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### All triangulations are reachable via sequences of edge-flips: an elementary proof

E. Osherovich, A.M. Bruckstein
2008 Computer Aided Geometric Design
A simple proof is provided for the fact that the set of all possible triangulations of a planar point set in a polygonal domain is closed under the basic diagonal flip operation.  ...  a set of points in a polygonal domain.  ...  This case is depicted in Figure 5 . Here we can simply flip the edge ViVj. Next we shall extend our proof to triangulations of points sets in polygonal domains.  ...

### Transforming Triangulations on Nonplanar Surfaces

C. Cortés, C. I. Grima, F. Hurtado, A. Márquez, F. Santos, J. Valenzuela
2010 SIAM Journal on Discrete Mathematics
We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips.  ...  The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.  ...  Let P be a triangulable polygon on the flat torus having an extreme earable vertex u. Then, any triangulation of P can be transformed by a sequence of flips into a triangulation having an ear in u.  ...

### An interactive human morphing system with self-occlusion enhancement

Zhiguang Liu, Howard Leung, Hubert P. H. Shum
2016 Proceedings of the 9th International Conference on Motion in Games - MIG '16
Figure 1 : Transforming a man with self-occlusion into one wolf beast  ...  Acknowledgments The work described in this paper was partially supported by a grant from City University of Hong Kong (Project No. 7004548) and the Engineering and Physical Sciences Research Council (EPSRC  ...  Previous methods for building compatible triangulation usually map the source and target polygons onto a convex domain, or employ the divide-and-conquer algorithm to keep partitioning them until each sub-polygon  ...

### A Polygonal Perspective of Nielsen Reduction and the Chord Slide Groupoid [article]

Alex James Bene
2010 arXiv   pre-print
In this survey, we discuss a variation called fatgraph Nielsen reduction which decomposes such a mapping class into elementary Nielsen transformations interpreted as rearrangements of polygon domains for  ...  S_g,1 described by systems of arcs in S_g,1.  ...  To this end, we begin by noting that any polygon domain P of S g,1 can be canonically extended to a triangulation T (P ) of S g,1 (based at p) by triangulating the polygon in a fan-like fashion as depicted  ...

### Ear-Clipping Based Algorithms of Generating High-Quality Polygon Triangulation [chapter]

Gang Mei, John C. Tipper, Nengxiong Xu
2012 Lecture Notes in Electrical Engineering
To apply the two algorithms on polygons with holes, "Bridge" edges are created to transform a polygon with holes to a degenerate polygon which can be triangulated by the two algorithms.  ...  A basic and an improved ear clipping based algorithm for triangulating simple polygons and polygons with holes are presented.  ...  Different from dividing a closed domain, Held [8] adopts 'Bridge' edges to transform a multiply-connected polygonal area into a single polygon.  ...

### Polygon Morphing and Its Application in Orebody Modeling

Hacer İlhan, Haşmet Gürçay
2012 Mathematical Problems in Engineering
These methods are used for generating a set of polygonal sections from two nonplanar polygonal sections which are nearly planar in 3D before constructing a three-dimensional object from these nonplanar  ...  The second one reduces the problem to the problem of morphing compatible planar triangulations and utilizes the representation of planar triangulations as a matrix constructed using barycentric coordinates  ...  sections in 2D using the triangulations. 3 Establish initial vertex for each transformed section in 2D. 4 Approximate each transformed section in 2D with Fourier parameters. 5 n t S − s 1 S 1 ·  ...

### Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization [chapter]

Alexander I. Bobenko, Stefan Sechelmann, Boris Springborn
2016 Advances in Discrete Differential Geometry
The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory.  ...  We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces.  ...  The images or other third party material in this chapter are included in the work's Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work's  ...

### POLYLLA: Polygonal meshing algorithm based on terminal-edge regions [article]

Sergio Salinas, Nancy Hitschfeld-Kahler, Alejandro Ortiz-Bernardin, Hang Si
2022 arXiv   pre-print
the third phase transforms each non simple polygon into simple ones.  ...  This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations.  ...  Our hypotheses are: (i) Terminal-edge regions can be transformed into simple polygons and used as basic cells, (ii) the domain geometry can be fitted using less elements than constrained Voronoi meshes  ...

### STL Triangular Mesh Generation Based on SAT Model

Yuwei Zhang, Yiqi Zhou, Xiaofeng Zhao
2013 Research Journal of Applied Sciences Engineering and Technology
Mesh generation is a fundamental technique in multiple domains. In this study, a STL triangular mesh generation method based on SAT model is proposed.  ...  Two novel triangulation methods, the constrained Delaunay algorithm and the grid subtraction algorithm, are employed on the multi-loop planer regions and the curved surfaces respectively.  ...  Then, the boundary polygons, which include a discrete point set on the curve, are projected to the parametric domain in relation to the surface equation.  ...

### On recent advances in 2D Constrained Delaunay triangulation algorithms [article]

Pranav Kant Gaur, S. K. Bose
2017 arXiv   pre-print
In this article, recent works on 2D Constrained Delaunay triangulation(CDT) algorithms have been reported.  ...  Since the review of CDT algorithms presented by de Floriani(Issues on Machine Vision, Springer Vienna, pg. 95--104, 1989), different algorithms for construction and applications of CDT have appeared in  ...  They first construct a triangulation of the points in input PSLG, then constraint edges are inserted using edge-flipping which results in a constrained triangulation which is then transformed into the  ...

### Very Fast Approximation of the Matrix Chain Product Problem

Artur Czumaj
1996 Journal of Algorithms
This problem can be Ž . solved in O n log n sequential time, while the best known parallel NC algorithm Ž 2 .  ...  This algorithm was later improved, analyzed, and transformed to the problem of finding a near-optimal triangulation of a convex w x polygon by Hu and Shing 11 .  ...  . , ARC of the arcs in the triangula-The problem of finding a near-optimal triangulation of a Ž . con¨ex polygon can be sol¨ed in O log n time on a CREW PRAM and in Ž .  ...

### MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1992 Lecture Notes Series on Computing
We especially focus on optimal triangulations of geometric domains in two-and three-dimensions.  ...  An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles.  ...  Conclusions We have described work in computational geometry motivated by nite element mesh generation.  ...

### MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1995 Lecture Notes Series on Computing
We especially focus on optimal triangulations of geometric domains in two-and three-dimensions.  ...  An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles.  ...  Conclusions We have described work in computational geometry motivated by nite element mesh generation.  ...

### An Affine-invariant Time-dependent Triangulation of Spatio-temporal Data [article]

Sofie Haesevoets, Bart Kuijpers
2008 arXiv   pre-print
This triangulation algorithm generates new geometric data objects that partition the given objects both in space and in time.  ...  the transformation functions).  ...  The polygons shaded in grey in (D) of Figure 5 are an example of such polygons. If a polygon does not belong to S, we do not triangulate it.  ...

### Updating and constructing constrained delaunay and constrained regular triangulations by flips

Jonathan Richard Shewchuk
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03
We modify these algorithms to succeed in practice for polygons whose vertices deviate from exact coplanarity.  ...  Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision  ...  The final step is to remove from T any tetrahedra that do not lie in the triangulation domain. Let us consider an algorithm for recovering a facet f . How does T transform into T f ?  ...
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