Poincaré metric

Okay, so imagine you are standing on a huge ball that is really bouncy. This ball is called a "surface," and it can be made out of lots of different materials, like rubber or paper or even dough! Now, if you want to measure how far you can walk on the surface of this ball, you need to use something called a "metric."

A metric is like a ruler that tells you how long something is. But instead of measuring the length of a thing, it measures the distance between two points on a surface. So, if you want to walk from one point to another on the surface of the ball, you need to use a metric to figure out how far you have gone.

Now, this is where the Poincaré metric comes in. It is a special metric that can be used on some surfaces that have special properties. One of these properties is that they are "hyperbolic," which means they curve inward like a saddle.

The Poincaré metric helps us figure out how far we have gone on these kinds of surfaces in a special way. Instead of using a flat ruler like we might do on a table, we use a curved ruler that follows the shape of the surface. This way, we can tell how far we have gone even when the surface is really curvy.

So, the Poincaré metric is like a special curved ruler that helps us measure distance on hyperbolic surfaces like a bouncy ball!