Projective harmonic conjugate


In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction:
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.
In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as .

Cross-ratio criterion

The four points are sometimes called a harmonic range as it is found that D always divides the segment AB internally in the same proportion as C divides AB externally. That is:
If these segments are now endowed with the ordinary metric interpretation of real numbers they will be signed and form a double proportion known as the cross ratio
for which a harmonic range is characterized by a value of −1. We therefore write:
The value of a cross ratio in general is not unique, as it depends on the order of selection of segments. But for a harmonic range in particular there are just three values of cross ratio: since −1 is self-inverse – so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the harmonic cross-ratio.
In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is
Note that when, then t is negative, and that it is positive outside of the interval.
The cross-ratio is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when, then c and d are harmonic conjugates with respect to a and b. So the division ratio criterion is that they be additive inverses.
Harmonic division of a line segment is a special case of Apollonius' definition of the circle.
In some school studies the configuration of a harmonic range is called harmonic division.

Of midpoint

When x is the midpoint of the segment from a to b, then
By the cross-ratio criterion, the harmonic conjugate of x will be y when. But there is no finite solution for y on the line through a and b. Nevertheless,
thus motivating inclusion of a point at infinity in the projective line. This point at infinity serves as the harmonic conjugate of the midpoint x.

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 harmonic conjugates by H. S. M. Coxeter, the diagonals are considered a pair of opposite sides:
It was Karl von Staudt that first used the harmonic conjugate as the basis for projective geometry independent of metric considerations:
To see the complete quadrangle applied to obtaining the midpoint, consider the following passage from J. W. Young:

Quaternary relations

Four ordered points on a projective range are called harmonic points when there is a tetrastigm in the plane such that the first and third are codots and the other two points are on the connectors of the third codot.
If p is a point not on a straight with harmonic points, the joins of p with the points are harmonic straights. Similarly, if the axis of a pencil of planes is skew to a straight with harmonic points, the planes on the points are harmonic planes.
A set of four in such a relation has been called a harmonic quadruple.

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

In the case where the conic is a circle, on the extended diameters of the circle, harmonic conjugates with respect to the circle are inverses in a circle. This fact follows from one of Smogorzhevsky's theorems:
That is, if the line is an extended diameter of k, then the intersections with q are harmonic conjugates.

Galois tetrads

In Galois geometry over a Galois field GF a line has q + 1 points, where ∞ =. In this line four points form a harmonic tetrad when two harmonically separate the others. The condition
characterizes harmonic tetrads. Attention to these tetrads led Jean Dieudonné to his delineation of some accidental isomorphisms of the projective linear groups PGL for q = 5, 7, and 9.
If q = 2n, then the harmonic conjugate of C is itself.

Iterated projective harmonic conjugates and the golden ratio

Let be three different points on the real projective line. Consider the infinite sequence of points where is the harmonic conjugate of with respect to for This sequence is convergent.
For a finite limit we have
where is the golden ratio, i.e. for large.
For an infinite limit we have
For a proof consider the projective isomorphism
with