachments

 Disk.jpg (75.07 KiB) Viewed 2147 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:17, edited 1 time in total.
$ifndef "Softimage"
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
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!
 Attachments

 Sphere.jpg (62.72 KiB) Viewed 2147 times

 Sphere2.jpg (58.41 KiB) Viewed 2147 times
Last edited by Daniel Brassard on 05 Dec 2011, 20:18, edited 1 time in total.
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 Hirazi Blue
 Administrator
 Posts: 4911
 Joined: 04 Jun 2009, 12:15
Re: ICE Topology and parametric equations (Warning math here!)
Let me say it one more time: Wow! ;)
...
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: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
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"
set "Softimage" "true"
$endif
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 Location: St. Thomas, Ontario
 Contact:
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 (halfpi) 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 (halfpi) 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 (16.38 KiB) Viewed 8954 times

 Sphere3.jpg (70.25 KiB) Viewed 8954 times

 Sphere4.jpg (76.8 KiB) Viewed 8954 times
Last edited by Daniel Brassard on 10 Dec 2011, 05:01, edited 1 time in total.
$ifndef "Softimage"
set "Softimage" "true"
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 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.
 Attachments

 Ellipsoid.jpg (72.84 KiB) Viewed 8951 times

 Ellipsoid2.jpg (60.84 KiB) Viewed 8951 times
$ifndef "Softimage"
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 Daniel Brassard
 Posts: 805
 Joined: 19 Mar 2010, 00:38
 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.
 Attachments

 Football.jpg (77.61 KiB) Viewed 8947 times

 Football2.jpg (62.51 KiB) Viewed 8947 times
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