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; it 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. 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 (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 and b on an affine line, the division ratio 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. 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) + 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.

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