It started out with an ancient looking piece of paper. Probably a few years' old. Scribblings over it and a drawing. My friend T. Y. Jun handed it to us. "This is the trigonometrical proof I came up with for the reflective property of ellipse a few years ago. Exceedingly bulky and unelegant. I think there should be a shorter proof. Let's work on that." Over a cup of ice-blended vanilla coffee in Pacific Coffee Co. Of course none of us were aware of the existence of the internet and the simplicity of googling for it. (AND of course that is sarcasm.) However, with the ego driving us to prove ourselves, we decided to prove the property ourselves.
But, to no avail. After 3 hours.
So we ( or I) decided to creep to the public computer in the cafe just 5 metres away, and googled it. Finally, the simple answers came to us. Simpler than anyone would have expected.
"We want to show that if something like a light ray leaves focus F (with position vector p) and strikes the ellipse at point A, then it will be reflected to focus F (with position vector q). Suppose that the ellipse is given by a smooth parameterization r = r(t) where r is the position vector of A and t is time. Then the velocity vector dr/dt and its opposite -dr/dt are parallel to the tangent line at A, and by the law of reflection (angle of incidence = angle of reflection), we must prove that a = b. Since a and b are each less than 180 degrees, this is equivalent to showing that cos a= cos b.
Before giving the proof, let's notice that if w = w(t) is a vector function of time t, then we can use the definition of vector magnitude and the dot product rule to compute
Now here's the proof, which uses the notation in the above figure:
By the standard definition of an ellipse, p - r + q - r = (constant).
Noticing that p and q are also constant (time derivative 0) we can take derivatives of both sides, using equation (*), to get
which is the same as
Since (p - r) / p - r and (q - r) / q - r are unit vectors, this means that
and hence, cos a = cos b..."
