Interpolating a scalar field in a 3D space

Question

I have a 3D space (x, y, z) with an additional parameter at each point (energy), giving 4 dimensions of data in total.

I would like to find a set of x, y, z points w

Answer 1

Since you have a spatial mesh with constant spacing, you can identify all neighbors on opposite sides of the isosurface. Choose some form of interpolation (q.v. Reed Copsey's answer) and do root-finding along the line between each such neighbor.

Answer 2

Why not try quadlinear interpolation?

extend Trilinear interpolation by another dimension. As long as a linear interpolation model fits your data, it should work.

Answer 3

There are quite a few options here...

In order to get your energy into your mesh, you'll need to use some form of interpolation. Shepard's method is a common, and reasonably simple, method to implement, and tends to work well if your data distribution is reasonable.

Once you have that done, you'll need to do some form of isosurface generation.

There are some libraries out there to make this easy. Most notably, VTK includes python wrappers and has all of the tools required to do both of these steps.

For details on how this could be done in VTK, you can check vtkShepardMethod and vtkContourFilter.