Conformal geometry

Conformal geometry is a type of geometry that is concerned with how shapes and distances change when a figure is stretched, squeezed, or turned around. Think of it this way: imagine you have a piece of cookie dough that you play with – you can stretch it, twist it, and bend it, but it remains a flat shape. Conformal geometry is like playing with cookie dough on paper, where you can see how the shape and distance changes as you manipulate it.

In this type of geometry, two figures are considered to be similar if and only if their corresponding angles are equal, and their respective side lengths are proportional. This notion of similarity is called "conformity." In other words, two shapes are "conformal" if they have the same angles and the same shape, but perhaps just a different size.

So, let's imagine that you have a rectangular sheet of paper. If you stretch one side of the paper, the angles between the sides will remain the same, but the sides themselves will change length. This is where conformal geometry comes in – it helps identify the new shape and distance of the paper after you've stretched it.

One important concept in conformal geometry is the idea of "conformal maps." These are like maps you might use to navigate from one place to another, but instead of showing streets and landmarks, they show how shapes and distances change in a particular way. A conformal map can be thought of as a set of instructions that guide you on how to manipulate a shape while keeping its shape and angles the same.

In summary, conformal geometry helps us understand how the shape and distance of a figure change when we stretch, squeeze, or turn it around. It involves the concept of "conformity," where two figures are considered similar if they have the same angles and shape, but potentially a different size. Conformal maps guide us on how to manipulate a shape while maintaining its conformity.