DesarguesInvolutionComplex

Desargues involution theorem in the complex projective space

The theorem of Desargues for families of conics passing through four points (see DesarguesInvolution.html ) is valid also for conics in the complex projective space PC².
The formal definition and the proof of the theorem are the same as in the real case.
A particular case arises for the family of real conics generated by two (real) conics c₁, c₂ which have no real intersection point at all. For any line (e), though, the intersections of the member conics with the line define point X₁, X₂ in involution.

The proof results from the theorem in the complex projective plane, by considering the conics c₁ and c₂ as conics of that plane. The circles (d) with diameters X₁X₂ on line (e) build a circle bundle. This is again a particular case of the above theorem.

Desargues involution theorem in the complex projective space

See Also