Cross Ratios in the Complex Plane

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The cross ratio of four points , , , in the extended complex plane is defined by , where pairs of zeros or infinities that can be canceled should be. You can drag all four locators in the graphic.

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The cross ratio is the quotient of two ratios, and . Suppose, for a moment, that the four points lie on a line. Then the ratio is a measure of the location of relative to and on the line, and similarly for .

Projecting the four points on a line from a central eye point to another line distorts the relative distances of the new points , , , ; in general and . However, the cross ratios remain equal: .

This is even true if the four points are not on a line, and the invariance holds more generally for any linear fractional transformation .

If the four points are unordered, there are six possible values (the red points; the cross ratio is bigger). They are shown as two triangles , , and , , , which are symmetric in the point . For orientation, and are drawn as small black points.

The cross ratio is real when the four points are on a circle or a line and is 2 for a square when the points are in cyclic order.

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Contributed by: George Beck (November 2015)
Open content licensed under CC BY-NC-SA


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