Möbius transformation

Imagine you have a piece of paper with a picture on it. You can move the paper around, turn it upside down, and even stretch or squish it. But what if you want to stretch part of the picture and leave the rest unchanged? That's where a möbius transformation comes in.

A möbius transformation is like a magical stretchy machine that can transform things on a flat surface. Imagine you have a rubber band that you can twist, stretch, and move around. When you stretch your rubber band, some parts will get longer, while others get shorter. A möbius transformation works in a similar way, as it can stretch, twist or bend the flat surface.

But how does it work? Let's say you want to transform a circle on the paper. A möbius transformation can stretch the circle into an oval, squeeze it into a line, or even transform it into another shape like a square or a triangle. This transformation preserves the angles between lines, but it can change the distance between points.

A möbius transformation can also change the position of shapes on the paper. For example, it can move a square from the top of the paper to the bottom, or rotate it around in circles. It is like moving pieces on a puzzle, but you can do it with math instead of your hands.

Möbius transformations are used in many areas of math, physics, and engineering. They can be used to transform shapes on a map, to study the dynamics of an airplane, or even to make special effects in movies. It may seem complicated, but it is just like playing with a rubber band, stretching and squishing it, and exploring all the different shapes you can create.