52 research outputs found
Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms
The thirty years old programme of Griffiths and Harris of understanding
higher-dimensional analogues of Poncelet-type problems and synthetic approach
to higher genera addition theorems has been settled and completed in this
paper. Starting with the observation of the billiard nature of some classical
constructions and configurations, we construct the billiard algebra, that is a
group structure on the set T of lines in simultaneously tangent to d-1
quadrics from a given confocal family. Using this tool, the related results of
Reid, Donagi and Knoerrer are further developed, realized and simplified. We
derive a fundamental property of T: any two lines from this set can be obtained
from each other by at most d-1 billiard reflections at some quadrics from the
confocal family. We introduce two hierarchies of notions: s-skew lines in T and
s-weak Poncelet trajectories, s = -1,0,...,d-2. The interrelations between
billiard dynamics, linear subspaces of intersections of quadrics and
hyperelliptic Jacobians developed in this paper enabled us to obtain
higher-dimensional and higher-genera generalizations of several classical genus
1 results: the Cayley's theorem, the Weyr's theorem, the Griffiths-Harris
theorem and the Darboux theorem.Comment: 36 pages, 11 figures; to be published in Advances in Mathematic
Poncelet Plectra: Harmonious Curves in Cosine Space
It has been shown that the family of Poncelet N-gons in the confocal pair
(elliptic billiard) conserves the sum of cosines of its internal angles.
Curiously, this quantity is equal to the sum of cosines conserved by its affine
image where the caustic is a circle. We show that furthermore, (i) when N=3,
the cosine triples of both families sweep the same planar curve: an equilateral
cubic resembling a plectrum (guitar pick). We also show that (ii) the family of
triangles excentral to the confocal family conserves the same product of
cosines as the one conserved by its affine image inscribed in a circle; and
that (iii) cosine triples of both families sweep the same spherical curve. When
the triple of log-cosines is considered, this curve becomes a planar,
plectrum-shaped curve, rounder than the one swept by its parent confocal
family.Comment: 15 pages, 13 figures, 7 video link
Polygons, conics and billiards
In this expository article we will describe some elementary properties of billiards and Poncelet maps and special attention is dedicated to N-periodic orbits. In general, problems involving billiards are easy to state and under-standing, and difficult or laborious to solve
Poncelet's theorem and Billiard knots
Let be any elliptic right cylinder. We prove that every type of knot can
be realized as the trajectory of a ball in This proves a conjecture of
Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use
Jacobi's proof of Poncelet's theorem by means of elliptic functions
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