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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

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


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?

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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 several other places where you will surely get help. – Mariano Suárez-Alvarez Jan 20 '10 at 20:11
I have rewritten the question for clarity. – Daniel Jan 20 '10 at 20:13
up vote 2 down vote accepted

The sets you mention (plane, ellipse) can be expressed as the zero sets of certain polynomial functions; you are asking about the set on which both of these polynomials vanish simultaneously -- this is a basic question in algebraic geometry. One common computational solution is to apply Buchberger's algorithm for computing a Gröbner basis for the corresponding ideal (the basis will give an explicit description of the zero set). An excellent resource on this subject -- even if you have no background in algebraic geometry -- is the book "Ideals, Varieties, and Algorithms" by David Cox, John Little, and Donal O'Shea. Additionally, many computer algebra systems (such as Mathematica) provide implementations of these kinds of algorithms.

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