Elliptic Billiard Capoeira: N=6 Zero-Area Focus-Antipedals and Constant Perimeter Focus-Inversives
Show in the family of 6-periodics (blue) in the a/b=2 elliptic billiard. 1) The inversive polygon (pink) wrt to the left focus (f1) has invariant perimeter and it is inscribed in a Pascal Limaçon (dashed pink). This invariance holds for all N and a/b. 2) The antipedal polygon (olive green) of the N-periodics wrt f1 at this choice of N and aspect ratio (6 and 2 respectively) has zero signed-area. Its locus is a fish curve with 3 self-intersections (dashed olive green). 3) The vertices of the pedal polygon of the inversive wrt f1 (orange) has a cusped limaçon-like locus (dashed orange). Not shown: the pedal polygons of N-periodics wrt to f1: this family is interscribed within a pair of non-concentric circles, see [1]. [1] D. Reznik et al, "Family of Pedal Polygons to N-Periodics", 2020. https://youtu.be/7TE3a5vEWuU
Show in the family of 6-periodics (blue) in the a/b=2 elliptic billiard. 1) The inversive polygon (pink) wrt to the left focus (f1) has invariant perimeter and it is inscribed in a Pascal Limaçon (dashed pink). This invariance holds for all N and a/b. 2) The antipedal polygon (olive green) of the N-periodics wrt f1 at this choice of N and aspect ratio (6 and 2 respectively) has zero signed-area. Its locus is a fish curve with 3 self-intersections (dashed olive green). 3) The vertices of the pedal polygon of the inversive wrt f1 (orange) has a cusped limaçon-like locus (dashed orange). Not shown: the pedal polygons of N-periodics wrt to f1: this family is interscribed within a pair of non-concentric circles, see [1]. [1] D. Reznik et al, "Family of Pedal Polygons to N-Periodics", 2020. https://youtu.be/7TE3a5vEWuU