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Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.
Comment: 18 pages, 9 figures, 3 tables, 8 videos
Comment: 18 pages, 9 figures, 3 tables, 8 videos
Externí odkaz:
http://arxiv.org/abs/2112.02545
Autor:
Moses, Peter, Reznik, Dan
If one erects regular hexagons upon the sides of a triangle $T$, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with $T$, (ii) the construction can be extended iteratively,
Externí odkaz:
http://arxiv.org/abs/2112.02157
The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of $N$ points. We study this problem where the points are the vertices of a triangle and the circle
Externí odkaz:
http://arxiv.org/abs/2110.13615
Autor:
Reznik, Dan, Garcia, Ronaldo
We describe some three-dozen curious phenomena manifested by parabolas inscribed or circumscribed about certain Poncelet triangle families. Despite their pirouetting motion, parabolas' focus, vertex, directrix, etc., will often sweep or envelop rathe
Externí odkaz:
http://arxiv.org/abs/2110.06356
We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families interscribed in (i) the confocal pair and (ii) an outer ellipse and an inner concentric circular caus
Externí odkaz:
http://arxiv.org/abs/2106.00715
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 p
Externí odkaz:
http://arxiv.org/abs/2103.11260
We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to eit
Externí odkaz:
http://arxiv.org/abs/2102.09438
Autor:
Garcia, Ronaldo, Reznik, Dan
We study self-intersected N-periodics in the elliptic billiard, describing new facts about their geometry (e.g., self-intersected 4-periodics have vertices concyclic with the foci). We also check if some invariants listed in "Eighty New Invariants of
Externí odkaz:
http://arxiv.org/abs/2011.06640
Publikováno v:
Journal of Dynamical and Control Systems. 29:157-184
We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to eit
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::ea3b4cd727e854363d7a00fea059c675
https://doi.org/10.1007/s10883-021-09580-z
https://doi.org/10.1007/s10883-021-09580-z
Autor:
Peter Moses, Dan Reznik
Publikováno v:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.
If one erects regular hexagons upon the sides of a triangle $T$, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with $T$, (ii) the construction can be extended iteratively,
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::654183a27a1af5124b44e5895c566b69
https://doi.org/10.1007/s13366-022-00651-1
https://doi.org/10.1007/s13366-022-00651-1