# Harmonic Distribution

## Harmonic Distribution

an arrangement of four points, *M*_{1}, *M*_{2}, *M*_{3}, and *M*_{4} on a straight line *ll*_{1} (see Figure 1) such that the point *M*_{3} lies within the interval *M*_{1}*M*_{2}, the point *M*_{4} is outside of this interval, and the ratios are equal. Usually the ratio of two intervals is considered to be positive if their directions on the straight line are the same, and negative when the directions are different. Accordingly, the harmonic distribution of the points *M*_{1}, *M*_{2}, *M*_{3}, and *M*_{4} can be described by the so-called cross-ratio of these

which is equal to −1 in the case of harmonic distribution. When a harmonic set of points is projected onto any straight line, it projects into a harmonic set of points. Consequently, harmonic distribution is one of the fundamental concepts of projective geometry. The following geometric configuration is associated with the harmonic distribution of the points *M*_{1}, *M*_{2}, *M*_{3}, and *M*_{4}. Let the points *M*_{1} and *M*_{2} be the points of intersection of the conjugate opposite sides of the quadrilateral *ABCD*. Then points *M*_{3} and *M*_{4} will be the points of intersection for the diagonals of this quadrilateral with the straight line *M*_{1}, *M*_{2}. (See Figure 1.)

The concept of harmonic distribution carries over into other geometric objects: four rays projecting a harmonic set of points from one point form a harmonic family of rays. A harmonic family of planes is defined in similar fashion.

E. G. POZNIAK