Cross Ratio

(also anharmonic ratio). The cross ratio of four collinear points M₁M₂, M₃, M₄ (Figure 1) is a number denoted by the symbol (M₁M₂M₃M₄ and equal to

In this case, the ratio M₁M₃/M₃M₂ is considered to be positive if the direction of the segments M₁M₃ and M₃M₂ is the same and negative if the directions are different. The cross ratio depends on the order of the numbering of the points, which may differ from the sequential order of the points on the straight line. In addition to the cross ratio of four points, there is also a cross ratio of four straight lines m₁, m₂, m₃, m₄, passing through the point O. This ratio is designated by the symbol (m₁m₂m₃m₄) and is equal to

where the angle (m_im_j) beween the straight lines m_i and m_j is considered with a sign.

Figure 1

If the points M₁, M₂, M₃, M₄ lie on the straight lines m₁, m₂, m₃, m₄ (Figure 1), then

(M₁M₂M₃m₄) = (m₁m₂m₃m₃m₄)

Hence, if the points M₁, M₂, M₃, M₄ and M₁’, M₂’, M₃’, M₄’ are the result of the intersection of four straight lines m₁, m₂, m₃, m₄ (Figure 1), then (M₁’ M₂’ M₃’ M₄’) = (M₁M₂M₃M₄). If, however, the straight lines m₁, m₂, m₃, m₄ and m₁’, m₂’, m₃’, m₄’ project one set of four points M₁, M₂, M_s, M₄ (Figure 2), then (m₁’m₂’m₃’m₄’) = (m₁m₂m₃m₄).

Figure 2

The cross ratio also remains unchanged by any projective transformation, that is, it is an invariant of such transformations, and cross ratios therefore play an important role in projective geometry. Sets of four points or lines for which the cross ratio is equal to 1 are of particular importance. Such sets are called harmonic.