2,709 research outputs found
Poncelet curves
No full textWe examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the sense that they produce all possible sufficiently regular pairs of such Poncelet curves. We arrive at similar results for sets of curves, analogous to the pencil of conic sections in the full Poncelet theorem. We also study the case of Poncelet curves that carry Poncelet polygons which are equiangular or even congruent
A characterization of complex plane Poncelet curves
No full textAbstractWe consider algebraic curves in the complex affine plane. A natural extension of the existing definition of Poncelet curves in the real plane to the complex plane is presented. Three equivalent polynomial equations in tangent coordinates are given for complex plane Poncelet curves: (a) the polynomial which generates the Bezoutian form with parameters—the foci of the curve; (b) the Darboux equation with parameters—the vertices of a Poncelet polygon; (c) the determinant equation involving matrices having certain specific properties. We use these polynomials in order to solve Poncelet-type problems. Namely, criteria are proved for real Poncelet curves to be generated by matrices that admit unitary bordering. These criteria answer the question when a convex Poncelet curve which is inscribed in a convex polygon is the boundary of a numerical range of a matrix.We also demonstrate that the general theorems of the first three sections may shorten the proofs of some known results
Poncelet a la presó de Saratov
No full textPoncelet va ser un militar francès que va formar part de les tropes napoleòniques que van perdre la batalla de Krasni a la campanya de Rússia. Durant el seu captiveri a la presó de Saratov, basant-se en el record de les lliçons inspiradores de Gaspard Monge a l'École Polytechnique, va reflexionar sobre els fonaments de la geometria projectiva i en va donar un punt de vista original que acabarà tenint una gran influència en el desenvolupament de la geometria al llarg del segle xix. Entre els seus resultats, el més conegut és l'anomenat porisma de Poncelet, que estudia l'existència de polígons amb vèrtexs en una cònica C i costats tangents a una altra cònica D en funció de la posició relativa de C i D. En aquest article repassem alguns fets de la vida de Poncelet i donem la idea d'una demostració del porisma
Effective shear speed in two-dimensional phononic crystals
No full textThe quasistatic limit of the antiplane shear-wave speed ('effective speed') c in 2D periodic lattices is studied. Two new closed-form estimates of c are derived by employing two different analytical approaches. The first proceeds from a standard background of the plane wave expansion (PWE). The second is a new approach, which resides in x-space and centers on the monodromy matrix (MM) introduced in the 2D case as the multiplicative integral, taken in one coordinate, of a matrix with components being the operators with respect to the other coordinate. On the numerical side, an efficient PWE-based scheme for computing c is proposed and implemented. The analytical and numerical findings are applied to several examples of 2D square lattices with two and three high contrast components, for which the new PWE and MM estimates are compared with the numerical data and with some known approximations. It is demonstrated that the PWE estimate is most efficient in the case of densely packed stiff inclusions, especially when they form a symmetric lattice, while in general it is the MM estimate that provides the best overall fitting accuracy.Peer reviewe
Effective Willis constitutive equations for periodically stratified anisotropic elastic media
No full textA method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series and explicit estimates for the rate of convergence are derived.Peer reviewedReceived July 22, 2010; accepted December 15, 2010; published online April 21, 2011. Manuscript dated December 21, 2013
Approximated Poncelet configurations
No full textIn this short note we present the approximate construction of closed Poncelet configurations using the simulation of a mathematical pendulum. Although the idea goes back to the work of Jacobi, only the use of modern computer technologies assures the success of the construction. We present also some remarks on using such problems in project based university courses and we present a Matlab program able to produce animated Poncelet configurations with given period. In the same spirit we construct Steiner configurations and we give a few teaching oriented remarks on the Poncelet grid theorem. (DIPF/authors
Poncelet Porisms and Beyond
No full textThe goal of the book is to present, in a complete and comprehensive way, areas of current research interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. The most important results and ideas, classical as well as modern, connected to the Poncelet theorem are presented, together with a historical overview analyzing the classical ideas and their natural generalizations. Special attention is paid to the realization of the Griffiths and Harris programme about Poncelet-type p
Polígonos de Poncelet
No full textDesde el año 2018, celebramos en un día especial del año los diversos logros de las mujeres en el ámbito matemático; de mujeres que durante siglos permanecieron en la sombra y tuvieron obstáculos incluso para poder acceder a la educación formal. Hoy en día, en pleno siglo XXI, a pesar de haber visitado el espacio, tener supercomputadoras a nuestro servicio diario, estar inmersos en el desarrollo y explotación de la inteligencia artificial y ver que los automóviles se empiezan a conducir solos, estos logros de las mujeres en algunos escenarios continúan siendo subvalorados y muy poco reconocidos; esto, sin duda, habla muy mal de la sociedad en que vivimos y habla peor del medio científico en el que tantos desarrollos se han logrado.Jean-Victor Poncelet fue un ingeniero francés que estudió en la École Polytechnique bajo la enseñanza de destacados matemáticos como Ampére y Monge. Tras graduarse, se unió al ejército de Napoleón y fue capturado durante la campaña en Rusia en 1812. Durante su cautiverio en Saratov, repasó y desarrolló conceptos matemáticos, lo que lo llevó a publicar su obra fundamental sobre geometría proyectiva. El artículo menciona también uno de sus teoremas más conocidos y su relación con los billares, así como algunas preguntas aún no resueltas sobre este trabajo, más de 200 años despuésJean-Victor Poncelet was a French engineer who studied at the École Polytechnique under the guidance of renowned mathematicians such as Ampére and Monge. After graduating, he joined Napoleon's army and was captured during the 1812 Russian campaign. While imprisoned in Saratov, he spent his time recalling and developing mathematical concepts, which later led to the publication of his foundational work on projective geometry. The article also discusses one of Poncelet’s most famous theorems, its connection to billiards, and some questions that remain unanswered more than 200 years after he formulated the theorem
À propos des variétés de Poncelet
No full text7 pagesInternational audienceWe recall the definition of Poncelet varieties that generalize the celebrated Poncelet curves introduced by Darboux. We show that any quadric and any smooth cubic in the projective space of dimension three is a Poncelet surface but that a general surface of degree greater than four is not
Explicit Constructions for Poncelet Polygons
No full textWe study the geometric structure of Poncelet -gons from a projective point of view. In particular we present explicit constructions of Poncelet -gons for certain and derive algebraic characterisations in terms of bracket polynomials. Via the connections of Poncelet polygons and -configurations, the results of this article can be used to construct a large class of specific movable -configurations, the trivial celestial 4-configurations, which up to this point were all thought to be rigid and to require regular polygons for their construction.35 pages, 20 figure
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