nLab harmonic ratio

Using cross ratio

A quadruple of points $(A,B,C,D)$ is called harmonic if the cross ratio $(A,B;C,D) = -1$; points $C$ and $D$ are then called harmonic conjugates and $(A,B;D,C) = -1$ as well. We say that this ratio is harmonic and that the quadruple $(A,B,C,D)$ is harmonic. For this reason, the cross ratio in general is often called the anharmonic ratio as it measures the difference from the harmonic special case.

In synthetic terms

Harmonic conjugates can be characterized in geometric terms, as usually done in axiomatic approaches.

To formulate this, recall that a complete quadrangle consists has 4 coplanar points called vertices no 3 of which are colinear and all 6 lines called sides which are incident with pairs of vertices. Two sides are opposite if they do not contain a common vertex. The three intersections of opposite sides are called diagonal points and the lines of the triangle incident with pairs of diagonal points are called the diagonal sides.

The following characterizations of harmonic quadruple of points are often cited (Palman 1984).

1. If $A$ and $B$ are distinct vertices of a complete quadrangle, $C$ a diagonal point on the line $A B$ and $D$ the intersection of $A B$ with the line through other two diagonal points.

2. $A$ and $B$ are are the diagonal points of a complete rectangle, and $C$ and $D$ are the intersections of the line $A B$ with those mutually opposite sides of the complete rectangle which pass through the third diagonal point.

$A B U V$ is a complete rectangle, $C,W,O$ are its diagonal points and hence $(A, B, C, D)$ is a harmonic quadruple, the cross ratio is $(A,B;C,D)=-1$.

For the second criterium, consider the complete quadrangle $O V W U$, then $A, B$ are two out of 3 diagonal points, the third pair of the mutually opposite sides in the criterium are $O W$ and $V U$ (their intersection is the third diagonal point).

Now, one has to prove that if one takes $A,B,C$ distinct points that there is a unique $D$ with above properties. In the synthetic approach this means that the choice of points $U,V$ with $U,V,C$ colinear (or equivalently, the choice of points $U, O$) does not affect the position of $D$ in the construction. This however uses the fact that the plane is Desarguesian. For the projective planes over a commutative field this is hence automatic.

Literature

Modern treatment is in Chapter 6 of

• Marcel Berger, Géométrie, Cassini (Engl. transl. Geometry I, Springer)

A synthetic treatment is for example in

• Dominik Palman, Projektivna geometrija, Školska knjiga, Zagreb 1984.

category: geometry