# Density function of projection of an oblique cylinder to a plane

I would like to project an oblique cylinder onto a plane. The cylinder looks like this:

Which I think can be written as:

$$\begin{cases} x=r\cos(\theta)+\frac{t}{h}\cot(\phi)\\ y=r\sin(\theta)\\ z=t \end{cases}$$

$$0\leq t\leq h$$, $$0\leq \theta\leq 2\pi$$

Here are three examples of what I mean by projection:

Basically, given an angle $$\gamma$$ I would like to pass a ray at angle $$\gamma$$ through each point on the cylinder to a plane and have that point on the plane represent the 'amount' of cylinder the ray passed through. The projections in the diagram are denoted by the arrows and the 'density functions' by the grayscale maps (which are just a rough guess of what I think the function would look like).

It's probably pretty clear by now that I don't have the vocabulary that I need to describe what I am trying to do, so even if you do not have a solution for me, it would already be a huge help if you could point me to some resources or keywords that I could read about to try and solve it myself.

Many thanks!