Projective harmonic conjugate

In projective geometry, the harmonic conjugate point of a triple of points on the real projective line is defined by the following construction:

Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively. If AN and BM meet at K, and LK meets AB at D, then D is called the harmonic conjugate of C with respect to A, B.[1]

What is remarkable is that the point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N. This fact follows from Desargues theorem; it can also be defined in terms of the cross-ratio as (A, B; C, D) = −1.

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Cross-ratio criterion

The four points are sometimes called a harmonic range on the real projective line. When this line is endowed with the ordinary metric interpretation via real numbers, then the projective tool of cross-ratio is in force. Given this metric context, the harmonic range is characterized by a cross-ratio of minus one:

(A,B;C,D) = -1. \,

Reordering the points results in the set of three cross-ratios, {−1, 1/2, 2}, which is less than the expected 6 (it is stabilized by exchanging the last 2 points), and is known classically as the harmonic cross-ratio.

The cross-ratio criterion implies that distances from any one of these points to the three remaining points form harmonic progression.

In terms of a double ratio, given points a and b on an affine line, the division ratio[2] of a point x is

t(x) = \frac {x - a} {x - b} .

Note that when a < x < b , then t(x) is negative, and that it is positive outside of the interval. The cross-ratio (c,d;a,b) = t(c)/t(d) is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when t(c) %2B t(d) = 0, then c and d are projective harmonic conjugates with respect to a and b. So the division ratio criterion is that they be additive inverses.

In some school studies the configuration of a harmonic range is called harmonic division.

From complete quadrangle

Another approach to the harmonic conjugate is through the concept of a complete quadrangle such as KLMN in the above diagram. Based on four points, the complete quadrangle has pairs of opposite sides and diagonals. In the expression of projective harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:

D is the harmonic conjugate of C with respect to A and B, which means that there is a quadrangle IJKL such that one pair of opposite sides intersect at A, and a second pair at B, while the third pair meet AB at C and D.[3]

It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:

...Staudt succeeded in freeing projective geometry from elementary geometry. In his Geometrie der Lage Staudt introduced a harmonic quadruple of elements independently of the concept of the cross ratio following a purely projective route, using a complete quadrangle or quadrilateral.[4]

Projective conics

A conic in the projective plane is a curve C that has the following property: If P is a point not on C, and if a variable line through P meets C at points A and B, then the variable harmonic conjugate of P with respect to A and B traces out a line. The point P is called the pole of that line of harmonic conjugates, and this line is called the polar line of P with respect to the conic. See the article Pole and polar for more details.

Inversive geometry

Main article: Inversive geometry

In the case where the conic is a circle, on the extended diameters of the circle, projective hamonic conjugates with respect to the circle are inverses in a circle. This fact follows from one of Smogorzhevsky's theorems:

If circles k and q are mutually orthogonal, then a straight line passing through the center of k and intersecting q, does so at points symmetrical with respect to k.

That is, if the line is an extended diameter of k, then the intersections with q are projective harmonic conjugates.

References

  1. ^ R. L. Goodstein & E. J. F. Primrose (1953) Axiomatic Projective Geometry, University College Leicester (publisher). This text follows synthetic geometry. Harmonic construction on page 11
  2. ^ Dirk Struik (1953) Lectures on Analytic and Projective Geometry, page 7
  3. ^ H. S. M. Coxeter (1942) Non-Euclidean Geometry, page 29, University of Toronto Press
  4. ^ B.L. Laptev & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 41, Birkhäuser Verlag ISBN 3764350482