ICE Topology and parametric equations
- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
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Re: ICE Topology and parametric equations
Egg Carton
This is a simple variation of the grid using sines.
The parameters are:
x = u
y = A * Sin(u) * Sin (v)
z = v
with
u_start = -10
u_end = 10
v_start = -10
v_end = 10
A = 2 (amplitude of the bump on the grid surface)
The ICE Tree and parametric surface is as follow.
This is a simple variation of the grid using sines.
The parameters are:
x = u
y = A * Sin(u) * Sin (v)
z = v
with
u_start = -10
u_end = 10
v_start = -10
v_end = 10
A = 2 (amplitude of the bump on the grid surface)
The ICE Tree and parametric surface is as follow.
- Attachments
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- Grid_Egg_Carton.jpg (62.36 KiB) Viewed 4114 times
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- Grid_Egg_Carton2.jpg (99.68 KiB) Viewed 4114 times
Last edited by Daniel Brassard on 11 Jan 2012, 03:13, 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
Folded Strip
This next surface uses sine and cosine functions to fold the grid. The surface is useful for making bow ties and candies.
The parameters are:
x = Sin(u)
y = Cos(u+v)
z = v
with
u_start = 0 ... (if you make this -2*PI you get the bow tie)
u_end = 2*PI
v_start = -PI
v_end = PI
The ICE Tree and parametric surface is as follow.
This next surface uses sine and cosine functions to fold the grid. The surface is useful for making bow ties and candies.
The parameters are:
x = Sin(u)
y = Cos(u+v)
z = v
with
u_start = 0 ... (if you make this -2*PI you get the bow tie)
u_end = 2*PI
v_start = -PI
v_end = PI
The ICE Tree and parametric surface is as follow.
- Attachments
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- Folded_Strip.jpg (57.25 KiB) Viewed 4113 times
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- Folded_Strip2.jpg (69.8 KiB) Viewed 4113 times
Last edited by Daniel Brassard on 11 Jan 2012, 03:18, 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
Folded Strip Variation
This next version of the folded strip uses the absolute function to fold the strip on itself along the z axis creating an interesting shape.
The parameters are:
x = Sin(u)
y = Cos(u+v)
z = abs(v) / 2
with
u_start = 0
u_end = 2*PI
v_start = -20
v_end = 20
The ICE Tree and parametric surface is as follow.
This next version of the folded strip uses the absolute function to fold the strip on itself along the z axis creating an interesting shape.
The parameters are:
x = Sin(u)
y = Cos(u+v)
z = abs(v) / 2
with
u_start = 0
u_end = 2*PI
v_start = -20
v_end = 20
The ICE Tree and parametric surface is as follow.
- Attachments
-
- Folded_Strip4.jpg (63.98 KiB) Viewed 4108 times
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- Folded_Strip3.jpg (89.34 KiB) Viewed 4110 times
Last edited by Daniel Brassard on 11 Jan 2012, 04:11, 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
Grid Ripple
This variation of the grid uses the sine function to create ripples on the surface. The amount of ripples is controlled by a scalar.
The parameters are:
x = u
y = Sin(u*v/Ripple)
z = v
with
u_start = -10
u_end = 10
v_start = -10
v_end = 10
Ripple = 5 (can be any amount except zero to avoid a "division by zero" error)
The ICE Tree and parametric surface is as follow.
This variation of the grid uses the sine function to create ripples on the surface. The amount of ripples is controlled by a scalar.
The parameters are:
x = u
y = Sin(u*v/Ripple)
z = v
with
u_start = -10
u_end = 10
v_start = -10
v_end = 10
Ripple = 5 (can be any amount except zero to avoid a "division by zero" error)
The ICE Tree and parametric surface is as follow.
- Attachments
-
- Grid_Ripple.jpg (63.49 KiB) Viewed 4110 times
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- Grid_Ripple2.jpg (108.84 KiB) Viewed 4110 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
Conical Spiral Disk (Cochlea)
Here is a variation of the spiral disk using u to deform the disk in a conical shape. The height of the disk on the y axis is controlled by a scalar.
The parameters are:
x = u * v * Sin(u)
y = Height * u
z = u * v * Cos(u)
with
u_start = -8*PI
u_end = 8*PI
v_start = 0
v_end = 1
Height = 1
The ICE Tree and parametric surface is as follow.
Here is a variation of the spiral disk using u to deform the disk in a conical shape. The height of the disk on the y axis is controlled by a scalar.
The parameters are:
x = u * v * Sin(u)
y = Height * u
z = u * v * Cos(u)
with
u_start = -8*PI
u_end = 8*PI
v_start = 0
v_end = 1
Height = 1
The ICE Tree and parametric surface is as follow.
- Attachments
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- Conical_Spiral_Disk.jpg (67.08 KiB) Viewed 4108 times
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- Conical_Spiral_Disk2.jpg (71.07 KiB) Viewed 4108 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
Torus Spiral (Coil)
A variation of the spiral helical torus with a constant width and no height. Useful to create clock springs, hoses and ornements.
The parameters are:
x = Cos(u) * (R0 + (R1 * Cos(v)))
y = R1 * Sin(v)
z = Sin(u) * (R0 + (R1 * Cos(v)))
with
u_start = 0
u_end = 8*PI
v_start = 0
v_end = 2*PI
R0 = Large radius = .5
R1 = Small radius = .5
The ICE Tree and parametric surface is as follow.
A variation of the spiral helical torus with a constant width and no height. Useful to create clock springs, hoses and ornements.
The parameters are:
x = Cos(u) * (R0 + (R1 * Cos(v)))
y = R1 * Sin(v)
z = Sin(u) * (R0 + (R1 * Cos(v)))
with
u_start = 0
u_end = 8*PI
v_start = 0
v_end = 2*PI
R0 = Large radius = .5
R1 = Small radius = .5
The ICE Tree and parametric surface is as follow.
- Attachments
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- Torus_Spiral_Coil.jpg (74.66 KiB) Viewed 4107 times
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- Torus_Spiral_Coil2.jpg (82.8 KiB) Viewed 4107 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
Whitney's Umbrella
An interesting surface with intersecting polygons. The parameters are so simple.
They are:
x = u
y = v * v
z = u * v
with
u_start = -1
u_end = 1
v_start = -1
v_end = 1
The ICE Tree and parametric surface is as follow.
An interesting surface with intersecting polygons. The parameters are so simple.
They are:
x = u
y = v * v
z = u * v
with
u_start = -1
u_end = 1
v_start = -1
v_end = 1
The ICE Tree and parametric surface is as follow.
- Attachments
-
- Whitney_umbrella.jpg (58.04 KiB) Viewed 4742 times
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- Whitney_umbrella2.jpg (67.09 KiB) Viewed 4742 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
Little Update, I am working on the next set of ice surface posts. Here is a preview of the braided torus for your enjoyment!
And no, this is not the torus wireframe, this is the actual surface!
And no, this is not the torus wireframe, this is the actual surface!
- Attachments
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- Torus_Braided2.jpg (73.32 KiB) Viewed 4691 times
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- Torus_Braided3.jpg (83.23 KiB) Viewed 4691 times
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- Torus_Braided4.jpg (54.72 KiB) Viewed 4691 times
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Re: ICE Topology and parametric equations
Great stuff. I hope to get this working in the near future after I finish a couple of projects I'm currently working on. Haven't had the chance to really take a look at ICE Modelling yet.
I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html
Great stuff - keep up the hard work!
-Dave
I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html
Great stuff - keep up the hard work!
-Dave
- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations
Hi Dave,
Thank you for your kind encouragement.
maybe emPolygoniser can help!
Maybe something to propose in the community project thread! That would be an interesting project!
Thank you for your kind encouragement.
For that, you will need some sort of ray marcher / marching cube / voxelizer method ....I can't wait till you generate 3d Mandelbulb!!
http://www.skytopia.com/project/fractal/mandelbulb.html
maybe emPolygoniser can help!
Maybe something to propose in the community project thread! That would be an interesting project!
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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
Little update!
Just created a serie of supershapes based on the super torus.
Here is a rendered preview.
The SuperCone, SuperSphere, SuperCylinder, SuperDisk and SuperTorus!
Each shape can adopt many variations by adjusting its parameters! As an example, the rounded cube in the middle is actually the SuperSphere!
Just created a serie of supershapes based on the super torus.
Here is a rendered preview.
The SuperCone, SuperSphere, SuperCylinder, SuperDisk and SuperTorus!
Each shape can adopt many variations by adjusting its parameters! As an example, the rounded cube in the middle is actually the SuperSphere!
- Attachments
-
- SuperShapes.jpg (59.34 KiB) Viewed 4610 times
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Re: ICE Topology and parametric equations
Very nice indeed! I've fallen behind in trying it myself (have to get some work done) but really admire what you've achieved here. Looking forward to more!
- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations
I have been experimenting with the super formula lately. Here are some Super Shapes based on Paul Bourke parameters on his website!
http://paulbourke.net/geometry/supershape3d/
All the shapes are created with the same compound by adjusting its parameters. You can get some very cool animation too!
Enjoy!
http://paulbourke.net/geometry/supershape3d/
All the shapes are created with the same compound by adjusting its parameters. You can get some very cool animation too!
Enjoy!
- Attachments
-
- Supershapes_3D.jpg (52.78 KiB) Viewed 4409 times
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- SuperShape_3d.jpg (84.04 KiB) Viewed 4409 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
SuperShape Compound
I have been busy at the studio lately and I have to prepare a portrait class, so in the meantime, i'll give you the supershape compound to play with. It is based on the sphere formula but it does not have to be.
It's not pretty, I have not done the PPG Logic yet, but it can show you the power of this compound.
There is still some work to do to improve it (better layout, the superformula break at high enough numbers) but we can create so many shapes with it that it is worth playing with, even at it's early stage.
Inside, you will find my early version of the super formula. Copy this node and put it in an other polar coordinate shape like the disk, cylinder or cone and see what you can do with it. It's not called the super formula for nothing!
So have fun, check Paul Bourke website and other sites that have posted super shapes settings and see what you can reproduce.
This compound is fun to play with and animate! Enjoy!
I have been busy at the studio lately and I have to prepare a portrait class, so in the meantime, i'll give you the supershape compound to play with. It is based on the sphere formula but it does not have to be.
It's not pretty, I have not done the PPG Logic yet, but it can show you the power of this compound.
There is still some work to do to improve it (better layout, the superformula break at high enough numbers) but we can create so many shapes with it that it is worth playing with, even at it's early stage.
Inside, you will find my early version of the super formula. Copy this node and put it in an other polar coordinate shape like the disk, cylinder or cone and see what you can do with it. It's not called the super formula for nothing!
So have fun, check Paul Bourke website and other sites that have posted super shapes settings and see what you can reproduce.
This compound is fun to play with and animate! Enjoy!
- Attachments
-
- SuperShape_Sphere.xsicompound
- (203.8 KiB) Downloaded 187 times
$ifndef "Softimage"
set "Softimage" "true"
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$endif
- Daniel Brassard
- Posts: 878
- Joined: 18 Mar 2010, 23:38
- Location: St. Thomas, Ontario
- Contact:
Re: ICE Topology and parametric equations
The SuperFormula
The superformula is well explained on Paul Bourke website and on Wikipedia here:
http://en.wikipedia.org/wiki/Superformula
In polar coordinates, with r the radius and φ the angle, the superformula is as per the first figure below from Wikipedia.
The formula appeared in a work by Johan Gielis in 2003. It was obtained by generalizing the superellipse in order to create organic shapes. The superformula can be used to describe many complex shapes and curves that are found in nature by inserting the formula into standard polar coordinate shapes such as the sphere, cylinder, disk, torus or cone.
Example – The sphere.
Remember the sphere second mapping:
x = r * Cos(u) * Cos (v)
y = r * Sin(v)
z = r * Sin(u) * Cos (v)
if you replace the r parameter with the superformula in u, v or both you get the supershapes in 3D presented in Paul Bourke website. The formula becomes:
In u:
R(u) = second figure
And in v
R(v) = third figure
This gives us three variations of the supershape sphere formula:
U only:
x = R(u) * Cos(u) * Cos (v)
y = Sin(v)
z = R(u) * Sin(u) * Cos (v)
v only:
x = Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = Sin(u) * R(v) * Cos (v)
both u and v, which is the general representation seen on various websites and the basis of the Supershape_Sphere ICE Compound:
x = R(u) * Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = R(u) * Sin(u) * R(v) * Cos (v)
The parameters:
M is the angular coefficient multiplier. If m = 0 we get the sphere representation. If m is an integer, the shape is closed. If m is a fraction, the shape is open but we can rotate u or v multiple times to create a petal/star shape with intersecting polygons. M being a multiplier will control the number of points in the shape (wave or star like shape), example m=5 will create a shape with five bumps or start points.
a and b are the radius controls. They can be any numbers except zero (to avoid a division by zero error). Negative numbers need to be converted to positive using an absolute to avoid imaginary numbers issue.
n1, n2 and n3 are the exponent controlling the supershape. This is where the magic happen. They can be any numbers, positive or negatives. Small values have the greatest impact. Note that if you make n1 equal to zero, x, y and z will be equal to zero.
So google up "superformula" and "supershape", feed those parameters you found into the supershape formula, give it some colours and see how flexible the superformula is.
Cheers!
The superformula is well explained on Paul Bourke website and on Wikipedia here:
http://en.wikipedia.org/wiki/Superformula
In polar coordinates, with r the radius and φ the angle, the superformula is as per the first figure below from Wikipedia.
The formula appeared in a work by Johan Gielis in 2003. It was obtained by generalizing the superellipse in order to create organic shapes. The superformula can be used to describe many complex shapes and curves that are found in nature by inserting the formula into standard polar coordinate shapes such as the sphere, cylinder, disk, torus or cone.
Example – The sphere.
Remember the sphere second mapping:
x = r * Cos(u) * Cos (v)
y = r * Sin(v)
z = r * Sin(u) * Cos (v)
if you replace the r parameter with the superformula in u, v or both you get the supershapes in 3D presented in Paul Bourke website. The formula becomes:
In u:
R(u) = second figure
And in v
R(v) = third figure
This gives us three variations of the supershape sphere formula:
U only:
x = R(u) * Cos(u) * Cos (v)
y = Sin(v)
z = R(u) * Sin(u) * Cos (v)
v only:
x = Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = Sin(u) * R(v) * Cos (v)
both u and v, which is the general representation seen on various websites and the basis of the Supershape_Sphere ICE Compound:
x = R(u) * Cos(u) * R(v) * Cos (v)
y = R(v) * Sin(v)
z = R(u) * Sin(u) * R(v) * Cos (v)
The parameters:
M is the angular coefficient multiplier. If m = 0 we get the sphere representation. If m is an integer, the shape is closed. If m is a fraction, the shape is open but we can rotate u or v multiple times to create a petal/star shape with intersecting polygons. M being a multiplier will control the number of points in the shape (wave or star like shape), example m=5 will create a shape with five bumps or start points.
a and b are the radius controls. They can be any numbers except zero (to avoid a division by zero error). Negative numbers need to be converted to positive using an absolute to avoid imaginary numbers issue.
n1, n2 and n3 are the exponent controlling the supershape. This is where the magic happen. They can be any numbers, positive or negatives. Small values have the greatest impact. Note that if you make n1 equal to zero, x, y and z will be equal to zero.
So google up "superformula" and "supershape", feed those parameters you found into the supershape formula, give it some colours and see how flexible the superformula is.
Cheers!
- Attachments
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- superformula.jpg (5.32 KiB) Viewed 4304 times
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- superformula_U.jpg (5.07 KiB) Viewed 4304 times
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- superformula_V.jpg (5.17 KiB) Viewed 4304 times
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Re: ICE Topology and parametric equations
hey Daniel, any chance you could share the braided torus? i'm really interested into this shape, would be very nice!
thanks again for your awesome work on this
Max
thanks again for your awesome work on this
Max
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