Projective harmonic conjugate

In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as (A, B; C, D) = −1.

The four points are sometimes called a harmonic range (on the real projective line) as it is found that D always divides the segment AB internally in the same proportion as C divides AB externally.

{\displaystyle {\overline {AC}}:{\overline {BC}}={\overline {AD}}:{\overline {DB}}\,.}

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 (sometimes double 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 (and there are six such selections possible).

But for a harmonic range in particular there are just three values of cross ratio: {−1, 1/2, 2}, 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, b on an affine line, the division ratio[2] of a point x is

Note that when a < x < b, then t(x) is negative, and that it is positive outside of the interval.

Setting the double ratio to minus one means that when t(c) + t(d) = 0, 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.

By the cross-ratio criterion, the harmonic conjugate of x will be y when t(y) = 1.

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.

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: 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.

[6] A set of four in such a relation has been called a harmonic quadruple.

[7] 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.

See the article Pole and polar for more details.

This fact follows from one of Smogorzhevsky's theorems:[8] That is, if the line is an extended diameter of k, then the intersections with q are harmonic conjugates.

it is easier to verify by substitution that the following expressions are the solutions, i.e.

(which makes it self-inverse) is known as the harmonic cross ratio.

is the chord of contact of the tangents to the ellipse from

One can also sho that the directrix of the ellipse is the polar of the focus.

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(2, q) for q = 5, 7, 9.

[10] Let P0, P1, P2 be three different points on the real projective line.

Consider the infinite sequence of points Pn, where Pn is the harmonic conjugate of Pn-3 with respect to Pn-1, Pn-2 for n > 2.

for large n. For an infinite limit we have For a proof consider the projective isomorphism with