Straight Skeletons of Three-Dimensional Polyhedra

Barequet, Gill; Eppstein, David; Goodrich, Michael T.; Vaxman, Amir

doi:10.1007/978-3-540-87744-8_13

Cited by 22 publications

(21 citation statements)

References 25 publications

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“…As a consequence, SK(v) can be computed in (roughly) O(m 4 3 ) time, when m is the degree of v. The next assertion, which has been stated informally in [12], shows the relevance of SK(v) for resolving the pointed vertex v.…”

Section: Lemma 41 In the Edge Events That Occur In The Construction mentioning

confidence: 71%

“…Barequet et al [12] proposed the interesting idea of using the incidence relations of the weighted straight skeleton SK(v) to resolve the vertex v. We will elaborate on the properties of this skeleton in some more detail here.…”

Section: Pointed Case Revisitedmentioning

confidence: 99%

“…Indeed, we encounter this problem already 'in the first moment', when polytope vertices of higher degree have to be resolved into several vertices with different incidence structure on the offset polytope surface: In general, only polytope vertices of degree 3 behave like polygon vertices, in the sense that they do not split when the object is shrunk infinitesimally. An elegant idea how to reduce the vertex resolution problem to a two-dimensional problem is proposed in Barequet et al [12], using the combinatorial structure of the weighted 1 straight skeleton [7,13,21] in a sectional plane that cuts off the polytope vertex in question. It remains unclear, though, how to generalize this method to vertices whose incident polytope faces positively span 3-space (for example, saddle point type vertices), as the use of more than one sectional plane leads to the necessity of merging planar straight skeletons, which is an unsolved problem.…”

mentioning

confidence: 99%

“…[27]. Similarly, the straight skeleton for axis-aligned (or orthogonal) polytopes is the medial axis in the L ∞ -metric, and has a quadratic behavior in size as well; the computation time increases by a polylogarithmic factor [12,25]. In these particular settings, the straight skeleton is a unique structure.…”

mentioning

confidence: 99%

“…No algorithmic results or nontrivial upper size bounds have been known for (more) general polytopes. However, a superquadratic lower bound of (n 2 α 2 (n)) on the skeleton size exists [12].…”

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confidence: 99%

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Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Aurenhammer

Walzl

2016

Discrete Comput Geom

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We give a concise definition of mitered offset surfaces for nonconvex polytopes in R 3 , along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification of their constructing events in the generic case. Our results extend to the weighted setting, to a larger class of polytope decompositions, and to general dimensions. For (weighted) straight skeletons of an n-facet polytope in R d , an upper bound of O(n d ) on their combinatorial complexity is derived. It relies on a novel layer partition for straight skeletons, and improves the trivial bound by an order of magnitude for d ≥ 3.

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Section: Lemma 41 In the Edge Events That Occur In The Construction mentioning

confidence: 71%

Section: Pointed Case Revisitedmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Aurenhammer

Walzl

2016

Discrete Comput Geom

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Voronoi Diagram of Orthogonal Polyhedra in Two and Three Dimensions

Emiris¹,

Katsamaki²

2019

Lecture Notes in Computer Science

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Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axisaligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly useful scenario in certain application domains. We construct the exact Voronoi diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoi diagram and yields a skeletal representation of the input object. We introduce a complete algorithm in 2D and 3D that follows the subdivision paradigm relying on a bounding-volume hierarchy; this is an original approach to the problem. The complexity is adaptive and comparable to that of previous methods. Under a mild assumption it is O(n/∆ + 1/∆ 2 ) in 2D or O(nα 2 /∆ 2 + 1/∆ 3 ) in 3D, where n is the number of sites, namely edges or facets resp., ∆ is the maximum cell size for the subdivision to stop, and α bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm.

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