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calculate the total of a billiard map

How to calculate the total of a billiard map $f(s,\theta)$ as a function of $\theta , \theta', H$ and the curvature $k_s , k_{s'}$. $H$ is the generating function. This is a exercise 9.2.3 in ...
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Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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1answer
112 views

Billiards in a circular table

This is a variation of Alhazen's Billiard Problem. Suppose we have a semicircular billiards table of radius r centered at the origin O, and a billiard ball placed somewhere on the 'x-axis' of the ...
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112 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 $C$ ...
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23 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: ...
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1answer
54 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 ...
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1answer
42 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 ...
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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 is:...
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2answers
60 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
49 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
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1answer
62 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 ...
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1answer
129 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 ...
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1answer
177 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 ...
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1answer
105 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
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2answers
5k 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). ...
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1answer
122 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 ...
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2answers
402 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 ...