New Invariants of Poncelet-Jacobi Bicentric Polygons
| Autor: | Dan Reznik, Ronaldo Garcia, Pedro Roitman |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Computational Geometry (cs.CG)
FOS: Computer and information sciences Pure mathematics General Mathematics Elliptic function FOS: Physical sciences Metric Geometry (math.MG) Limiting Dynamical Systems (math.DS) Mathematical Physics (math-ph) Computer Science::Computational Geometry Perimeter Corollary Mathematics - Metric Geometry FOS: Mathematics Mathematics::Metric Geometry Computer Science - Computational Geometry Invariant (mathematics) Dynamical billiards Mathematics - Dynamical Systems 51M04 51N20 51N35 68T20 Constant (mathematics) Internal and external angle Mathematical Physics Mathematics |
| Popis: | 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). 17 pages, 6 figures, 1 table with 18 video links |
| Databáze: | OpenAIRE |
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