How to make Isosurfaces with "tickness" ? Goto page 1, 2  Next
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nextstep

Joined: 06 Jan 2007
Posts: 539

 Posted: Wed Jan 10, 2007 6:04 am    Post subject: How to make Isosurfaces with "tickness" ? Hi all, The ultimate formulas for tickness is : G[x, y, z] = F[x, y, z] * F[x - (T/R)*dF()/dx, y - (T/R)*dF()/dy, z - (T/R)*df()/dz] Where : dF()/dx == partial derivative of F() to the variable x. R = sqrt[(dF()/dx)^2 + (dF()/dy)^2 + (dF()/dz)^2] T = Tickness value Applied to Schwartz-P : F():(cos(x) + cos(y) + cos(z))*((cos(x + sin(x)/2.3) + cos(y + sin(y)/2.3) + cos(z + sin(z)/2.3))) [x, y, z]: -7, 7 _________________Cheers, Abderrahman
tomot

Joined: 21 Jul 2007
Posts: 23
Location: Vancouver

 Posted: Fri Jul 27, 2007 4:02 pm    Post subject: nextstep: does one need to use a program such as Mathmatica or Maple to do the calculus that produce thickness for other surfaces? if the answer is No to the above question; How do I implement the code for your "Ultimate thickness formula" into a the .k3ds file Please explain a little bit more about the process. thanks!
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Jul 29, 2007 1:04 am    Post subject:

Hi,
 Code: does one need to use a program such as Mathematica or Maple to do the calculus that produce thickness for other surfaces?

If you want to make tick surfaces, you can use some 3D program (AFAIK Hexagon can do it) : This is how most 3D artists are generating them. Using my formula can be hard especially if you're not familiar with this kind of exercises . Also, there is no way to implement it in a .k3ds file because K3DSurf can't generate the appropriate formulas for your surface.
In short, i suggest you to look for informations on which 3D program (like Hexagon, Maya...) that have implemented this kind of feature. Hope it's clear enought...
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Abderrahman
inode

Joined: 27 Jan 2007
Posts: 127
Location: Austria

 Posted: Sun Nov 04, 2007 11:39 am    Post subject: Another simple way to make a surface thick In some cases it's very easy to created a thick surface by squaring those part of the formular which is describing the main surface. For example the formula of a hyperboloid is: x*x - y*y + z Squaring it and subtract it from a "thickness value" will give the desired formula: 4 -(x*x - y*y + z)^2 Gerd example 2: Sphere: 1-(x^2+y^2+z^2) Hollow Sphere: 0.05-(1-(x^2+y^2+z^2))^2 example 3: Rounded Cube: 1-(x^22 + y^22 + z^22) Hollow Rounded Cube: 0.8-(1-(x^22 + y^22 + z^22))^2
Schmiegel

Joined: 28 Nov 2009
Posts: 26

 Posted: Mon Nov 30, 2009 10:03 pm    Post subject: Thanks for allowing me to participate in this forum I was digging out this old thread because it deals best with the problem I'm having. I would like to do 3d renders of isosurfaces. Following the above instructions I succeed to create an object having thickness - however if I try this for infinite surfaces (my favourite is the gyroid) I end up with an open object resulting from K3D clipping a limited cube out of infinity (see also image at top). So I cannot intersect etc the resulting object in 3D apps. Now, the thread is old - is hopefully somebody aware of a freeware app or method that currently can do the job of adding a third dimension to 2D isosurfaces - or can close the 'open' faces of an clipped object. P.S.: I have seen the hint for Hexagon but i'm just a 3d playchild so it is a bit expensive ...
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Mon Nov 30, 2009 11:16 pm    Post subject:

Hi all,
Welcome Schmiegel in the forum
There is a way to make a closed and tick isosurface: in K3DSurf, the example "CloseIso" shows how to do it for a simple surface and the example "CloseIso_2" shows how to do it for a tick isosurface.
From K3DSurf:
CloseIso
 Quote: /* To make a closed Isosurface, you can use the "if" instruction like in this example with Schwartz : if( (x^10 + y^10 +z^10 < 200000), // We use a Cube as a condition -(cos(x) + cos(y) + cos(z) ) , // Schwartz (x^10 + y^10 +z^10 - 200000 ) // Cube */ if((x^10 + y^10 +z^10 < 200000), -(cos(x) + cos(y) + cos(z) ) , (x^10 + y^10 +z^10 - 200000))

CloseIso_2
 Quote: /* And now, to make a tick and closed Schwartz Isosurface, we use the two formulas described above : */ if((x^10 + y^10 +z^10 < 3*(3.5^10)), (cos(x) + cos(y) + cos(z))*((cos(x + sin(x)/(2*sqrt(sin(x)^2 + sin(y)^2 + sin(z)^2))) +cos(y + sin(y)/(2*sqrt(sin(x)^2 + sin(y)^2 + sin(z)^2))) +cos(z + sin(z)/(2*sqrt(sin(x)^2 + sin(y)^2 + sin(z)^2))))) , (x^10 + y^10 +z^10 - 3*(3.5^10)))

No need for a budget software to do this
Hope this can help.
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Abderrahman
Schmiegel

Joined: 28 Nov 2009
Posts: 26

 Posted: Tue Dec 01, 2009 10:49 pm    Post subject: Merci beaucoup Taha! This was the information I was looking for. And it worked for me. As I am an optimist I made that: http://www.flickr.com/groups/1304045@N23/pool/ I hope it fills soon with some art. I have added there my first result - going in a totally different direction as expected but that's typical for play children Thanks again!
sudhirsinghgill

Joined: 31 Jul 2012
Posts: 1
Location: India

 Posted: Wed Aug 01, 2012 8:33 am    Post subject: Thickening surface Friends I have easy method to thicken the surface. First of all special thanks to admin to approve my request to join you guys. I also tried the method of using function to thicken the surface and somewhat successful. I tried many software for that most of them are not easy and failure. Now I uses Blender to thicken the surface. I import the OBJ exported from the K3DSurf and then use Solidify Modifier with thickness. This works perfectly and then I export it as STL/OBJ. Regards, Sudhir Gill _________________How quickly they forget that all it takes to change the course of history is the will of a single man.
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Wed Oct 08, 2014 1:21 am    Post subject:

Hi all,
For the Gyroid, the formula for thickness T = 1:

 Quote: { "Iso3D": { "Name": [ "Gyroid" ], "Component": [ "Gyroid" ], "Fxyz": [ "(cos(x)*sin(y)+cos(y)*sin(z)+cos(z)*sin(x))*(cos(x-(-sin(x)*sin(y)+cos(x)*cos(z))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(y-(-sin(y)*sin(z)+cos(y)*cos(x))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(y-(-sin(y)*sin(z)+cos(y)*cos(x))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(z-(-sin(z)*sin(x)+cos(z)*cos(y))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(z-(-sin(z)*sin(x)+cos(z)*cos(y))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(x-(-sin(x)*sin(y)+cos(x)*cos(z))/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2)))" ], "Xmax": [ "10" ], "Xmin": [ "-10" ], "Ymax": [ "10" ], "Ymin": [ "-10" ], "Zmax": [ "10" ], "Zmin": [ "-10" ] } }

TickGyroid by taha_ab, on Flickr

I didn't try to make the formula more compact but I'm open to all your suggestions
In the near future, MathMod's scripting language will support a better formulation for this kind of equations.
F(x,y,z) = (cos(x)*sin(y)+cos(y)*sin(z)+cos(z)*sin(x))
dF/dx = (-sin(z)*sin(x) + cos(z)*cos(y))
dF/dy = (-sin(y)*sin(z) + cos(y)*cos(x))
dF/dz = (-sin(z)*sin(x) + cos(z)*cos(y))

R = sqrt[(dF()/dx)^2 + (dF()/dy)^2 + (dF()/dz)^2]
T = Tickness value = 1
Also you can play with the formula by making it a "closed" isosurface
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Abderrahman
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Wed Oct 08, 2014 10:04 am    Post subject:

Hi all,
when extracting the Gyroid equation from the above implicit surface, you will have a triply periodique latice equation (F[x - (T/R)*dF()/dx, y - (T/R)*dF()/dy, z - (T/R)*df()/dz] )

You can play with the this equation by given T some values between {-1, 1}
This another example is made with two triply periodic latice, for (T =1 , T=G=-1) and (T =.2 , T=G=-.2)
Now, the big question, what will happen if you make T = T(x,y,z) ? That should be a good way to make quite complicated pattern on a surface

 Quote: { "Iso3D": { "Name": [ "Gyroid" ], "Component": [ "Gyroid" ], "Const": [ "T = 1", "G = -1" ], "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 8" ], "Fxyz": [ "((cos(x-(-sin(x)*sin(y)+cos(x)*cos(z))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(y-(-sin(y)*sin(z)+cos(y)*cos(x))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(y-(-sin(y)*sin(z)+cos(y)*cos(x))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(z-(-sin(z)*sin(x)+cos(z)*cos(y))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(z-(-sin(z)*sin(x)+cos(z)*cos(y))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(x-(-sin(x)*sin(y)+cos(x)*cos(z))*T/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2)))) * ( (cos(x-(-sin(x)*sin(y)+cos(x)*cos(z))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(y-(-sin(y)*sin(z)+cos(y)*cos(x))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(y-(-sin(y)*sin(z)+cos(y)*cos(x))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(z-(-sin(z)*sin(x)+cos(z)*cos(y))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))+cos(z-(-sin(z)*sin(x)+cos(z)*cos(y))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))*sin(x-(-sin(x)*sin(y)+cos(x)*cos(z))*G/sqrt((-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2))) ) " ], "Xmax": [ "10" ], "Xmin": [ "-10" ], "Ymax": [ "10" ], "Ymin": [ "-10" ], "Zmax": [ "10" ], "Zmin": [ "-10" ] } }

PeriodiqueLatice1 by taha_ab, on Flickr

PeriodiqueLatice by taha_ab, on Flickr
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Abderrahman
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Fri Oct 10, 2014 2:20 am    Post subject:

Hi,
A more compact version of the last script:
 Quote: { "Iso3D": { "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 8" ], "Component": [ "GyroidLatice" ], "Const": [ "T = 1", "G = -1" ], "Funct": [ "R=sqrt( (-sin(x)*sin(y)+cos(x)*cos(z))^2+(-sin(y)*sin(z)+cos(y)*cos(x))^2+(-sin(z)*sin(x)+cos(z)*cos(y))^2 )", "df=(-sin(x)*sin(y)+cos(x)*cos(z))", "Gyroid=(cos(x) * sin(y) + cos(y) * sin(z) + cos(z) * sin(x))" ], "Fxyz": [ "Gyroid(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* Gyroid(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Name": [ "GyroidLatice" ], "Xmax": [ "8" ], "Xmin": [ "-8" ], "Ymax": [ "8" ], "Ymin": [ "-8" ], "Zmax": [ "8" ], "Zmin": [ "-8" ] } }

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Abderrahman
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Fri Oct 10, 2014 3:56 am    Post subject:

Hi,
With the Diamond implicit surface, I get something a little bit different:
 Quote: { "Iso3D": { "Name": [ "DiamondLatice" ], "Component": [ "DiamondLatice" ], "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 8" ], "Const": [ "T = 1", "G = -1" ], "Funct": [ "df=(cos(x)*sin(y)*sin(z) + cos(x)*cos(y)*cos(z) - sin(x)*sin(y)*cos(z) - sin(x)*cos(y)*sin(z))", "R=sqrt( df(x,y,z,t)^2 + df(y,z,x,t)^2 + df(z,y,x,t)^2)", "Diamond=(sin(x)*sin(y)*sin(z) + sin(x)*cos(y)*cos(z) + cos(x)*sin(y)*cos(z) + cos(x)*cos(y)*sin(z))" ], "Fxyz": [ "Diamond(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* Diamond(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Xmax": [ "8" ], "Xmin": [ "-8" ], "Ymax": [ "8" ], "Ymin": [ "-8" ], "Zmax": [ "8" ], "Zmin": [ "-8" ] } }

DiamondLatice by taha_ab, on Flickr

If the first script doesn't work, try this one:

 Quote: { "Iso3D": { "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 8" ], "Component": [ "DiamondLatice" ], "Const": [ "T = 1", "G = -1" ], "Funct": [ "df=(cos(x)*sin(y)*sin(z) + cos(x)*cos(y)*cos(z) - sin(x)*sin(y)*cos(z) - sin(x)*cos(y)*sin(z))", "R=sqrt(( cos(x)*sin(y)*sin(z) + cos(x)*cos(y)*cos(z) - sin(x)*sin(y)*cos(z) - sin(x)*cos(y)*sin(z))^2 + ( cos(y)*sin(z)*sin(x) + cos(y)*cos(z)*cos(x) - sin(y)*sin(z)*cos(x) - sin(y)*cos(z)*sin(x))^2 + ( cos(z)*sin(x)*sin(y) + cos(z)*cos(x)*cos(y) - sin(z)*sin(x)*cos(y) - sin(z)*cos(x)*sin(y))^2)", "Diamond=(sin(x)*sin(y)*sin(z) + sin(x)*cos(y)*cos(z) + cos(x)*sin(y)*cos(z) + cos(x)*cos(y)*sin(z))" ], "Fxyz": [ "Diamond(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* Diamond(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Name": [ "DiamondLatice" ], "Xmax": [ "8" ], "Xmin": [ "-8" ], "Ymax": [ "8" ], "Ymin": [ "-8" ], "Zmax": [ "8" ], "Zmin": [ "-8" ] } }

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Abderrahman
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sat Oct 11, 2014 10:58 pm    Post subject: Linoid Latice

Hi all,
This is what you get when given the Linoid some thickness:

 Quote: { "Iso3D": { "Name": [ "LinoidLatice" ], "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 3" ], "Component": [ "LinoidLatice" ], "Const": [ "T = .05", "G = -.05" ], "Funct": [ "df=((1/2)*(2*cos(2*x)*cos(y)*sin(z) + sin(2*y)*cos(z)*cos(x) - sin(2*z)*sin(x)*sin(y)) -(1/2)*(-2*sin(2*x)*cos(2*y) - 2*cos(2*z)*sin(2*x)))", "R=sqrt( df(x,y,z,t)^2 + df(y,z,x,t)^2 + df(z,y,x,t)^2)", "Linoid=((1/2)*(sin(2*x)*cos(y)*sin(z) + sin(2*y)*cos(z)*sin(x)+ sin(2*z)*cos(x)*sin(y)) -(1/2)*(cos(2*x)*cos(2*y)+cos(2*y)*cos(2*z)+cos(2*z)*cos(2*x))+0.15)" ], "Fxyz": [ "Linoid(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* Linoid(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Xmax": [ "3" ], "Xmin": [ "-3" ], "Ymax": [ "3" ], "Ymin": [ "-3" ], "Zmax": [ "3" ], "Zmin": [ "-3" ] } }

LinoidLatice by taha_ab, on Flickr
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Abderrahman
nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Oct 12, 2014 3:10 am    Post subject:

Hi,
For the Neovius implicit surface, you get this:
 Quote: { "Iso3D": { "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 9" ], "Component": [ "NeoviusLatice" ], "Const": [ "T = .1", "G = -.1" ], "Funct": [ "df=(-3*sin(x) -4*sin(x)*cos(y)*cos(z))", "R=sqrt( df(x,y,z,t)^2 + df(y,z,x,t)^2 + df(z,y,x,t)^2)", "Neovius=(3*(cos(x)+cos(y)+cos(z))+4*cos(x)*cos(y)*cos(z))" ], "Fxyz": [ "Neovius(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* Neovius(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Name": [ "NeoviusLatice" ], "Xmax": [ "9" ], "Xmin": [ "-9" ], "Ymax": [ "9" ], "Ymin": [ "-9" ], "Zmax": [ "9" ], "Zmin": [ "-9" ] } }

NeoviusLatice by taha_ab, on Flickr
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nextstep

Joined: 06 Jan 2007
Posts: 539

Posted: Sun Oct 12, 2014 8:46 am    Post subject:

Hi,
For the Schwarz P surface, we get this:

 Quote: { "Iso3D": { "Name": [ "SchwarzSkelet" ], "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 9" ], "Component": [ "SchwarzPSkelet" ], "Const": [ "T = .9", "G = -.9" ], "Funct": [ "df=(-sin(x))", "R=sqrt( df(x,y,z,t)^2 + df(y,z,x,t)^2 + df(z,y,x,t)^2)", "SchwarzP=(cos(x) + cos(y) + cos(z))" ], "Fxyz": [ "SchwarzP(x-df(x,y,z,t)*T/R(x,y,z,t), y - df(y,z,x,t)*T/R(x,y,z,t), z - df(z,x,y,t)*T/R(x,y,z,t) ,t)* SchwarzP(x-df(x,y,z,t)*G/R(x,y,z,t), y - df(y,z,x,t)*G/R(x,y,z,t), z - df(z,x,y,t)*G/R(x,y,z,t) ,t)" ], "Xmax": [ "9" ], "Xmin": [ "-9" ], "Ymax": [ "9" ], "Ymin": [ "-9" ], "Zmax": [ "9" ], "Zmin": [ "-9" ] } }

SchwarzP by taha_ab, on Flickr

For MathMod's old version you should use this script:
 Code: { "Iso3D": { "Name": [ "SchwarzSkelet" ], "Cnd": [ " (sqrt(x^2 + y ^2 + z ^2)) < 9" ], "Component": [ "SchwarzPSkelet" ], "Const": [ "T = .9", "G = -.9" ], "Funct": [ "df1=(-sin(x))", "df2=(-sin(y))", "df3=(-sin(z))", "R=sqrt( df1(x,y,z,t)^2 + df2(x,y,z,t)^2 + df3(x,y,z,t)^2)", "SchwarzP=(cos(x) + cos(y) + cos(z))" ], "Fxyz": [ "SchwarzP(x-df1(x,y,z,t)*T/R(x,y,z,t), y - df2(x,y,z,t)*T/R(x,y,z,t), z - df3(x,y,z,t)*T/R(x,y,z,t) ,t)* SchwarzP(x-df1(x,y,z,t)*G/R(x,y,z,t), y - df2(x,y,z,t)*G/R(x,y,z,t), z - df3(x,y,z,t)*G/R(x,y,z,t) ,t)" ], "Xmax": [ "9" ], "Xmin": [ "-9" ], "Ymax": [ "9" ], "Ymin": [ "-9" ], "Zmax": [ "9" ], "Zmin": [ "-9" ] } }

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