Self-overlapping curves: Analysis and applications

Mukherjee, Uddipan

doi:10.1016/j.cad.2013.08.037

Cited by 5 publications

(2 citation statements)

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“…Notably, many problems related to identifying self‐intersections are NP‐complete [EM09]. Despite this, efficient algorithms frequently exist; for example, Mukherjee [Muk14] gave a quadratic algorithm (in the number of points on the discrete curve) to determine the mapping from a disk to an arbitrarily stretched, potentially self‐overlapping curve, also known as computing an immersion of the disk. In another vein, Li [Li11] used Gauss diagrams from knot theory to characterize self‐intersecting two‐dimensional projections of three‐dimensional polygons, in order to understand whether there are one or multiple ways to perform mesh repair algorithms like Brunton et al.…”

Section: Related Workmentioning

confidence: 99%

A Robust Grid‐Based Meshing Algorithm for Embedding Self‐Intersecting Surfaces

Gagniere,

Han,

Chen

et al. 2023

Computer Graphics Forum

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The creation of a volumetric mesh representing the interior of an input polygonal mesh is a common requirement in graphics and computational mechanics applications. Most mesh creation techniques assume that the input surface is not self‐intersecting. However, due to numerical and/or user error, input surfaces are commonly self‐intersecting to some degree. The removal of self‐intersection is a burdensome task that complicates workflow and generally slows down the process of creating simulation‐ready digital assets. We present a method for the creation of a volumetric embedding hexahedron mesh from a self‐intersecting input triangle mesh. Our method is designed for efficiency by minimizing use of computationally expensive exact/adaptive precision arithmetic. Although our approach allows for nearly no limit on the degree of self‐intersection in the input surface, our focus is on efficiency in the most common case: many minimal self‐intersections. The embedding hexahedron mesh is created from a uniform background grid and consists of hexahedron elements that are geometrical copies of grid cells. Multiple copies of a single grid cell are used to resolve regions of self‐intersection/overlap. Lastly, we develop a novel topology‐aware embedding mesh coarsening technique to allow for user‐specified mesh resolution as well as a topology‐aware tetrahedralization of the hexahedron mesh.

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Section: Related Workmentioning

confidence: 99%

A Robust Grid‐Based Meshing Algorithm for Embedding Self‐Intersecting Surfaces

Gagniere,

Han,

Chen

et al. 2023

Computer Graphics Forum

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“…Notably, many problems related to identifying self-intersections are NP-complete [Eppstein and Mumford 2009]. Despite this, efficient algorithms frequently exist; for example, Mukherjee [2014] gave a quadratic algorithm (in the number of points on the discrete curve) to determine the mapping from a disk to an arbitrarily stretched, potentially self-overlapping curve, also known as computing an immersion of the disk. In another vein, Li [2011] used Gauss diagrams from knot theory to characterize self-intersecting two-dimensional projections of three-dimensional polygons, in order to understand whether there are one or multiple ways to perform mesh repair algorithms like [Brunton et al 2009].…”

Section: Self-intersecting Curves and Surfacesmentioning

confidence: 99%

A Robust Grid-Based Meshing Algorithm for Embedding Self-Intersecting Surfaces

Gagniere¹,

Han²,

Hyde³

et al. 2022

Preprint

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nnnnnnn is designed for efficiency by minimizing use of computationally expensive exact/adaptive precision arithmetic. Although our approach allows for nearly no limit on the degree of self-intersection in the input surface, our focus is on efficiency in the most common case: many minimal self-intersections. The embedding hexahedron mesh is created from a uniform background grid and consists of hexahedron elements that are geometrical copies of grid cells. Multiple copies of a single grid cell are used to resolve regions of selfintersection/overlap. Lastly, we develop a novel topology-aware embedding mesh coarsening technique to allow for user-specified mesh resolution as well as a topology-aware tetrahedralization of the hexahedron mesh.CCS Concepts: • Computing methodologies → Computer graphics; • Mathematics of computing → Mesh generation.

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