How to determine the reflection point on an ellipse Here is my problem. There are  two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point P is reflected by the ellipse and arrivates at Q. The question is how to determine the reflection point on the ellipse. I mean is there any analytical method to calculate the coordinate of the reflection point?


Hint:
We can use a method similar to that of Max Payne, but representing the ellipse by its parametric equations $x = a \cos t$, $y = b \sin t$. By differentiation, the tangent vector is $(- a \sin t, b \cos t)$, hence the normal vector $(b \cos t, a \sin t)$.
Now we express the equality of the angles to the normal, by the equality of dot products
$$\frac{(P_x - a \cos t) b \cos t + (P_y - b \sin t) a \sin t}{\sqrt{(P_x - a \cos t)^2+(P_y - b \sin t)^2}} = \frac{(Q_x - a \cos t) b \cos t + (Q_y - b \sin t) a \sin t}{\sqrt{(Q_x - a \cos t)^2+(Q_y - b \sin t)^2}}.$$
This can be rewritten as a nasty trigonometric polynomial of degree four by squaring and reducing to the common denominator, then rationalized to an octic polynomial.
