Gradients based on tangent function?

I have the following screen capture from SculptyPaint, which is used to create a mesh of 1024 points on a grid using RGB channels to represent a position within 3-space on the X, Y and Z axis:



I would like to use SculptyPaint to recreate basic shapes, deform the shapes and then use the same deformation process to create the surface texture to be applied to the mesh.

As you can see, the green channel is a simple linear profile gradient. The blue is just the red channel 50% offset. The outside half of the red channel is an inversion of the inside half.



How do I get the inside half? I can 'see' the profile shape in GIMP just by using threshold and running it up and down. This is a round gradient that starts as an ellipse, but as it gets towards the edges it stretches at the corners until it just matches the outline of a box at 0x808080h (flat grey).

If only I understood how to create a gradient as you see it in the above image, I could control the gradient to make other shapes as well as deformations and the surface texture map that would go over the shape. Thanks to anyone who can guide me in this matter.