Riemann Sphere and Möbius Transformation

Hubert Shutrick

In projective geometry, the xy-plane is supplemented by adding an extra line and homogeneous coordinates are introduced to cope with this line at infinity (see also Chasles' Theorem). If points in the plane are described by a complex number z = x + Iy, then, once again, homogeneous coordinates are convenient. A pair (z₁, z₂) of complex numbers that are not both zero determine a point in the plane and (az₁, az₂) determines the same point for any complex number a that is not zero. If z₂ ≠ 0, then the point is z = z₁/z₂ and (z₁, 0) is an extra point, the point at infinity. The plane with this extra point is called the complex projective line.

Since stereographic projection sets up a one-to-one correspondence between points of the plane and points of the sphere except N - the North pole - and since it has nice properties, it is convenient to take the sphere as a model for the complex projective line (see Hopf fibration).

It is natural in projective geometry to consider linear transformations and they turn out to be important in this context:

z'₁ = az₁ + bz₂
z'₂ = cz₁ + dz₂

is a linear transformation if the determinant ad - bc is not zero, i.e. it is invertible. Cross-ratio can be defined just as in the real case and it is invariant under linear transformations. In the real case, the linear transformations were considered as changes of base giving different parametrizations of the line, but here we usually keep the coordinate system fixed and consider that the transformations move the points giving homeomorphisms (i.e. 1-1 and onto transformations continuous in both directions) of the complex line.

Converting to the non-homogeneous coordinate z, the linear transformations become

and are called Möbius transformations and, in this context, the sphere is called the Riemann sphere. The condition ad - bc ≠ 0, ensures that z' is not constant.

Möbius transformations transform circles and lines into circles and lines which on the sphere become simply circles into circles. To prove this property, divide the fraction in the case c ≠ 0 to get

z' = f + g / (cz + d)

The transformation can then be considered as the composite

z to cz to cz + d to 1 / (cz + d) to g / (cz + d) to z'.

Multiplying by c = re^iθ rotates a circle about the origin through theta and blows it up away from the origin or shrinks it towards by a factor r, adding a constant merely translates it and inverting, which takes re^iθ to 1/re^{- Iθ}, is inversion with respect to the unit circle at the origin followed by complex conjugation. All these transformations are known to take circles to circles unless the circle goes through -d/c when the circle goes to a line. The case c = 0 works in the same way. (For the action of the inverse function f(z) = 1/z on circles, see the article on applications of complex numbers in geometry.)

Corresponding to the fact that any three different points determine a change of base on the real projective line, a Möbius transformation can be determined by choosing three points that are to be the images of 0, 1 and infinity, which is N on the sphere.

Möbius transformations have had many important applications, some of which are described in the internet.

Complex Numbers

1. Algebraic Structure of Complex Numbers
2. Division of Complex Numbers
3. Useful Identities Among Complex Numbers
4. Useful Inequalities Among Complex Numbers
5. Trigonometric Form of Complex Numbers
6. Real and Complex Products of Complex Numbers
7. Complex Numbers and Geometry
8. Remarks on the History of Complex Numbers
   • First Geometric Interpretation of Negative and Complex Numbers
9. Complex Numbers: an Interactive Gizmo
10. Cartesian Coordinate System
11. Fundamental Theorem of Algebra
12. Complex Number To a Complex Power May Be Real
13. One can't compare two complex numbers
14. Riemann Sphere and Möbius Transformation
15. Problems
    • Product of Diagonals in Regular N-gon
    • Sum of the Nth Roots of Unity
    • Napoleon's Relatives
    • Distance between the Orthocenter and Circumcenter

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