A straightforward algorithm for computing the medial axis of a simple polygon

Yao, Chengfu; Rokne, Jon G.

doi:10.1080/00207169108803978

Cited by 27 publications

(18 citation statements)

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“…Finally, Na et al [33] propose to build an arbitrary spherical Voronoi diagram from two planar Voronoi diagrams of the same type and some clever merging of both diagrams, relying on early results from Brown [34]. As shown in Yao et al [30], reflex edges generate multiple rays in the generalized Voronoi diagram to build rays associated with vertices. Reflexivity, or concavity, is not conserved from a two dimensional Voronoi diagram to a three dimensional diagram.…”

Section: The Real 'Most Normal' Normalmentioning

confidence: 97%

“…Note the similarity between the original sweeping of Fortune [35], Yao et al [30] outside in sweeping, a boundary layer mesh extrusion [10][11][12][13] and a fast marching method [22]. They all rely on the hyperbolic character of the Eikonal equation.…”

Section: The Real 'Most Normal' Normalmentioning

confidence: 98%

“…In the current work, the diagram is therefore computed in three dimensions. As we are interested in only one part of the Voronoi diagram on the whole sphere, the algorithm proposed in Yao et al [30] extended to the spherical Voronoi diagram seems to be the algorithm of choice.…”

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

“…Regarding the original algorithm proposed in [30], the main idea is to build the diagram outside in. Note the similarity between the original sweeping of Fortune [35], Yao et al [30] outside in sweeping, a boundary layer mesh extrusion [10][11][12][13] and a fast marching method [22].…”

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

“…Yao et al [30] design a simpler algorithm to compute it. Regarding the Voronoi diagram on the sphere, Dinis et al [31] and Zheng et al [32] generalize the point Voronoi diagram to the sphere.…”

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

See 4 more Smart Citations

On the ‘most normal’ normal—Part 2

Aubry

Mestreau

Dey

et al. 2015

Finite Elements in Analysis and Design

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Section: The Real 'Most Normal' Normalmentioning

confidence: 97%

Section: The Real 'Most Normal' Normalmentioning

confidence: 98%

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

Section: The Real 'Most Normal' Normalmentioning

confidence: 99%

See 3 more Smart Citations

On the ‘most normal’ normal—Part 2

Aubry

Mestreau

Dey

et al. 2015

Finite Elements in Analysis and Design

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Boundary layer mesh generation on arbitrary geometries

Aubry

Dey

Mestreau

et al. 2017

Numerical Meth Engineering

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Volume boundary layer mesh generation on non-smooth geometries with multiple normals is considered in this work. A new highlight is given to corner and ridge boundary layer mesh generation through the generalized Voronoi diagram in general, and the spherical Voronoi diagram in particular. This provides the keystone to allow for the first time boundary layer mesh generation at arbitrary corner configurations. The work proposed in [18] is revisited and compared with the new approach. A detailed description of the new algorithm is provided. The spherical Voronoi diagram delivers geometrically the optimal normals as well as topologically the optimal connectivities, as far as normality is concerned. Numerical examples illustrate the accuracy and robustness of the method. This approach seems to handle arbitrary geometry boundary layer mesh generation, in theory as well as in practice.This example represents a cube inside another cube and is illustrated in Figure 5. Some archetypal convex corners are displayed. The difficulty here relies in accurately meshing the mixed convex concave corners. Because of the planar faces, it is easier to appreciate the exact position of the extruded surface. The intersection between the convex ridges and the planar faces are exact, thanks to the location of the Voronoi bisectors for the direction, and the geodesic distance for the computation of the offset modulus. The spherical Voronoi diagram for each of these mixed corners corresponds to the one presented before in Figure 3. The relevant part consists of the spherical parabolic bisectors between the reflex vertex and the bottom face. As mentioned in Section 4, because of the particular surface discretization, this bisector has been split in various parts. Algorithm 3 gathers all these bisectors to create only one group. The group is then meshed along its bisector given the angle provided for discretizing the multiple normals. This group is a vertex/edge bisector. Then, Algorithm 4 sew these edges with the multiple normals from the ridge on one side, and to the triangle faces associated with the bisectors on the other side. The final mesh has 23 Kpoints and 125 Kelems. The boundary layer has 10 layers and reaches up to layer 5 before producing inverted elements. Sharov exampleThis example is extracted from [12] and is displayed in Figure 6. Four ridges abut on the complex corner, two concave and two convex. There is no single normal that ensures visibility for the complex corner of this geometry. Therefore, multiple normals are mandatory for this example. The Voronoi diagram is quite complex for this example, and presents two reflex vertices arising from the two convex ridges. It displays the geometry vertex/edge bisectors and the vertex/vertex bisectors. For this example, Algorithm 3 identifies three different groups, one for the vertex/vertex bisector and two BOUNDARY LAYER MESH GENERATION ON ARBITRARY GEOMETRIES 167 Figure 5. A cube inside a cube.for the edge/vertex bisectors associated with the two different reflex vertices. The zo...

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Conformal Mapping in Linear Time

Bishop

2010

Discrete Comput Geom

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A straightforward algorithm for computing the medial axis of a simple polygon

Cited by 27 publications

References 9 publications

On the ‘most normal’ normal—Part 2

On the ‘most normal’ normal—Part 2

Boundary layer mesh generation on arbitrary geometries

Conformal Mapping in Linear Time

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