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 $$\Gamma $$ Γ is realized as the 1-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $$\Gamma $$ Γ is realized as the 1-skeleton of a polyhedron inscribed in the sphere and $$\Gamma $$ Γ admits a Hamiltonian cycle. This answers a question asked by Steiner in 1832. 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.

中文翻译：

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刻在二次曲面上的多面体

我们研究了内接于二次曲面的三空间中的凸多面体。在投影变换之前，存在三个这样的曲面：球面、双曲面和圆柱面。我们的主要结果是平面图 $$\Gamma $$Γ 被实现为内接在双曲面或圆柱体中的多面体的 1-骨架当且仅当 $$\Gamma $$Γ 被实现为一个内接在球体内的多面体和 $$\Gamma $$ Γ 承认一个哈密顿循环。这回答了 Steiner 在 1832 年提出的一个问题。 Rivin 通过研究双曲空间中理想多面体的几何形状，表征了内接在球体内的凸多面体。我们通过参数化反德西特几何和半管几何中凸理想多面体的空间来研究双曲面和圆柱的情况。正如圆柱体可以看作是球体和双曲面的退化，半管几何自然是双曲几何和反德西特几何的极限。我们提倡统一的观点来研究这三个案例。