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.

Cross-ratio criterion

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:

A

C

¯

:

B

C

¯

=

A

D

¯

:

D

B

¯

.

{\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)

(

A

,

B

;

C

,

D

)

=

A

C

¯

A

D

¯

/

B

C

¯

−

D

B

¯

,

{\displaystyle (A,B;C,D)={\frac {\overline {AC}}{\overline {AD}}}\left/{\frac {\overline {BC}}{-{\overline {DB}}}}\right.,}

for which a harmonic range is characterized by a value of −1. We therefore write:

(

A

,

B

;

C

,

D

)

=

A

C

¯

A

D

¯

×

B

D

¯

B

C

¯

=

−

1.

{\displaystyle (A,B;C,D)={\frac {\overline {AC}}{\overline {AD}}}\times {\frac {\overline {BD}}{\overline {BC}}}=-1.}

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 of a point x is

t

(

x

)

=

x

−

a

x

−

b

.