📂Complex Anaylsis

Cross Ratio in Complex Analysis

Definition ¹

On the extended complex plane, for four distinct points $ z_{1} , z_{2} , z_{3} , z_{4} \in \overline{ \mathbb{C} }$, the following is defined as the Cross Ratio: $$ (z_{1} , z_{2} , z_{3} , z_{4} ) = {{( z_{1} - z_{4})( z_{3} - z_{2})} \over {(z_{1} - z_{2}) ( z_{3} - z_{4}) } } $$

Description

If we change the form a bit to $\displaystyle (z_{1} , z_{2} , z_{3} , z ) = {{( z_{3} - z_{2}) } \over {(z_{1} - z_{2})} } \cdot {{ ( z - z_{1}) } \over { ( z - z_{3}) } }$, $$ (z_{1} , z_{2} , z_{3} , z_{1} ) = 0 \\ (z_{1} , z_{2} , z_{3} , z_{2} ) = 1 \\ (z_{1} , z_{2} , z_{3} , z_{3} ) = \infty $$ it holds true, and since at least three points appear, it’s not hard to guess that it has uses in dealing with circles or lines.

The key property is as follows.

Theorem

The cross ratio is invariant under bilinear transformation.

Proof

If we set the bilinear transformation as $f$ and the cross ratio as $g(z) = (z_{1} , z_{2} , z_{3} , z )$, then $$ g ( f^{-1} (w_{1}) ) = g (z_{1}) = 0 \\ g ( f^{-1} (w_{2}) ) = g (z_{2}) = 1 \\ g ( f^{-1} (w_{3}) ) = g (z_{3}) = \infty $$ that is, $g \circ f^{-1}$ becomes the cross ratio for $w_{1} , w_{2} , w_{3} , w_{4} \in \overline{ \mathbb{C} }$. Therefore, $$ ( z_{1} , z_{2} , z_{3} , z_{4} ) = g(z_{4}) = g ( f^{-1} (w_{4} ) ) = ( w_{1} , w_{2} , w_{3} , w_{4} ) $$

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