Poncelet’s porism: a long story of renewed discoveries, I

@article{Centina2016PonceletsPA,
  title={Poncelet’s porism: a long story of renewed discoveries, I},
  author={Andrea Del Centina},
  journal={Archive for History of Exact Sciences},
  year={2016},
  volume={70},
  pages={1-122}
}
  • A. Centina
  • Published 2016
  • Mathematics
  • Archive for History of Exact Sciences
In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern… Expand
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  • Mathematics, Computer Science
  • Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
  • 2021
TLDR
It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles, and that when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). Expand
Poncelet-Darboux, Kippenhahn, and Szeg\H{o}: interactions between projective geometry, matrices and orthogonal polynomials
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that allExpand
Closed chains of conics carrying poncelet triangles
We investigate closed chains of conics which carry Poncelet triangles. In particular, we show that every chain of conics which carries Poncelet triangles can be closed. Furthermore, for $$k=3$$k=3Expand
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TLDR
It is shown that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles. Expand
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We describe a newly-developed, free, browser-based application, for the interactive exploration of the dynamic geometry of Poncelet families of triangles. The main focus is on responsive display ofExpand
On two extensions of Poncelet theorem
Abstract In this note we provide two extensions of a particular case of Poncelet theorem.
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References

SHOWING 1-10 OF 121 REFERENCES
I. On the porism of the in-and-circumscribed polygon
  • A. Cayley
  • Mathematics
  • Proceedings of the Royal Society of London
  • 1862
The Porism referred to is as follows, viz. two conics may be so related to each other, that a polygon may be inscribed in the one, and circumscribed about the other conic, in such manner that anyExpand
Mathematical Thought from Ancient to Modern Times
This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creationExpand
Hidden Harmony―Geometric Fantasies: The Rise of Complex Function Theory
The book is a history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place. It is the first history of mathematics devoted to complexExpand
James Joseph Sylvester: Jewish Mathematician in a Victorian World
Here, in this first biographical study of James Joseph Sylvester, Karen Hunger Parshall makes a signal contribution to the history of mathematics, Victorian history, and the history of science. AExpand
Toward a History of Nineteenth-Century Invariant Theory
Publisher Summary Before Gauss considered binary forms in his Disquisitiones Arithmeticae , Joseph-Louis Lagrange had encountered and dealt with the problem of transformation of homogeneousExpand
Bicentennial of the Great Poncelet Theorem (1813-2013): Current Advances
The paper gives a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of Geometry,Expand
The British development of the theory of invariants (1841–1895)
The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s,Expand
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 angularExpand
The rise of Cayley's invariant theory (1841–1862)
Abstract In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation.Expand
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