The Perfect Pool Table

If you don't want a ball to have any preference for which way it should roll, what shape should a pool table's surface be? It seems like the answer should be just a flat rectangle, but we don't live on a flat Earth [citation needed] and so the curvature of the planet is relevant. This is easier to see if you imagine a tiny planet with a massive pool table.

On the first planet the centre of the table would be closer to the centre of the planet and so the gravity would be stronger. This would make all of the balls cluster into the middle, away from the pockets. To combat this you could design a table which was part of a sphere with a slightly larger radius than the planet.

How much would this variation be on a planet the size of Earth?

Well, pool tables have the dimensions 2.84m by 1.42m (I'd never appreciated the 2:1 ratio before). The standard height is somewhere between 80cm and 75cm so let's say that the tallest part of the table should be the centre at exactly 80cm. The question can be framed as finding the greatest dip for a point on the table below that 80cm perfectly flat rectangle.

The lowest points are necessarily in the corners, which by a bit of Pythagoras are 1.587m from the centre of the table. The radius of the Earth at its smallest is about 6357km and this is where our pool table will be most bent. For more on the changing radius of our planet see here. We can draw a diagram of our situation as a right angled triangle like so:

To find the drop in the table we have to find x, which we can do with Pythagoras. 

That drop is quite small, only 0.2 micrometres. That's about 2 or 3 times thicker than a red blood cell. I think the verdict here has to be that the surface of the pool table is going to vary more than this anyway and so it doesn't matter.