foci of bicircular quartics

 this activity is a page of geogebra-book elliptic functions & bicircular quartics & . . .(07.08.2023)

07.08.2023

Four different points in the complex plane can always be mapped by a suitable Möbius transformation to 4 points in normal form with . Depending on the order of the points the complex double ratio . Absolutely invariant is .
  • 4 distinct points are the foci of confocal bicircular quartics, if their absolute invariant is real.
  • If is real and non-negative, the foci are concyclic.
  • If is real and not positiv, then the foci in 2 pairs lie mirror-inverted on 2 orthogonal circles.
  • If , then both are true: the foci lie harmonically.
In the appplet above, the absolute invariant is real if lies on one of the axes, or on the unit circle, or on one of the bisecting circles or straight lines! The highest symmetry is in the tetrahedron case: coincides with one of the points, in which 3 angle bisectors intersect: .
– Walter Füchte

Information: foci of bicircular quartics