The Poncelet annuli
نویسندگان
چکیده
منابع مشابه
Poncelet Theorems
The aim of this note is to collect some more or less classical theorems of Poncelet type and to provide them with short modern proofs. Where classical geometers used elliptic functions (or angular functions), we use elliptic curves (or degenerate elliptic curves decomposing into two rational curves). In this way we unify the geometry underlying these Poncelet type statements. Our starting point...
متن کاملThe Poncelet Grid
Given a convex polygon P in the projective plane we can form a finite “grid” of points by taking the pairwise intersections of the lines extending the edges of P . When P is a Poncelet polygon we show that this grid is contained in a finite union of ellipses and hyperbolas and derive other related geometric information about the grid.
متن کاملThe Pentagram Integrals for Poncelet Families
The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if P1 and P2 are two polygons in the same same Poncelet family, and f is a monodromy invariant for the pentagram map, then f(P1) = f(P2). Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence ...
متن کاملA Poncelet Theorem for Lines
Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective plane P. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2, · · · , A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, · · · , A2n. Then we prove : Theorem Let Ln be a configuration of n lines ...
متن کاملThe Poncelet Grid and Billiards in Ellipses
1. INTRODUCTION: THE PONCELET CLOSURE THEOREM. The Poncelet closure theorem (or Poncelet porism) is a classical result of projective geometry. Given nested ellipses γ and , with γ inside , one plays the following game: starting at a point x on , draw a tangent line to γ until it intersects at point y, repeat the construction , starting with y, and so on. One obtains a polygonal curve inscribed ...
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ژورنال
عنوان ژورنال: Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
سال: 2013
ISSN: 0138-4821,2191-0383
DOI: 10.1007/s13366-013-0142-x