The Talented Mr. Inversive Triangle in the Elliptic Billiard

Reznik, Dan; Garcia, Ronaldo; Helman, Mark

doi:10.48550/arxiv.2012.03020

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Appendix B Polar Image Of Bicentric Pair1

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2021

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“…N=3 case. Referring to Figure 5, the perimeter L † of the inversive polygon for the N = 3 family, originally derived in [15,Prop. 4] is given by:…”

Section: Appendix B Polar Image Of Bicentric Pairmentioning

confidence: 99%

New Invariants of Poncelet-Jacobi Bicentric Polygons

Roitman,

Garcia,

Reznik

2021

Preprint

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The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).

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“…N=3 case. Referring to Figure 5, the perimeter L † of the inversive polygon for the N = 3 family, originally derived in [15,Prop. 4] is given by:…”

Section: Appendix B Polar Image Of Bicentric Pairmentioning

confidence: 99%