ag.algebraic geometry - 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

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

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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?