Stereographic Projection

What is stereographic projection

Stereographic projection maps each point on a sphere onto a plane, which is useful in making a map of the earth’s surface. We can project points \(P\) on the surface of sphere from the sphere's north pole \(N\) to point \(P'\) in a plane tangent to the south pole \(S\), as is shown in the following picture.

(https://mathworld.wolfram.com/StereographicProjection.html)
We can use Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane. For the sphere \(\{(x,y,z) \in R^3 | x^2+y^2+z^2=a^2\}\), we can set the point \((0,0,a)\) as the north pole, then project the points on the sphere onto the plane \(z=-a\).

The projection and its inverse are given by the following formulae: \[\begin{aligned} (X, Y)&=\left(\frac{2ax}{a-z},\frac{2ay}{a-z}\right),\\ (x, y, z)&=\left(\frac{4a^2X}{X^2+Y^2+4a^2},\frac{4a^2Y}{X^2+Y^2+4a^2} , \frac{a( X^2+Y^2-4a^2)}{X^2+Y^2+4a^2} \right). \end{aligned}\]

A design of stereographic projection

Let’s consider the unit sphere \(\{(x,y,z) \in R^3 | x^2+y^2+z^2=1\}\), we can set the point \((0,0,1)\) as the north pole, then project the points on the sphere onto the plane \(z=-1\). The projection and its inverse are given by the following formulae: \[\begin{aligned} (X, Y)&=\left(\frac{2x}{1-z},\frac{2y}{1-z}\right),\\ (x, y, z)&=\left(\frac{4X}{X^2+Y^2+4},\frac{4Y}{X^2+Y^2+4} , \frac{X^2+Y^2-4)}{X^2+Y^2+4} \right). \end{aligned}\] Let’s consider a group of circles on the \(X-Y\) plane. The centers of the circles are denoted by \[ (b cos \theta, b sin \theta),\] \[\theta = i*\pi/6, \quad i=0,1,2,3,…,11, \quad b =0, 0.6, 1.2, 1.8, 2.4, 3, 3.6.\] The radii of the circles are set to 0.1.

Basically there are no big challenges in implementing this design, as we can refer to the example OpenSCAD programs for this project. We just need to use the representation of polar coordinates to make the design simple. And the radii of the circles should be small, so that two adjacent circles near the south pole will not intersect with each other.

Why to use this design

This design of the stereographic projection was chosen for the following reasons:

1. The design shows that the stereographic projection is conformal. In geometry, the stereographic projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. The stereographic projection has one import feature that it is conformal, i.e. it preserves angles at which curves meet. By using our design of stereographic projection, we can verify that a longitude is projected as a straight line, and a latitude is projected as a circle on the plane. Thus stereographic projection creates an accurate representation in specific circumstances that allow for simple observation and study. A map with stereographic projection works best when looking at the planet from a polar perspective. Maps of the Arctic and Antarctic are particularly excellent when using this approach.
2. The design shows that the stereographic projection is neither isometric nor area-preserving. The stereographic projection preserves neither distances nor the areas of figures. By using our design of stereographic projection, we can find that the areas of holes on different latitudes are different on the sphere, while the areas of holes projected on the plane are the same. What’s more, the distances between adjacent holes on the sphere vary along a longitude, while the corresponding distances of adjacent holes along the longitude projected on the plane are the same. In the above design, we investigate the properties of stereographic projection. This helps us to better understand the concept of stereographic projection.