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Concept
Projective harmonic conjugate
Summary
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points 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 and B.
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 (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 internally in the same proportion as C divides 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 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 harmonic conjugates with respect to a and b. So the division ratio criterion is that they be additive inverses.
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