Poncelet Triangles: a Theory for Locus Ellipticity
Metric Geometry
2021-12-14 v4 Computational Geometry
Robotics
Algebraic Geometry
Dynamical Systems
Abstract
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 caustic. Previously, determining if a locus was a conic was done on a case-by-case basis. In the confocal case, we also derive conditions under which a locus degenerates to a segment or a circle. We show the locus' turning number is +/- 3, while predicting its monotonicity with respect to the motion of a vertex of the triangle family.
Cite
@article{arxiv.2106.00715,
title = {Poncelet Triangles: a Theory for Locus Ellipticity},
author = {Mark Helman and Dominique Laurain and Dan Reznik and Ronaldo Garcia},
journal= {arXiv preprint arXiv:2106.00715},
year = {2021}
}
Comments
10 pages, 5 figures, 2 tables, and 6 video links