Article (Scientific journals)
Polyhedra inscribed in a quadric
Danciger, Jeffrey; Maloni, Sara; SCHLENKER, Jean-Marc
2020In Inventiones Mathematicae, 221 (1), p. 237-300
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Abstract :
[en] We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $\Gamma$ is realized as the $1$--skeleton of a polyhedron inscribed in the sphere and $\Gamma$ admits a Hamiltonian cycle. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.
Disciplines :
Mathematics
Author, co-author :
Danciger, Jeffrey;  University of Texas at Austin > mathematics
Maloni, Sara;  Brown University > Mathematics
SCHLENKER, Jean-Marc ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
yes
Language :
English
Title :
Polyhedra inscribed in a quadric
Publication date :
May 2020
Journal title :
Inventiones Mathematicae
ISSN :
1432-1297
Publisher :
Springer, Germany
Volume :
221
Issue :
1
Pages :
237-300
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBilu :
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