# Cross Ratio

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## cross ratio

[′krȯs ‚rā·shō]*A*,

*B*,

*C*, and

*D*, the ratio (

*AB*)(

*CD*)/(

*AD*)(

*CB*), or one of the ratios obtained from this quantity by a permutation of

*A*,

*B*,

*C*, and

*D*.

## Cross Ratio

(also anharmonic ratio). The cross ratio of four collinear points *M*_{1}*M*_{2}, *M*_{3}, *M*_{4} (Figure 1) is a number denoted by the symbol (*M*_{1}*M*_{2}*M*_{3}*M*_{4} and equal to

In this case, the ratio *M*_{1}*M*_{3}/*M*_{3}*M*_{2} is considered to be positive if the direction of the segments *M*_{1}*M*_{3} and *M*_{3}*M*_{2} 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 *m*_{1}, *m*_{2}, *m*_{3}, *m*_{4}, passing through the point *O*. This ratio is designated by the symbol (*m*_{1}*m*_{2}*m*_{3}*m*_{4}) and is equal to

where the angle (*m*_{i}*m*_{j}) beween the straight lines *m*_{i} and *m*_{j} is considered with a sign.

If the points *M*_{1}, *M*_{2}, *M*_{3}, *M*_{4} lie on the straight lines *m*_{1}, *m*_{2}, *m*_{3}, *m*_{4} (Figure 1), then

(*M*_{1}*M*_{2}*M*_{3}*m*_{4}) = (*m*_{1}*m*_{2}*m*_{3}*m*_{3}*m*_{4})

Hence, if the points *M*_{1}, *M*_{2}, *M*_{3}, *M*_{4} and *M*_{1}’, *M*_{2}’, *M*_{3}’, *M*_{4}’ are the result of the intersection of four straight lines *m*_{1}, *m*_{2}, *m*_{3}, *m*_{4} (Figure 1), then (*M*_{1}’ *M*_{2}’ *M*_{3}’ *M*_{4}’) = (*M*_{1}*M*_{2}*M*_{3}*M*_{4}). If, however, the straight lines *m*_{1}, *m*_{2}, *m*_{3}, *m*_{4} and *m*_{1}’, *m*_{2}’, *m*_{3}’, *m*_{4}’ project one set of four points *M*_{1}, *M*_{2}, *M*_{s}, *M*_{4} (Figure 2), then (*m*_{1}’*m*_{2}’*m*_{3}’*m*_{4}’) = (*m*_{1}*m*_{2}*m*_{3}*m*_{4}).

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.

E. G. POZNIAK