The Poncelet Grid
نویسنده
چکیده
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.
منابع مشابه
N ov 2 00 5 The Poncelet grid and the billiard in an ellipse
1 The closure theorem and the Poncelet grid The Poncelet closure theorem (or Poncelet porism) is a classical result of pro-jective geometry. Given two nested ellipses, γ and Γ, one plays the following game: choose a point x on Γ, draw a tangent line to γ until it intersects Γ at point y, repeat the construction, starting with y, and so on. One obtains a polygonal line, inscribed into Γ and circ...
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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 ...
متن کاملPoncelet 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 کاملMarden theorem and Poncelet-Darboux curves
The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between the two theorems. We establish a dynamical equivalence between the full Marden theorem and the Poncelet-Darboux theorem. By introducing a class of isofocal def...
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 ...
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تاریخ انتشار 2005