TL;DR
This paper introduces Fregier ellipses, generalizing Fregier points, and explores their invariants within Poncelet configurations, linking Euclidean geometry with dynamical systems and affine transformations.
Contribution
It defines Fregier ellipses, relates them to Poncelet's theorem, and demonstrates invariant area sums in elliptical billiard configurations.
Findings
Invariant sum of areas for Fregier circles in Poncelet configurations
Special angles pi/3 and 2pi/3 are significant in the analysis
Connection between Fregier ellipses and affine maps of elliptical billiards
Abstract
We introduce Fregier ellipses which generalize the Fregier point in euclidean geometry. Subject is related to Dynamical Systems and Poncelet's closure theorem aka Poncelet porism and displaying geometric invariants (area,angles). Special angles pi/3 and 2pi/3 are emphasized and we show that we have invariant sum of areas for Fregier circles in general Poncelet configuration, occuring from an affine map of the elliptical billiard.
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Taxonomy
TopicsMathematics and Applications · Advanced Mathematical Theories and Applications · Advanced Mathematical Theories