Cross Ratios in the Complex Plane

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.


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.


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



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.