Elliptic Billiards (2 Viewers)

[seanieg89]

Here is a question that is vaguely related to my research. I am not sure how difficult it is to prove, but some of you might find it interesting.

Consider an elliptical region bounded by a mirror. A light ray is emitted from one focus of this ellipse in any direction, and proceeds to bounce off the mirrored boundary of the region. In particular, the law of reflection implies that the path traced out by this ray of light will pass through alternating focii with each reflection.

Prove that this path "converges" in some sense to the major axis of our ellipse.

A diagram of what I mean by this can be found at http://cage.ugent.be/~hs/billiards/bilfocan.gif.

 

[Nooblet94]

Here is a question that is vaguely related to my research. I am not sure how difficult it is to prove, but some of you might find it interesting.

Consider an elliptical region bounded by a mirror. A light ray is emitted from one focus of this ellipse in any direction, and proceeds to bounce off the mirrored boundary of the region. In particular, the law of reflection implies that the path traced out by this ray of light will pass through alternating focii with each reflection.

Prove that this path "converges" in some sense to the major axis of our ellipse.

A diagram of what I mean by this can be found at http://cage.ugent.be/~hs/billiards/bilfocan.gif.

i.e. "plz do my phd 4 me"

Seriously though, I haven't the slightest idea how to even try and prove this, but it does look quite interesting. Looking forward to seeing a proof!

 

[seanieg89]

i.e. "plz do my phd 4 me"

Seriously though, I haven't the slightest idea how to even try and prove this, but it does look quite interesting. Looking forward to seeing a proof!

Haha I wish. No this is a well known fact, and there are definitely books containing a proof. It's only loosely related to my actual research.

 

[Carrotsticks]

I remember reading this stuff when I was procrastinating during HSC. I remember that depending on where the initial shot is made, if you were to look at the paths of INFINITE number of reflections, you can get one of the following cases (may have missed out on one or two, memory a bit rusty):

1. seanieg89's situation where the path converges to the major axis.

2. It makes another ellipse of its own WITHIN that ellipse via an Envelope: http://en.wikipedia.org/wiki/Envelope_(mathematics)

3. It makes a hyperbola sharing the SAME foci as the ellipse within the ellipse itself (mind was blown when I read this)

4. It follows a set path and eventually retraces its own steps (kinda like a regular n-gon within a circle, but within an ellipse because a circle is a special case of an ellipse)

I may have left a few out, but that's the general jist of it.

And to think that some people say that Maths is boring...

 

[Carrotsticks]

Never mind, just realised that the cases are in the same link that seanieg89 provided.

For those wanting extra reading on the topic, this may be of interest: http://en.wikipedia.org/wiki/Caustic_(mathematics)

seanieg89, I have an idea of a proof, but not sure how practical it would be.

Suppose we have the initial line passing through the focus (either of them). It will reflect at the same angle (Reflective Property) and we'll call this angle .

When it bounces off and hits the 'circumference' of the ellipse again, it will do so at another angle etc etc.

And we have to prove that for all natural K, we have a monotonically decreasing sequence such that:



Since:



We can say it's bounded from below, monotonically decreasing, and therefore theta_k converges to 0 as k --> infinity.

However, my apprehension lies in the fact that just because theta_k --> 0, does not necessarily imply that the lines converge to the major axis. For all we know, the lines could converge by bounding up and down and get more and more vertical along the MINOR axis (this still satisfies the conditon theta_k = 0, just the wrong way around).

A way of overcoming this is to observe the behaviour of the parameter phi (a cos phi, b cos phi) as k --> infinity.

My second idea was to prove that with each iteration, the parameter phi in the expression (a cos phi, b cos phi) gets closer and closer to 0 or pi (since when phi = 0, we have the positive x intercept and when phi = pi, we have the negative).

Just throwing some ideas out there in case anybody wants to give it a try. Have some other stuff to do now.

 

[IamBread]

However, my apprehension lies in the fact that just because theta_k --> 0, does not necessarily imply that the lines converge to the major axis. For all we know, the lines could converge by bounding up and down and get more and more vertical along the MINOR axis (this still satisfies the conditon theta_k = 0, just the wrong way around).

Remember it has to go through both of the foci, and bouncing along the minor axis does not satisfy this, so it must be along the major axis.

 

[Carrotsticks]

Remember it has to go through both of the foci, and bouncing along the minor axis does not satisfy this, so it must be along the major axis.

Oh I actually forgot about that! Well if that's the case, then I suppose that would be a way of proving the convergence to the major axis. All we need to do now is prove the sequence of reflective angles is monotonically decreasing. However, I am not 100% sure if that covers an 'if and only if' situation, it might only be one side of it. I only made that observation based on the .gif file in the first post.

However, proofs are often easier said than done.

 

[IamBread]

If I was to let the simulation or whatever run for a while, and then stop it when the beam is at a foci, fire a different laser back along the exact path, would it 'diverge' back to the original start point of the first beam, or go straight to 'converging'?

 

[Carrotsticks]

If I was to let the simulation or whatever run for a while, and then stop it when the beam is at a foci, fire a different laser back along the exact path, would it 'diverge' back to the original start point of the first beam, or go straight to 'converging'?

It really depends on where your first beam shoots.

If between the foci, you will get a caustic in the form of a Hyperbola ie: never passes through foci again.

If outside the foci, you will get a caustic in the form of an Ellipse or an 'almost regular' polygon ie: still never passes through foci.

But if your initial beam shines through the focus and say you let it run for 100 iterations, then fire a beam 'backwards' down the same path, it will indeed go back to its original point, then continue to converge to the major axis as usual.

 

[seanieg89]

Showing the reflective angles decrease doesn't necessarily imply they converge to zero, although that approach does seem fruitful.

I was thinking more along the lines of finding a recursive relation involving the parameters describing the points where the ray meets the boundary. If the subsequence of even terms in this sequence converges to either 0 or pi, we are done. I'm sure there are nice geometric properties of the ellipse that allow us to avoid a brute-force coordinate bash.

 

[seanieg89]

And yes the initial beam is fired FROM a focus. Or equivalently, from the boundary directly towards a focus.