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geometry > Euclidean geometry > projective harmonic conjugate

geometry > affine geometry > projective harmonic conjugate

geometry > projective geometry > projective harmonic conjugate

Preferred term

projective harmonic conjugate

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 $(A, B; C, D) = -1$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Projective_harmonic_conjugate)

harmonic conjugate point

division harmonique

French
conjugaison harmonique

URI

http://data.loterre.fr/ark:/67375/PSR-NFS3J16J-V

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