What is a stereographic projection?

Question

I don't get what it is. It has something to do with a tesseract

Answer 1

It's a way of projecting a points on a spherical surface on a 2D plane.

If you consider that the south pole of the sphere is touching the 2D plane, the projection of any point on the sphere will be the point of intersection of a straight line through the north pole of the sphere and the point in question.

There is a nice diagram at Mathworld.

http://mathworld.wolfram.com/StereographicProjection.html

Answer 2

We let be a sphere in Euclidean three space. We want to obtain a picture of the sphere on a flat piece of paper or a plane. Whenever one projects a higher dimensional object onto a lower dimensional object, some type of distortion must occur. There are a number of different ways to project and each projection preserves some things and distorts others. Later we will explain why we choose stereographic projection, but first we describe it.

Description

We shall map the sphere onto the plane containing its equator. Connect a typical point on the surface of the sphere to the north pole by a straight line in three space. This line will intersect the equatorial plane at some point . We call the projection of .

Using this recipe every point of the sphere except the North pole projects to some point on the equatorial plane. Since we want to include the North pole in our picture, we add an extra point , called the point at infinity, to the equatorial plane and we view as the image of under stereographic projection.