Non-harmonic ratio

Japanese: 非調和比 - ひちょうわひ（英語表記）non-harmonic ratio

When there are four points A, B, C, and D arranged in the order A, C, B, and D on a line,

is called the four-point anharmonic ratio or cross ratio, and is represented as (AB,CD). Here, AC is the signed length between A and C. Therefore, if we introduce coordinates onto this line and make the coordinates of the four points x ₁ , x ₂ , x ₃ , and x ₄ respectively, then

In addition, when four lines a, b, c, and d that pass through A, B, C, and D on a plane intersect at a point O,

is called the inharmonic ratio of four straight lines and is expressed as (ab, cd). Here, it is the angle with a sign made by the straight lines a and c. Inharmonic ratios have the interesting property that their value does not change due to projective transformation, which can be seen in (1) of the figure : (AB, CD) = (ab, cd)
=(A′B′,C′D′)
=(a′b′,c′d′)
This is because when an aerial photograph is taken, the anharmonic ratios of four points on a line on the ground and four points photographed on a photographic paper are equal, so if the distances between three points on the ground are known, the anharmonic ratio can be used to find the distance to a fourth point. The four points whose anharmonic ratio is -1 are called harmonic sequence points, and the four lines are called a harmonic bundle. The anharmonic ratios of four points on a circle are defined as the anharmonic ratios of the four lines a, b, c, and d in Figure (2) by taking an arbitrary point P on the circle that is different from these, that is, (AB,CD)=(ab,cd). It is easy to see that (AB,CD) defined in this way has a constant value regardless of the position of point P on the circle. The anharmonic ratios of four points on a line are invariant under projective correspondence, so they play a fundamental role in projective geometry, and the anharmonic ratios of four points on a circle are invariant under inversion, so they play a fundamental role in conformal geometry.

[Tachibana Shunichi]

[Reference] | Projective Geometry

Anharmonic ratio explanation diagram [figure]
©Shogakukan ">

Anharmonic ratio explanation diagram [figure]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

1直線上にA、C、B、Dの順番に並ぶ4点A、B、C、Dがあるとき

を4点の非調和比または複比といい、これを(AB,CD)で表す。ここでACはA、C間の符号をもった長さである。したがって、この直線上に座標を導入して、4点の座標をそれぞれx₁、x₂、x₃、x₄とすれば

となる。また1平面上にあるA、B、C、Dを通る4直線a、b、c、dが1点Oで交わるとき

を4直線の非調和比といい(ab,cd)と表す。ここでは直線a、cがなす符号をもつ角とする。非調和比は射影変換によって値が変わらないというおもしろい性質をもっているが、それは図の(1)で
　　(AB,CD)=(ab,cd)
　　　　　 =(A′B′,C′D′)
　　　　　 =(a′b′,c′d′)
が成り立つためである。航空写真をとったとき、地上にある1直線上の4地点と印画紙上に写された4点の非調和比は等しいので、地上の3地点相互間の距離がわかっていれば、非調和比を利用して第4の地点までの距離を知ることができる。非調和比の値が－1であるような4点を調和列点、4直線を調和線束という。円周上の4点の非調和比は、これらと異なる任意の点Pを円周上にとり、図の(2)の4直線a、b、c、dの非調和比として、すなわち、(AB,CD)=(ab,cd)として定義する。このように定義された(AB,CD)は円周上の点Pの位置に関係なく一定の値をもつことが容易にわかる。直線上の4点の非調和比は射影対応で不変であるから射影幾何学で、また円周上の4点の非調和比は反転で不変であるから共形幾何学で基本的な役割をする。