Consider a cube filled with random particles. The cube is rotated around the z-axis through its center, with the rotation being proportional to the height within the cube. At the bottom of the cube, there is no rotation, and as we move upward, the rotation gradually increases. The rotation creates this shape within the cube, as shown in the provided images.:
and that's how it looks like when we add particles inside the cube and rotate them:
Now when the rotated particles in 3D are projected onto a 2D plane, and then count the number of points in each pixel. We can then visualize this 2D histogram as a heatmap, which will give an image where the colour of each pixel corresponds to the number of particles projected onto that pixel after the rotation. The image formed looks like this:
How can I describe the geometry of the side AREAS (in pink) as a piecewise function where each piece represents the height of the pixels as a function of x,y and rotation $\theta$ for each region? It should have 12 pieces except for the middle region in yellow whose height has a maximum value "c", so it is mostly a flat surface.
What is the best way to describe the geometries of the side areas? to look something like this?
The goal is to multiply the equations of the sides by their inverse to elevate their values, so the region of the side becomes as high as the middle flat surface with the value "c"