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Results tagged with billiards
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user 6094
Billiards are a class of dynamical systems in which a point particle moves uniformly in a domain $D\subset \mathbb{R}^d$ except for mirror-like reflections from the boundary. Varying $D$ leads to examples satisfying many ergodic properties. Billiards enhance visual explanations of dynamical concepts to students and the general public. There are many applications in physics and image processing. The free motion and/or reflection rule may be generalized.
10
votes
Accepted
Current state of Straus's illumination problem
As far as I know, the specific question you ask (entirely illuminable from at least one point) remains open.
(Tokarsky showed that placing a light in some spots can leave some points dark.)
But you ma …
- 151k
8
votes
Accepted
Existence of periodic orbits in rational billiards
Schwartz, Richard Evan. Mostly surfaces. Vol. 60. American Mathematical Society, 2011.
On p.219ff of Schwartz's book, he sketches "an elementary proof, due to Boshernitsyn, that every rational po …
- 151k
3
votes
Can every $\mathbb{Z}^2$ disk be pinball-reached?
"From rational billiards to dynamics on moduli spaces." Apr. 2015.
(arXiv abstract.) …
- 151k
6
votes
Reflection of light from function graph
Just empirically, I believe the OP's $e^{-x}$ example has the property that
ray reflections quickly become increasingly vertical,
and so will not reach arbitrarily large $x$:
(I did not, …
- 151k
2
votes
Which polygons have *simple* periodic billiard paths?
An example that may be difficult for Aaron's line-of-symmetry argument:
- 151k
1
vote
Well-definedness of single-particle smooth billiards flow
This is just a guess, I will admit
(apologies if I'm among the misunderstandings you anticipated),
but perhaps Peter Guber's paper,
Convex billiards.
Geometriae Dedicata. …
- 151k
5
votes
Can every $\mathbb{Z}^2$ disk be pinball-reached?
I add this image just to illustrate that matters seem more complicated (as Douglas Zare has emphasized)
when the radii of the disks approaches $\frac{1}{2}$:
- 151k
2
votes
Optic fibers after Joseph O'Rourke
Suppose that the normal frame along $\gamma$ twists. Are you sure it still constitutes an optic fiber?
I am imagining, for example, smoothly twisting an ellipse of nonzero eccentricity.
- 151k
25
votes
Light rays bouncing in twisted tubes
Here is my interpretation of Anton's idea to capture the ray.
I found it almost impossible to illustrate; I could only show three sections of the
tube, separated by two rotations, the first counterclo …
- 151k
14
votes
Light rays bouncing in twisted tubes
Here is the phase portrait of $(\theta, x \; \mathrm{mod} \; 2)$ for the rays that Dimitri suggested
in his response to Q2
(if I have interpreted his suggestion correctly), using the same
data as disp …
- 151k