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Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I am interested in the behavior of a light ray aimed directly down the central axis of one end of the tube, as it bounces with perfect reflection from the interior wall of the tube. This could be viewed as a model of an optic fiber, although I want to treat the light ray as a billiard ball and not a dispersing wave.

Q1. Does the light ray always emerge from the other end?

I believe the answer is Yes, although there can be close calls:

I would be interested in a succinct, convincing proof (or a counterexample!). Perhaps I should stipulate that if the ray hits a boundary singularity (as it nearly does above), it dies; otherwise it could pass through the center of curvature and reflect to its own reversal.

Q1 Answer: Yes if $c$ is $C^\infty$ (Dimitri Panov), Not necessarily if $c$ is $C^2$ (Anton Petrunin).

I explored one particular, maximally convoluted, snake-like tube, composed of alternating semicircles of radius 2 (so the central curve $c$ has curvature 1):

The lightray behaves seemingly chaotically, although when I look at the angles the rays make with the $(+x)$-axis, there is a striking distribution. Here is a histogram for a tube composed of 1000 semicircles:

Q2. Can you offer an explanation for the observed distribution of ray orientations? Why is the angle $\pm 17^\circ$ so prominent, and there are no ray angles whose absolute value lies within $[19^\circ,111^\circ]$? There are approximately 1.55 ray bounces per semicircle: Why? Is this approaching $\frac{3}{2}$? Or $\frac{\pi}{2}$?

Ideas/insights welcomed—Thanks!

Q2 Answer: Dimitri Panov's remarks and the phase portrait below show that likely the ray trajectory is quasiperiodic, which explains the angle histogram, which is, in a sense, a projection of the phase portrait.

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Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I am interested in the behavior of a light ray aimed directly down the central axis of one end of the tube, as it bounces with perfect reflection from the interior wall of the tube. This could be viewed as a model of an optic fiber, although I want to treat the light ray as a billiard ball and not a dispersing wave.

Q1. Does the light ray always emerge from the other end?

I believe the answer is Yes, although there can be close calls:

I would be interested in a succinct, convincing proof (or a counterexample!). Perhaps I should stipulate that if the ray hits a boundary singularity (as it nearly does above), it dies; otherwise it could pass through the center of curvature and reflect to its own reversal.

Q1 Answer: Yes if $c$ is $C^\infty$ (Dimitri Panov), Not necessarily if $c$ is $C^2$ (Anton Petrunin).

I explored one particular, maximally convoluted, snake-like tube, composed of alternating semicircles of radius 2 (so the central curve $c$ has curvature 1):

The lightray behaves seemingly chaotically, although when I look at the angles the rays make with the $(+x)$-axis, there is a striking distribution. Here is a histogram for a tube composed of 1000 semicircles:

Q2. Can you offer an explanation for the observed distribution of ray orientations? Why is the angle $\pm 17^\circ$ so prominent, and there are no ray angles whose absolute value lies within $[19^\circ,111^\circ]$? There are approximately 1.55 ray bounces per semicircle: Why? Is this approaching $\frac{3}{2}$? Or $\frac{\pi}{2}$?

Ideas/insights welcomed—Thanks!

Q2 Answer: Dimitri Panov's remarks and the phase portrait below show that likely the ray trajectory is quasiperiodic, which explains the angle histogram, which is, in a sense, a projection of the phase portrait.

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Imagine a smooth curve $c$ sweeping out a unit-radius disk that is orthogonal to the curve at every point. Call the result a tube. I want to restrict the radius of curvature of $c$ to be at most 1. I am interested in the behavior of a light ray aimed directly down the central axis of one end of the tube, as it bounces with perfect reflection from the interior wall of the tube. This could be viewed as a model of an optic fiber, although I want to treat the light ray as a billiard ball and not a dispersing wave.

Q1. Does the light ray always emerge from the other end?

I believe the answer is Yes, although there can be close calls:

I would be interested in a succinct, convincing proof (or a counterexample!). Perhaps I should stipulate that if the ray hits a boundary singularity (as it nearly does above), it dies; otherwise it could pass through the center of curvature and reflect to its own reversal.

I explored one particular, maximally convoluted, snake-like tube, composed of alternating semicircles of radius 2 (so the central curve $c$ has curvature 1):

The lightray behaves seemingly chaotically, although when I look at the angles the rays make with the $(+x)$-axis, there is a striking distribution. Here is a histogram for a tube composed of 1000 semicircles(updated with finer detail):

Q2. What is Can you offer an explanation for the observed distribution of ray orientations? Why is the angle $\pm 17^\circ$ so prominent, and there are no ray angles whose absolute value lies within $[19^\circ,111^\circ]$? There are approximately 1.55 ray bounces per semicircle: Why? Is this approaching $\frac{3}{2}$? Or $\frac{\pi}{2}$?

Ideas/insights welcomed—Thanks!

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