Project Project

Introduction

Stereographic projection is a method for visualizing the unit sphere in $(n+1)$-dimensional space in $n$-dimensional space while preserving some interesting structure. This can be extremely useful when, for example, you have an object living in $\mathbb{S}^3 \subseteq \mathbb{R}^4$, but you want to be able to actually see it. This comes up in my work with Milnor Fibrations, where the fibers do in fact live in $\mathbb{S}^3$. This article will explain how stereographic projection works in general, and at the end there are some neat, interactive examples.

How does this sort of projection preserve what kind of structure, and how does it work exactly? Read more to find out.

Stereographic Projection

First off, I want to make notation clear. Somewhat counterintuitively, the $n$-sphere, $\mathbb{S}^n$, is the collection of points unit distance from the origin in $\mathbb{R}^{n+1}$. Thus a 1-sphere is a circle in 2-dimensional space, a 2-sphere is an ordinary sphere (the surface of a ball) in 3-dimensional space, a 3-sphere lives in $\mathbb{R}^4$, etc. This is a reasonable convention because locally (i.e., in a small neighborhood) the $n$-sphere looks like $\mathbb{R}^n$. (Talk to a member of your local Flat Earth Society if you’d like clarification on this issue.) Now on to projection!

Let’s try projecting $\mathbb{S}^1$ to $\mathbb{R}^1$, so $n=1$. Think of $\mathbb{R}^1$ as the $x$-axis in $\mathbb{R}^2$. To project a point $p\in\mathbb{S}^1$ onto $\mathbb{R}^1$, draw a line from the top of the circle, a point we will call $\infty$, through $p$ making sure to extend your line so that it passes through $\mathbb{R}^1$. The point at which this line intersects $\mathbb{R}^1$ is $p^\prime.$

In the gif below, the green dot is $p$ and the blue dot is $p^\prime.$

It is hard to tell, but the frame where the green dot would be at the top of the circle is missing (intentionally). It turns out that our name for that point was appropriate; it would map to $\pm \infty$ on $\mathbb{R}^1.$ This doesn’t make sense for the projection though, so one way to solve this is just to remove $\infty$ from the domain, via the final mapping

This 2D to 1D example probably seems boring, but it clearly shows the process. If you are interested in the aesthetics of 3D to 2D stereographic projection, I recommend you check out Henry Segerman’s models that use a light source at $\infty$ to project a pattern on $\mathbb{S}^2$ onto the ground: (7,3,2) triangles, (5,3,2) triangles, (7,3,2) triangles (small).

Interactive Examples

The 3D model below demonstrates a circle on $\mathbb{S}^2$ being mapped to $\mathbb{R}^2$ via stereographic projection (created by Kyle Ormsby). Note that the circle is preserved!

Interactive 3D Model
For some more interactive examples, I’ve created a Mathematica notebook that demonstrates stereographic projection into $\mathbb{R}^3, \mathbb{R}^2$ and, of course, $\mathbb{R}^1.$ Check it out here: stereographic projection.