Projecting surface of sphere onto plane
We can represent any point on a sphere by using a complex number.
When we were looking at complex functions we saw that for inversions:
(w = 1/z) then:
- Straight lines through the origin map to themselves.
- Straight lines not through the origin map to circles.
- Circles through the origin map to straight lines.
- Circles not through the origin map to circles.
- Applying the inverse function twice restores the original.
- The inverse function can be decomposed into a conjugate and a reflection in a circle.
So this function can be used to map between circles and lines. We can also extend the concept to 3 dimensions which allows us to map the surface of a sphere to the plane.
The mapping between the surface of the sphere and the plane can be represented by the Möbius transformation of the form:
| M(z)= | az + b |
| cz + d |
Where:
- z = complex variable
- a,b,c & d = complex constants
To simplify things we will now place the plane through the centre of the sphere and we change the notation slightly:
- π = points on surface of sphere represented as a plane (complex number).
- x,y,z = three dimensional coordinates.
This gives the representation of a point on the sphere as:
| π(x,y,z)= | x + i z |
| 1-y |
Examples:
point at south pole = (0,-1,0)
| π(x,y,z)= | 0 | = | 0+ 0 i |
| 2 |
point at north pole = (0,1,0)
| π(x,y,z)= | 0 | = | ∞ |
| 0 |
point at equator = (1,0,0)
| π(x,y,z)= | 1 | = | 1 + 0 i |
| 1 |
point at equator 90° = (0,0,1)
| π(x,y,z)= | i 1 | = | 0 + 1 i |
| 1 |
