# Unusual ray tracing

Background

Ray tracing is very common in computational geometry and the problem is then to find the point of intersection between the equation of a line and the equation of a plane in 3D.

The parametric form of the line is given by

$\mathbf{p}_\mathrm{line}=\mathbf{p}_\mathrm{a} + \xi (\mathbf{p}_\mathrm{b}-\mathbf{p}_\mathrm{a})$

and the plane can be defined by

$\mathbf{p}_\mathrm{plane} \cdot \mathbf{n}+\mathrm{d}=0$,

where $\mathbf{p}_\mathrm{plane}$ is a point on the plane and $\mathbf{n}$ is the normal vector to the plane.

Combining these two equations $(\mathbf{p}_\mathrm{line}=\mathbf{p}_\mathrm{plane})$ gives a convenient expression for the desired point from

$\xi=\frac{-\mathrm{d}-\mathbf{p}_\mathrm{a} \cdot \mathbf{n}}{(\mathbf{p}_\mathrm{b}-\mathbf{p}_\mathrm{a}) \cdot \mathbf{n}}$.

Question

I now consider the problem of finding the intsersection(s) between an ellipse and a plane in 3D. Is there an effective way to perform this without an iterative scheme?

• Your question is probably a borderline one for this forum. And it's also unclear. Do you want light rays to travel on ellipses, is that what you mean? – Ryan Budney Jan 20 '10 at 19:24
• Yes. The details depend on how you describe the ellipse, but but most sane choices this reduces to the solution of a quadratic equation. This is mostly not MO-material, really. The faq lists at mathoverflow.net/faq#whatnot several other places where you will surely get help. – Mariano Suárez-Álvarez Jan 20 '10 at 20:11
• I have rewritten the question for clarity. – Daniel Jan 20 '10 at 20:13