The reflective property of the ellipse: why light from one focus always reaches the other

I suddenly remembered an article about the optical property of the ellipse that I posted somewhere else a long time ago. Even today I find that explanation intuitive and fun. Back then I thought it was the "essential" explanation, but a classmate pointed out that "intuitive" is probably the more accurate word.

The article talks about ellipses, but the same idea applies more or less to conic sections in general.

The optical property of the ellipse

Among the properties of conic sections, you often see a picture like the one below: a ray of light leaving one focus of an ellipse reflects off the wall of the ellipse and passes through the other focus.

Other conic sections, such as the parabola and the hyperbola, have similar properties — they are all about what happens to light coming from a focus.

But why?

Of course you can prove it mathematically: at the point of reflection (P in the figure), the incoming ray and the outgoing ray make equal angles with the tangent at that point. I'll skip the proof here. It takes a few turns, but it's not hard to follow.

The problem is that after working through a pile of math, I still wasn't satisfied with the answer. To me, a mathematical proof just takes one step back into mathematics and restates the original problem in mathematical terms. I could keep asking: why such a coincidence — why do the angle of incidence and the angle of reflection come out exactly equal after the calculation?

What I wanted to know was "why does the ellipse have this property", not "prove that the ellipse has this property" — just as Wang Yin (a Chinese programming-language researcher and blogger) puts it in his essay on causes and proofs. What I'm really getting at is the difference between the efficient cause and the final cause.

So could I have predicted this in advance, without going through such a complicated proof?

A more interesting explanation (from mechanics)

I puzzled over this, and over all the curious properties of the ellipse, for a long time in high school.

Then one day I noticed how my math textbook said to draw an ellipse: fix a string to two nails, pull the string taut with the tip of a pen, and go once around the nails — the result is an ellipse.

Thinking about it carefully, this description is interesting because it carries two hidden facts.

The first is obvious, and is essentially the definition of the ellipse: for any point on the ellipse, the sum of its distances to the two foci is constant.

The second is subtler, and you can see it by looking at the drawing process physically: as you draw, at any point of tangency the forces from the two ends of the string on the pen tip are equal. So the angle of incidence must equal the angle of reflection — otherwise the string wouldn't be in balance (it would slide). From this you can foresee that the calculation will come out this way.

At that point the "why" finally had an "intuitive" explanation.

Some fun generalizations

The same reasoning extends to the other conic sections, and even the ellipse itself has some extensions — for example, changing the distance between the two nails. What happens when the two nails coincide? The answer is obvious.