Friday, December 5, 2014

[UE4 Material] Graphing Functions


Being interested in math and shaders I recently got into shadertoy and made a shader to draw some interesting looking implicit functions. This was fun but it was also just a matter of time until I'd do something similar using Unreal Engine.
So here we go, a tutorial about graphing a mathematical function with the UE3/4 material editor alone. There are no textures involved, everything is just done in the pixel shader.


In order to actually draw the graphs I'm using a method suggested by Iñigo Quílez:
http://www.iquilezles.org/www/articles/distance/distance.htm

The shader code described isn't very complex but requires the derivatives to be computed.
So in total there are 4 things required for graphing a function in the material editor:
  1. A function.
    Any kind of function will work but the elements need to be reordered so one
    side of the equation is zero. E.g.  y=x²  will become  0=x²-y.
  2. Derivatives of the function.
    It's recommended to to manually do the differentiation. However if the math
    is out of reach you can use finite differences instead.
  3. A coordinate system.
    Usually this will be the UVs, screen coordinates or the worldposition.
  4. A grid.
    This is optionally but helps to read the graph. I'd suggest to draw the x/y-axis at the very least.



Material Setup

I'm starting off with a rather simple function:
0 = x³-y²

The derivatives of this function are as follows:
d/dx = 3*x²
d/dy = -2*y

The coordinate system is based on the UVs. Since these range from 0-1 for the preview meshes a negative offset of -0.5 is added to the texture coordinates.



If you prefer an automated process over manual differentiation you can use forward differencing to calculate the derivatives. This will calculate the function 2 additional times with a slight offset on the x- and y-axis.
Keep in mind that this might increase the instruction count a lot, depending on the functions complexity.




With the function and its derivatives available we can now draw the graph:



Visalizing the x/y-axis is pretty similar to drawing the graph but much easier at the same time since it's directly based on the absolutes of the coordinates.



For a real grid with multiple sub-lines use this setup instead:




Putting it all together


To replicate this network of nodes just copy and paste the code below to your own material:

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   Begin Object Class=MaterialExpressionComment Name="MaterialExpressionComment_0"
   End Object
   Begin Object Name="MaterialExpressionComment_0"
      SizeX=1136
      SizeY=432
      Text="Draw Grid"
      MaterialExpressionEditorX=1648
      MaterialExpressionEditorY=-3088
      MaterialExpressionGuid=28366D4648509726F5EC76959F7B50DC
      Material=PreviewMaterial'/Engine/Transient.PreviewMaterial_1'
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   MaterialExpressionComment=MaterialExpressionComment'MaterialExpressionComment_0'
   NodePosX=1648
   NodePosY=-3088
   NodeWidth=1136
   NodeHeight=432
   NodeComment="Draw Grid"
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   Begin Object Class=MaterialExpressionComment Name="MaterialExpressionComment_1"
   End Object
   Begin Object Name="MaterialExpressionComment_1"
      SizeX=880
      SizeY=448
      Text="Draw Graph"
      MaterialExpressionEditorX=1936
      MaterialExpressionEditorY=-2640
      MaterialExpressionGuid=28366D4648509726F5EC76959F7B50DC
      Material=PreviewMaterial'/Engine/Transient.PreviewMaterial_1'
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   MaterialExpressionComment=MaterialExpressionComment'MaterialExpressionComment_1'
   NodePosX=1936
   NodePosY=-2640
   NodeWidth=880
   NodeHeight=448
   NodeComment="Draw Graph"
   NodeGuid=C58F85374D951E9D5D1C24B70B2F0FAF
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   End Object
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      SizeY=256
      Text="Derivatives"
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      MaterialExpressionEditorY=-2400
      MaterialExpressionGuid=34357E4D47B94A25FCB75C98A5F1DFF2
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   MaterialExpressionComment=MaterialExpressionComment'MaterialExpressionComment_2'
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   NodePosY=-2400
   NodeWidth=304
   NodeHeight=256
   NodeComment="Derivatives"
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      PinFriendlyName=" "
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Posted by at

1 comment:

  1. WerwackSeptember 17, 2015 at 3:24 PM

    Hi Oliver,

    Thank you for sharing this. The approach is really interesting.
    I have not implemented it but I took several ideas to develop my own function shader. I needed something like that to check the values of variables in a shader. My shader is limited to bijective functions though.
    http://werwackfx.com/index.php/graphictools/ue4-main/ue4-articles/35-shaderwkfunctiondrawer

    Best regards

    ReplyDelete

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