Constructing L∞ Voronoi Diagrams in 2D and 3D

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dc.contributor.authorBukenberger, Dennis R.en_US
dc.contributor.authorBuchin, Kevinen_US
dc.contributor.authorBotsch, Marioen_US
dc.contributor.editorCampen, Marcelen_US
dc.contributor.editorSpagnuolo, Michelaen_US
dc.date.accessioned2022-06-27T16:19:54Z
dc.date.available2022-06-27T16:19:54Z
dc.date.issued2022
dc.description.abstractVoronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p -> ∞ lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm.en_US
dc.description.number5
dc.description.sectionheadersTiling and Nesting
dc.description.seriesinformationComputer Graphics Forum
dc.description.volume41
dc.identifier.doi10.1111/cgf.14609
dc.identifier.issn1467-8659
dc.identifier.pages135-147
dc.identifier.pages13 pages
dc.identifier.urihttps://doi.org/10.1111/cgf.14609
dc.identifier.urihttps://diglib.eg.org:443/handle/10.1111/cgf14609
dc.publisherThe Eurographics Association and John Wiley & Sons Ltd.en_US
dc.rightsAttribution 4.0 International License
dc.rights.urihttps://creativecommons.org/licenses/by-nc/4.0/
dc.subjectCCS Concepts: Computing methodologies --> Mesh geometry models; Mesh models; Volumetric models
dc.subjectComputing methodologies
dc.subjectMesh geometry models
dc.subjectMesh models
dc.subjectVolumetric models
dc.titleConstructing L∞ Voronoi Diagrams in 2D and 3Den_US
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