Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Aurenhammer, Franz; Walzl, Gernot

doi:10.1007/s00454-016-9811-5

Cited by 7 publications

(14 citation statements)

References 33 publications

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“…That is, in general there exists more than one surface that can be regarded as the mitered offset surface of T . See Aurenhammer and Walzl [3], who credit Erickson for this observation.…”

Section: Offsettingmentioning

confidence: 92%

“…However, very little is known on such a suitable preprocessing for computing (families of) mitered offsets in 3D. Aurenhammer and Walzl [3] provide the first complete analysis of straight skeletons in three dimensions. However, it is not obvious how to implement their approach in full generality such that it can cope with arbitrary polyhedral objects in 3D.…”

Section: Offsettingmentioning

confidence: 99%

“…Straight skeletons of polyhedral objects in 3D were studied by Barequet et al [4] and, more recently, by Aurenhammer and Walzl [3]. However, while combinatorial complexities have been established for the straight skeleton of polytopes, no runtime bounds have been investigated.…”

Section: Straight Skeleton In 3dmentioning

confidence: 99%

“…This is due to the fact that the offset facets of four (or more) triangular input facets need not intersect in a point. See Aurenhammer and Walzl [3] for more details.…”

Section: Wavefront Propagation and Eventsmentioning

confidence: 99%

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Skeletal Structures for Modeling Generalized Chamfers and Fillets in the Presence of Complex Miters

Held¹,

Palfrader²

2018

CAD&A

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We present a simple algorithm to compute the straight skeleton and mitered offset surfaces of a polyhedral terrain in 3D, i.e., of a z-monotone piecewise-linear surface. Like its 2D pedant, the 3D straight skeleton is the result of a wavefront propagation process, which we simulate in order to construct the skeleton in (worst-case) time O(n 4 log n), where n is the number of vertices of the terrain. Any mitered offset surface can then be obtained from the skeleton in time linear in the combinatorial size of the skeleton.

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“…That is, in general there exists more than one surface that can be regarded as the mitered offset surface of T . See Aurenhammer and Walzl [3], who credit Erickson for this observation.…”

Section: Offsettingmentioning

confidence: 92%

Section: Offsettingmentioning

confidence: 99%

Section: Straight Skeleton In 3dmentioning

confidence: 99%

“…This is due to the fact that the offset facets of four (or more) triangular input facets need not intersect in a point. See Aurenhammer and Walzl [3] for more details.…”

Section: Wavefront Propagation and Eventsmentioning

confidence: 99%

See 2 more Smart Citations

Skeletal Structures for Modeling Generalized Chamfers and Fillets in the Presence of Complex Miters

Held¹,

Palfrader²

2018

CAD&A

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“…For x-monotone rectilinear polygons, a linear time algorithm was recently introduced [7]. In 3D, an analogous equivalence of the straight skeleton of orthogonal polyhedra and the L ∞ Voronoi diagram exists [4] and a complete analysis of 3D straight skeletons is provided in [3]. Specifically for 3D orthogonal polyhedra, in [4] they offer two algorithms that construct the skeleton in O(min{V 2 log V, k log O(1) V }), where k = O(V 2 ) is the number of skeleton features.…”

Section: Introductionmentioning

confidence: 99%

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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