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Cross-ratio

For four distinct points of the Riemann sphere, their cross-ratio is

[z_{1}, z_{2}, z_{3}, z_{4}] = \frac{( z _{4} - z _{2} ) ( z _{1} - z _{3} )}{( z _{4} - z _{3} ) ( z _{1} - z _{2} )},

with the evident limiting conventions at infinity. Every Möbius transformation preserves it.

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