Pseudo-Edge Unfoldings of Convex Polyhedra

Barvinok, Nicholas; Ghomi, Mohammad

doi:10.1007/s00454-019-00082-1

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…-Let T the number of triangles and E the number of edges in some geodesic triangulation of (S, m). We have E = 3 2 T , so that the Euler formula…”

Section: Flat Metricsmentioning

confidence: 99%

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A remark on spaces of flat metrics with cone singularities of constant sign curvatures

Fillastre¹,

Izmestiev²

2017

Séminaire De Théorie Spectrale Et Géométrie

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By a result of W. P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n − 3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions n − 3 and 1 2 (n − 1). By a result of W. Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a local structure of complex pseudo-spheres. The complex structure comes again from the area of the metric. The form can be degenerate; its signature depends on the curvatures prescribed. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.

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“…-Let T the number of triangles and E the number of edges in some geodesic triangulation of (S, m). We have E = 3 2 T , so that the Euler formula…”

Section: Flat Metricsmentioning

confidence: 99%

“…This construction is similar to unfolding a convex polytope. On the problem of a non-self-intersecting unfolding see [3].…”

Section: Positive Curvaturesmentioning

confidence: 99%

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

Fillastre¹,

Izmestiev²

2017

Séminaire De Théorie Spectrale Et Géométrie

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“…• There exists a convex polyhedron, equipped with 3-vertex-connected planar graph of geodesics partitioning the surface into regions metrically equivalent to convex polygons, that cannot be cut and unfolded along graph edges [6].…”

Section: Introductionmentioning

confidence: 99%

Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra

Demaine,

Eppstein

2020

Preprint

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We present new examples of topologically convex edgeununfoldable polyhedra, i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet cannot be cut along their edges and unfolded into one planar piece without overlap. One family of examples is acutely triangulated, i.e., every face is an acute triangle. Another family of examples is stacked, i.e., the result of face-to-face gluings of tetrahedra. Both families achieve another natural property, which we call very ununfoldable: for every k, there is an example such that every nonoverlapping multipiece edge unfolding has at least k pieces.

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Edge-unfolding nested prismatoids

Radons

2024

Computational Geometry

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Pseudo-Edge Unfoldings of Convex Polyhedra

Cited by 3 publications

References 18 publications

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

A remark on spaces of flat metrics with cone singularities of constant sign curvatures

Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra

Edge-unfolding nested prismatoids

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