Complex_Cross

Complex cross ratio (ABCD) = (( A-C)/(B-C))/((A-D)/(B-D))

The complex cross ratio of four points in the (complex) plane is defined to be:
(ABCD) = ((A-C)/(B-C))/((A-D)/(B-D)), where points are identified with complex numbers.
1) (ABCD) is real if the points are all on a circle or line.
2) Assuming the points on a conic (ellipse), project them on a line e, from a point P of the conic. Let A*, B*, C*, D* the corresponding projections. Then set cx= (ABCD) and cr = (A*B*C*D*).

Below the two cross ratios are calculated using the User-Tool [ComplexCrossRatio] found and compiled in the file [EUC_Scripts\EUC_User_Tools\ComplexCrossRatio.txt]. The two cross ratios are points of the complex plane. The second being real and lying on the real axis.

cx and cr remain invariant on
a) modifying the line (position and/or orientation),
b) modifying the point P on the ellipse (switch to select-on-contour-tool (CTRL+2), catch and modify points P, A, B, C, D).

The case of constancy cr = (A*B*C*D*) on a varying line e and/or the location of P on the conic depends on a basic property of homographic relations: they preserve the cross ratio. This is discussed in CrossRatio.html .
The complex cross ration cx trivially remains constant since, varying P does not affect its value ((A-C)/(B-C))/((A-D)/(B-D)).
A picture of another interesting case, related to four tangents of a conic, is given in the file: FourTangentsCrossRatio.html .

Complex cross ratio (ABCD) = (( A-C)/(B-C))/((A-D)/(B-D))

See Also