Translation Surfaces: Geometric Definition & Billiard Systems

• MHB
• Joppy
In summary, a geometric definition of a translation surface is given. It is explained that this surface can be generated from unfolding a billiard trajectory, and that it is topologically equivalent to a torus.
Joppy
MHB
In this Wiki article, a geometric definition of a translation surface is given.

I'm lost in at the first line were it is stated that a given collection of polygons need not be convex. How is this possible? I am trying to understand translation surfaces from the perspective of dynamical systems, specifically, billiard systems. In this setting we can 'unfold' the trajectory of a point particle. But surely this unfolding process only works for trajectories confined to convex regions?

I suspect my confusion comes from the fact that generating a translation surface from unfolding a billiard trajectory, and generating one given the definition from Wiki are different things. I also don't understand what is meant by $s_j = s_i + \vec{v}_i$. Are we saying that for every side in a plane of polygons, there exist some other side which lies in the same direction?
Thanks

Hey Joppy,

Here's my 2 cents.

Since it's about billiards, I imagine that we define a set of neighboring rectangles, each representing the table. Now we can shoot the ball, and instead of reflecting it against an edge, we let it roll into the next rectangle.
That does mean that various edges are actually the same, so they are identified with each other through a translation.

And instead of convex rectangles, we might also have non-convex polygons.

I like Serena said:
Hey Joppy,

Here's my 2 cents.

Since it's about billiards, I imagine that we define a set of neighboring rectangles, each representing the table. Now we can shoot the ball, and instead of reflecting it against an edge, we let it roll into the next rectangle.
That does mean that various edges are actually the same, so they are identified with each other through a translation.

Thanks! Yes that's how I understand it to work for convex polygonal billiards. I am curious as to how this works, if at all, for billiard tables which are not convex.
I like Serena said:
Now turn those rectangles into rectangular prisms and we have a translation space.

Do you mean we fold up this mesh of rectangles into a prism? For example, if we have a horizontal trajectory inside the unit square, and we let it roll into four squares (so that we have four squares side by side), do we fold them back up to obtain a cube with two faces missing?

Supposedly, using this method of unfolding, some trajectories will yield a torus as the translation surface. I would like to know in what sense the translation surface is a torus (for this case).

For a billiard table we wouldn't be folding squares up into a cube.
Instead we have 4 neighboring rectangles. And when the ball rolls over a rightmost edge, it will magically appear on the corresponding leftmost edge.Same for top and bottom.

Now imagine a rectangle with a halfsize rectangle removed from a corner. We put again 4 such shapes next each to other in a mirrored layout. There we go.

As for a torus, we effectively get that when we have just a single rectangle. That is, it's topologically equivalent to a torus.

I like Serena said:
For a billiard table we wouldn't be folding squares up into a cube.
Instead we have 4 neighboring rectangles. And when the ball rolls over a rightmost edge, it will magically appear on the corresponding leftmost edge.Same for top and bottom.

Now imagine a rectangle with a halfsize rectangle removed from a corner. We put again 4 such shapes next each other in a mirrored layout. There we go.

As for a torus, we effectively get that when we have just a single rectangle. That is, it's topologically equivalent to a torus.

Thanks! :)

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1. What is a translation surface?

A translation surface is a mathematical object that can be thought of as a flat sheet with a pattern of lines on it. These lines represent the direction and magnitude of translation that can be applied to the surface.

2. How is a translation surface defined geometrically?

A translation surface is defined geometrically as a two-dimensional surface that can be tiled by parallelograms in a way that preserves the direction and magnitude of translation between adjacent tiles. This means that the surface must have a consistent pattern of lines that can be extended infinitely in all directions.

3. What is the significance of billiard systems in relation to translation surfaces?

Billiard systems are used to study the dynamics of particles bouncing around on a translation surface. This can provide insights into the geometric properties and behavior of the surface, as well as applications in physics and engineering.

4. Are translation surfaces only found in mathematical theory or do they have real-world applications?

Although translation surfaces were initially studied in a purely mathematical context, they have since found applications in various fields such as physics, engineering, and computer graphics. They can be used to model the behavior of light, sound, or other waves, and have been used in the design of musical instruments and other structures.

5. What are some current areas of research in the field of translation surfaces?

Some current areas of research in the field of translation surfaces include studying the behavior of billiard systems on surfaces with more complicated geometric structures, investigating the relationship between translation surfaces and other mathematical objects, and exploring new applications for translation surfaces in various fields.

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