Harmonic Distribution

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Harmonic Distribution


an arrangement of four points, M1, M2, M3, and M4 on a straight line ll1 (see Figure 1) such that the point M3 lies within the interval M1M2, the point M4 is outside of this interval, and the ratios Harmonic Distribution 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 M1, M2, M3, and M4 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 M1, M2, M3, and M4. Let the points M1 and M2 be the points of intersection of the conjugate opposite sides of the quadrilateral ABCD. Then points M3 and M4 will be the points of intersection for the diagonals of this quadrilateral with the straight line M1, M2. (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.


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The HHG experiments with Mn plasma using 3.5 fs pulses were also performed with stabilized CEP (e.g., at phases values of [phi] = 0 and [pi]/2) and no considerable differences were found in that case (Figure 13), though some variation of harmonic distribution was observed for the lower order harmonics (compare Figures 13(b) and 13(c)).
The song was written by poet Arif Al-Hashil, produced by Busatieen Sports Club, vocals by singer Mohammed Al-Bakri, lyrics by Ibrahim Al-Dossary and harmonic distribution by Ahmed Asadi.
The down-scale prototype circuit with 216 [V.sub.DC] ([V.sub.LVDC]) to 400 [V.sub.DC] ([V.sub.MVDC]) DC wind turbine converter is built to verify the harmonic distribution of the turbine converter.
The ANOVA supported the hypothesis that the harmonic distributions were equal.