In the analysis of blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. Poncelets theorem a talk by prof joe harris youtube. These conservation theorems are collectively called. Harris, on cayleys explicit solution to poncelets porism, lenseignement math. On the rotation index of bar billiards and poncelets porism cieslak, w. Ponceletdarboux curves, their complete decomposition and. This classical fact was known to motzkin and perhaps goes back even further. If a closed nedge polygonal line may be inscribed in the outer ellipse so as to circumscribe i. The aim of this note is to collect some mostly classical versions of this theorem, namely. Our main aim is to present an effective criterion for a complete decomposition of a transversal ponceletdarboux curve of degree n.
In the severe winter of 1812, he and his fellows were marched 800 km across the russian steppe to the prisoner camp of saratov. Our aim in this short note is to involve vector bundle techniques to propose a new proof of this celebrated result. Griffiths published a purely algebraic geo metric proof of poncelets result variations on a theorem of abel. Mirko radic, extreme areas of triangles in poncelets closure theorem, forum geometricorum, 4 2004 2326. We also show that, up to projective transformations, these two chains are unique. How we measure reads a read is counted each time someone views a. There are several proofs of the theorem, none of which is elementary.
Recent years have seen a flourishing revitalization of poncelets closure theorem. Poncelet theorem is equivalen t to poncelet s theorem for tw o conics in the plane. Riemann surfaces elliptic functions the modular function elliptic curves poncelet and cayley theorems. As a mathematician, his most notable work was in projective geometry, although an early collaboration with charles julien brianchon provided a. The theorem was suggested by poncelet in 1822 and proved by steiner in 1833. More precisely, we will show that poncelets theorem is a purely combinatorial consequence of pascals theorem.
Both poncelet and steiner were ardent supporters of synthetic geometry and disliked analytic methods to the extent of attacking those who used them. Arithmetic concerning poncelets closure theorem jaap top rug. Before we give several forms of the latter, let us introduce some. Auniversalmeasureforapencilof conics and the great. Professor joe harris from harvard university gave a talk at the columbia undergraduate math society on poncelets closure theorem. The story continues by introducing the reader to poncelets theorema beautiful result in projective geometry that ties together two conics and, in particular, two ellipses, one circumscribed by a polygon that is inscribed in the second. The ponceletsteiner theorem and double rulers springerlink. A vector bundle proof of poncelets closure theorem. In fact, as will be seen below, the poncelet theorem and addition theorem are essentially equival ent, so that at least in principle poncelet gave a synthetic derivation of the group law on an elliptic curve. Suppose that e0 is an ellipse in the plane and e1 is another ellipse. A poncelet theorem for lines jean vall es abstract. Among the triangles with the same incircle and circumcircle, we determine the ones with maximum and miniumum areas. The book demonstrates that poncelets the orem serves as a prism through which one can learn and appreciate a lot of beautiful mathematics.
Poncelets theorem cayleys theorem nongeneric cases the real case of poncelets theorem related topics. Basic notions of projective geometry conics intersection of two conics complex analysis. Therefore, it is with apologies to both poncelet and steiner that we will use analytic geometry in proving the theorem that bears both their names. Every pentagon is a poncelet polygon, and in fact there is a suitable labeling convention with respect to which t is the identity on c 5. Eulers formula that relates the circumradius, the inradius and the distance between the circumcenter and the incenter of a triangle serves the basis for the poncelet porism for triangles. A copy of the license is included in the section entitled gnu free documentation license. Let k and c be nondegenerate conics in general position. This file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. Theorem of the day poncelets porism suppose that two ellipses lie in the euclidean plane, with one totally enclosed by the other. With similar arguments as in the proof of poncelet s theorem for triangles corollary 2. Blaschke products, poncelets theorem, and operator theory pamela gorkin bucknell university october, 2015. The basic theorem relating the fundamental theorem of calculus to multidimensional in. Take a poncelet 3ellipse and the blaschke ellipse with same foci. A simple proof of poncelets theorem on the occasion of.
This process closes after n steps for some initial point p 0 2c if, and only if, it closes after n steps for any initial point on c. The aim of this paper is to give a simple proof of poncele ts theorem in the real projective plane. Because of the appeal of the poncelet theorem it seems reasonable to look for higherdimensional analogues. In geometry, poncelets porism, sometimes referred to as poncelets closure theorem, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all. In connection with these relations some special conics related to the angular points and sides of the quadrilateral are discussed. We present a proof of poncelets theorem in the real projective plane which relies only on pascals theorem. In this article we first discuss the early history of poncelet s closure theorem. In the great poncelet theorem, in place of two conics. Poncelets theorem poncelets closure theorem in jail at saratov, he wrote his famous closure theorem. Poncelets theorem says that if you shoot according to this rule. Pdf bicentennial of the great poncelet theorem 1820. Let k and c be nondegenerate conics in gen eral position. This book provides a thematic introduction to a variety of mathematical topics from the perspective of a single problem. We shall also name the coordinates x, y, z in the usual way.
Once we know this, proving poncelets theorem reduces to checking the cases e 0. We shall use a righthanded coordinate system and the standard unit coordinate vectors, k. Finally, the occurrence of conics in a poncelet grid 14 of lines constructed on a pencil of circles is explained with a simple proof. All euclidean geometric constructions can be carried out with a straightedge alone if, in addition, one is given the radius of a single circle and its center. Suppose there is an nsided polygon inscribed in k and. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1.
The theorem is of great depth in that it relates to a large and diverse body of mathematics. Theorem poncelet 2 given two triangles on a conic k, there exists a conic c circumscribed by them. If the file has been modified from its original state, some details such as the timestamp. Jeanvictor poncelet was a prisoner of war in napoleons russian campaign when 18 he proved this theorem porism meaning something like if its true in one case then its true in many or in. Stokes theorem 1 chapter stokes theorem in the present chapter we shall discuss r3 only. Factorization of homogeneous polynomials degenerate conics of a. If there is one polygon inscribed into some smooth conic and circumscribed about another one, then there are infinitely many such polygons. On the marked length spectrum of generic strictly convex billiard tables huang, guan, kaloshin, vadim, and sorrentino, alfonso, duke mathematical journal, 2018. Our aim is to prove a poncelet type theorem for a line con guration on the complex projective plane p2. Some relations in a complete quadrilateral are derived. The pentagram integrals for poncelet families sciencedirect. A group theoretic interpretation of poncelets theorem forum. A poncelet point in c n is an equivalence class of poncelet polygons.
More precisely, we say that a polygon with 2n sides joining 2n. Ponceletsteiner theorem hi everyone, a friend of mine asked me if i knew of any place where he could find the proof to the ponceletsteiner theorem, so since i knew of none, i thought someone here must know. Poncelet published his theorem poncelets porism in 1822, after he returned to france following his captivity as war prisoner in russia. The great poncelet theorem from projective geometry of conics has many deep interrelations and consequences. Poncelets theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. In the town of saratov where he was prisonner, poncelet, continuing the work of euler and steiner on polygons simultaneously. Fixed point theorems and applications vittorino pata dipartimento di matematica f.
A simple proof of poncelets theorem on the occasion of its. It concerns closed polygons inscribed in one conic and circumscribed about another. Poncelets theorem mathematical association of america. These are also the ones with maximum and minimum perimeters and sums of altitudes. Closed chains of conics carrying poncelet triangles pdf.
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