Maths - Stereographic Projection - Riemann Sphere

This page overlaps with the page here, I need to combine them.

Projecting surface of sphere onto plane

We can represent any point on a sphere by using a complex number.

stereograpic projection

When we were looking at complex functions we saw that for inversions:
(w = 1/z) then:

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 (Möbius transformations are described on this page)

M(z)= az + b
cz + d


To simplify things we will now place the plane through the centre of the sphere and we change the notation slightly:

This gives the representation of a point on the sphere as:

π(x,y,z)= x + i z

stereographic xyz


point at south pole = (0,-1,0)

π(x,y,z)= 0 = 0+ 0 i

point at north pole = (0,1,0)

π(x,y,z)= 0 =

point at equator = (1,0,0)

π(x,y,z)= 1 = 1 + 0 i

point at equator 90° = (0,0,1)

π(x,y,z)= i 1 = 0 + 1 i

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see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Visual Complex Analysis - If you already know the basics of complex numbers but want to get an in depth understanding using an geometric and intuitive approach then this is a very good book. The book explains how to represent complex transformations such as the Möbius transformations. It also shows how complex functions can be differentiated and integrated.


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