Geometry • Topic 31

Cross Ratios

The cross ratio is the fundamental invariant of projective geometry. If you project a line onto another line (from a point $O$ ), distances change, but the cross ratio stays the same.

Definition

Definition (Cross Ratio)

For four collinear points

A, B, C, D

, the cross ratio is defined as:

(A, B; C, D) = \frac{A C}{CB} \cdot \frac{D B}{A D}

(Note: Sign conventions vary. Often directed segments are used).

Standard form using directed segments:

(A, B; C, D) = \frac{A C}{CB} / \frac{A D}{D B}

Invariance

Theorem (Projective Invariance)

If lines

O A, OB, OC, O D

intersect a transversal line

ℓ^{'}

A^{'}, B^{'}, C^{'}, D^{'}

, then:

(A, B; C, D) = (A^{'}, B^{'}; C^{'}, D^{'})

This defines a cross ratio for concurrent lines (a pencil of lines) as well.

Properties

If $(A, B; C, D) = - 1$ , the points form a Harmonic Range.
Permuting the points ( $A, B, C, D$ ) results in related values like $1/ λ$ or $1 - λ$ .

Practice Problems

Exercise (Problem 1)

Calculate the cross ratio of points at

x = 0, 1, 3, 4

on the number line.

Exercise (Problem 2)

Prove that if

(A, B; C, D) = 1

, then points

C

and

D

must coincide.

Exercise (Problem 3)

Given three points

A, B, C

on a line, construct a fourth point

D

such that

(A, B; C, D) = - 1

using only a straightedge.