# Stereographic Projection 3D-Printed Physical Model

By Xah Lee. Date: . Last updated: .

Stereographic Projection and mobius transformation. If you want to understand Einstein's theory of relativity, you must understand mobius transformation. To understand mobius transformation, you must first understand Riemann sphere, complex numbers, geometric inversion.

## Moebius Transformation

In these videos, can you see that, the plane, and sphere, are mapped. That is, each point on the plane has a corresponding point on the sphere, and vice versa (except the north pole of the sphere, which should map to some “infinite” point on the plane)

Now, this idealized plane, with a point at infinity in the stereographic projection sense, is called Riemann sphere. Riemann sphere is a plane, but it has the word “sphere” in it because the essence of this plane is really a sphere, as you see in the videos. You can think of it as a plane or a sphere.

In the video, you see a grid on the plane and it gets mapped to the sphere. Now, if you only look at the sphere, and think of the sphere as a plane, then, you see that the lines from the grid become arcs of circles. There is a function in math that does exactly that transformation. It's called geometric inversion.

Inversion in geometry is a transformation. Let P be a given point. Let c be a circle centered on O and radius r. The inverse of P with respect to c is a point Q on the line[O,P] such that distance[O,P] * distance[O,Q] == r^2.

If you combine {geometric inversion, rotation, translation, reflection} into one function, that function is called Möbius transformation. And the formula/definition of this function, expressed with complex numbers, is `f[z]:= (a*z+b)/(c*z+d)` with `a*d - b*c ≠ 0`.