Math 2.0

Functions of a complex variable Definition

Cross-ratio

Builds on Riemann sphere

Definition

Let z1,z2,z3,z4z_1, z_2, z_3, z_4 be four distinct complex numbers. Their cross-ratio is (z1,z2;z3,z4)=(z3−z1)(z4−z2)(z3−z2)(z4−z1).(z_1, z_2; z_3, z_4) = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}. Suppose that one of the four points is the point ∞\infty of the Riemann sphere and the other three are distinct complex numbers. Then the cross-ratio is the same formula without the two factors that contain ∞\infty.

Remarks

Other books use other orders of the four points in the formula, so one must check the convention. For each Möbius transformation ff, the cross-ratio (f(z1),f(z2);f(z3),f(z4))(f(z_1), f(z_2); f(z_3), f(z_4)) is equal to (z1,z2;z3,z4)(z_1, z_2; z_3, z_4). With this convention, the map z↦(z,z2;z3,z4)z \mapsto (z, z_2; z_3, z_4) is the Möbius transformation that sends z2z_2, z3z_3, z4z_4 to 11, 00, ∞\infty. The cross-ratio is a real number exactly when the four points are on one circle or on one line.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectComplex plane, concept, no evidence levelComplex planeconceptRiemann sphere, definition, no evidence levelRiemann spheredefinitionCross-ratio, definition, no evidence levelCross-ratiodefinition
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  1. Complex numbers object
  2. Complex plane concept
  3. Riemann sphere definition
  4. Cross-ratio definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cross-ratio, Definition
  2. Web https://en.wikipedia.org/wiki/Cross-ratio, Transformational approach
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