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(This uses a bit of physics)

So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their velocities after the collision will be perpendicular.

The proof goes like this: Let $v$ be the starting velocity, and let $v_1$ and $v_2$ be the velocities of the two pool balls after the collision. By conservation of momentum, $v_1+v_2=v$ and by conservation of energy, $|v_1|^2+|v_2|^2=|v|^2$. The first says that $v_1,v_2$ and $v$ form a triangle, and the second tells us that this triangle is right.

Is this still true for hyperbolic pool, pool played on the hyperbolic plane? (See http://www.geometrygames.org/HyperbolicGames/ for a hyperbolic pool app.)

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Maybe use that the hyperbolic metric is conformally related to the Euclidean metric. –  Neal Dec 7 '12 at 4:12
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1 Answer 1

Speed $v$ is the movement of a body by an infinitesimally small distance in an infinitesimally small amount of time. And as an infinitesimally small portion of hyperbolic space looks just like euclidean space, I'd say the argument holds there as well. So yes, the trajectories should be perpendicular as well, at least for point-like balls.

My only concern is whether you are allowed to assume the radii of your pool balls to be negligible, or if not, whether this changes anything. As there is no simple parallel transport in hyperbolic geometry, you cannot simply move force vectors between the point of collision and the centers of your balls. This might change things.

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