Poncelet theorem pdf file

The aim of this note is to collect some mostly classical versions of this theorem, namely. In this article we first discuss the early history of poncelet s closure theorem. Professor joe harris from harvard university gave a talk at the columbia undergraduate math society on poncelets closure theorem. We also show that, up to projective transformations, these two chains are unique. Our aim in this short note is to involve vector bundle techniques to propose a new proof of this celebrated result. Our aim is to prove a poncelet type theorem for a line con guration on the complex projective plane p2. Both poncelet and steiner were ardent supporters of synthetic geometry and disliked analytic methods to the extent of attacking those who used them.

Angle measure and other measures the thing that made poncelets theorem so easy to prove for concentric circles is that all the line segments with endpoints on the outer circle that are tangent to the inner. Let k and c be nondegenerate conics in general position. The great poncelet theorem from projective geometry of conics has many deep interrelations and consequences. Blaschke products, poncelets theorem, and operator theory pamela gorkin bucknell university october, 2015. 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. Fixed point theorems and applications vittorino pata dipartimento di matematica f. Poncelets theorem cayleys theorem nongeneric cases the real case of poncelets theorem related topics. 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. Auniversalmeasureforapencilof conics and the great.

If there is one polygon inscribed into some smooth conic and circumscribed about another one, then there are infinitely many such polygons. A theorem of carnot valid for a triangle is extended to a quadrilateral. In connection with these relations some special conics related to the angular points and sides of the quadrilateral are discussed. In 1870 weyr proved a poncelet theorem in space more precisely for two quadrics that implies the one above when one quadric is a cone. Poncelets theorem a talk by prof joe harris youtube. The ponceletsteiner theorem and double rulers springerlink. The book demonstrates that poncelets the orem serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. 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. As a mathematician, his most notable work was in projective geometry, although an early collaboration with charles julien brianchon provided a. Stokes theorem 1 chapter stokes theorem in the present chapter we shall discuss r3 only. In the analysis of blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. We shall use a righthanded coordinate system and the standard unit coordinate vectors, k. There are several proofs of the theorem, none of which is elementary.

Theorem of the day poncelets porism suppose that two ellipses lie in the euclidean plane, with one totally enclosed by the other. Some relations in a complete quadrilateral are derived. Suppose there is an nsided polygon inscribed in k and. In the severe winter of 1812, he and his fellows were marched 800 km across the russian steppe to the prisoner camp of saratov. More precisely, we say that a polygon with 2n sides joining 2n. Let c and d be two conics on the complex projective plane such that there exists one ngon inscribed in d and circumscribed around c. Take a poncelet 3ellipse and the blaschke ellipse with same foci. We present a proof of poncelets theorem in the real projective plane which relies only on pascals theorem. More precisely, we will show that poncelets theorem is a purely combinatorial consequence of pascals theorem. Before we give several forms of the latter, let us introduce some. Poncelets theorem mathematical association of america.

More precisely, we say that a polygon with 2n sides joining 2n vertices a1, a2. 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. 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. On the rotation index of bar billiards and poncelets porism cieslak, w. 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. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. A copy of the license is included in the section entitled gnu free documentation license.

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. Theorem poncelet 2 given two triangles on a conic k, there exists a conic c circumscribed by them. Riemann surfaces elliptic functions the modular function elliptic curves poncelet and cayley theorems. A poncelet theorem for lines jean vall es abstract. 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. On the marked length spectrum of generic strictly convex billiard tables huang, guan, kaloshin, vadim, and sorrentino, alfonso, duke mathematical journal, 2018. Factorization of homogeneous polynomials degenerate conics of a. How we measure reads a read is counted each time someone views a. Recent years have seen a flourishing revitalization of poncelets closure theorem. Because of the appeal of the poncelet theorem it seems reasonable to look for higherdimensional analogues. Griffiths published a purely algebraic geo metric proof of poncelets result variations on a theorem of abel. The aim of this paper is to give a simple proof of poncele ts theorem in the real projective plane. Let k and c be nondegenerate conics in gen eral position. Once we know this, proving poncelets theorem reduces to checking the cases e 0.

Poncelets theorem says that if you shoot according to this rule. Suppose that e0 is an ellipse in the plane and e1 is another ellipse. 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. Closed chains of conics carrying poncelet triangles pdf. The theorem was suggested by poncelet in 1822 and proved by steiner in 1833. With similar arguments as in the proof of poncelet s theorem for triangles corollary 2. In the town of saratov where he was prisonner, poncelet, continuing the work of euler and steiner on polygons simultaneously. Ponceletdarboux curves, their complete decomposition and. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. If the file has been modified from its original state, some details such as the timestamp.

We shall also name the coordinates x, y, z in the usual way. 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. Finally, the occurrence of conics in a poncelet grid 14 of lines constructed on a pencil of circles is explained with a simple proof. These conservation theorems are collectively called. 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. In the great poncelet theorem, in place of two conics. Among the triangles with the same incircle and circumcircle, we determine the ones with maximum and miniumum areas. This book provides a thematic introduction to a variety of mathematical topics from the perspective of a single problem. These are also the ones with maximum and minimum perimeters and sums of altitudes. Poncelet published his theorem poncelets porism in 1822, after he returned to france following his captivity as war prisoner in russia. Blaschke products, poncelets theorem, and operator theory. Basic notions of projective geometry conics intersection of two conics complex analysis. A group theoretic interpretation of poncelets theorem forum.

Poncelets theorem poncelets closure theorem in jail at saratov, he wrote his famous closure theorem. A simple proof of poncelets theorem on the occasion of. Poncelets theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. Our aim is to prove a poncelet type theorem for a line configuration on the complex projective. A simple proof of poncelets theorem on the occasion of its. A vector bundle proof of poncelets closure theorem. This classical fact was known to motzkin and perhaps goes back even further. A poncelet point in c n is an equivalence class of poncelet polygons.