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3
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0answers
64 views

Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
1
vote
0answers
17 views

Is there an elementary proof for the envelope of an infinitely-bouncing billiard ball?

I was reading this page here http://cage.ugent.be/~hs/billiards/billiards.html And was wondering if there existed an elementary proof of proving that the envelopes form certain conic sections (ie: ...
3
votes
1answer
48 views

For points A, B, does there there a billiard such that any trajectory from A will reflect twice and then reach B?

I'm looking for a kind of generalisation of an ellipse; a shape with a more complicated optical property. I'm not sure how to rigorously define this shape, or prove that it exists, or find an equation ...
1
vote
1answer
38 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
2
votes
0answers
33 views

Arc length of stadium billiard

I've been trying to figure out the most straightforward way of doing this for a while, and would like to get some advice on new approaches, as the one I was using didn't work out at all. So here it ...
0
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0answers
47 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
2
votes
2answers
56 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
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0answers
40 views

Constants of motion for non-chaotric billiards

As I understand it, for a mechanical system, each symmetry leads to a constant of motion. For integrable systems, the number of constants of motion equals the number of degrees of freedom. So take 2D ...
2
votes
1answer
55 views

Turning a rhombus billiard into an equivalent barrier billiard

I am reading a research paper and the authors map the rhombus billiard (angles $60$-$120$ degrees) to an equivalent barrier billiard. They start with a rhombus standing upright and reflect in a ...
4
votes
1answer
97 views

Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
6
votes
1answer
154 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
0
votes
1answer
90 views

Tool for generating mathematical billiards

i'm currently looking for some program which can generate arbitrary mathematical billiards and trajectories inside them. Any help would be much appreciated! Thanks/ Mats
21
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2answers
4k views

Perfectly centered break of a perfectly aligned pool ball rack

This question is asked on Physics SE and MathOverflow by somebody else. I don't think it belongs there, but rather here (for reasons given there in my comments there; edit: now self-removed). ...
3
votes
1answer
107 views

The forever moving billiard ball

Suppose I have a rectangular table, dimensions $x$ by $y$, and a billiard ball is positioned in the very center. For descriptive convenience, let us impose a coordinate system on this table with an ...
5
votes
2answers
374 views

Polygonal billiards and uniform distribution

According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle ...