A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes

Perdomo, Francisco; Plaza, Ángel

doi:10.1016/j.amc.2012.08.029

Cited by 6 publications

(5 citation statements)

References 8 publications

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“…Hence it has an accumulation point which corresponds to the fixed point z = 1 2 . This completes the argument of the paper [1] for the case n = 4.…”

Section: Then Im W(z)mentioning

confidence: 49%

See 1 more Smart Citation

Corrigendum to “A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes” [Applied Mathematics and Computation 219 (4) (2012) 2342-2344]

Perdomo

Plaza

2015

Applied Mathematics and Computation

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“…Hence it has an accumulation point which corresponds to the fixed point z = 1 2 . This completes the argument of the paper [1] for the case n = 4.…”

Section: Then Im W(z)mentioning

confidence: 49%

“…Even though, the subsequent argument in [1] is correct for the case n > 4. But, the case n = 4 needs a closer look.…”

Section: Then Im W(z)mentioning

confidence: 99%

Corrigendum to “A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes” [Applied Mathematics and Computation 219 (4) (2012) 2342-2344]

Perdomo

Plaza

2015

Applied Mathematics and Computation

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“…By considering a dynamical system of sequentially refined triangles in hyperbolic geometry, it was shown in Perdomo and Plaza (2013) that trisection guarantees high-quality refined meshes; whereas the n -section method for n ≥ 4 can produce degenerate triangles (Perdomo and Plaza, 2012). Other advanced refinement strategies are discussed in Brandts et al (2009), Choi et al (2003), and Korotov and Křížek (2001).…”

Section: Accuracy-improving Adaptive Mesh Refinementmentioning

confidence: 99%

Asymptotically optimal feedback planning using a numerical Hamilton-Jacobi-Bellman solver and an adaptive mesh refinement

Yershov

2015

The International Journal of Robotics Research

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We present the first asymptotically optimal feedback planning algorithm for nonholonomic systems and additive cost functionals. Our algorithm is based on three well-established numerical practices: 1) positive coefficient numerical approximations of the Hamilton-Jacobi-Bellman equations; 2) the Fast Marching Method, which is a fast nonlinear solver that utilizes Bellman’s dynamic programming principle for efficient computations; and 3) an adaptive mesh-refinement algorithm designed to improve the resolution of an initial simplicial mesh and reduce the solution numerical error. By refining the discretization mesh globally, we compute a sequence of numerical solutions that converges to the true viscosity solution of the Hamilton-Jacobi-Bellman equations. In order to reduce the total computational cost of the proposed planning algorithm, we find that it is sufficient to refine the discretization within a small region in the vicinity of the optimal trajectory. Numerical experiments confirm our theoretical findings and establish that our algorithm outperforms previous asymptotically optimal planning algorithms, such as PRM* and RRT*.

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“…See in this sense [3,8,9] and references therein. While previous demonstrations employed ad hoc methods of Euclidean geometry, the methodology of the proofs given here is generalizable to other partition methods [17][18][19]. The proofs are not necessarily shorter, but they are structured in an systematic way.…”

Section: Introductionmentioning

confidence: 99%

“…One of methods used in the literature on triangular mesh refinement is to normalize triangles [10,17,22]. The method consists of applying several possible isometries and dilations to a triangle, in such way one side can be identify with the segment whose endpoints are (0 0) and (1 0 The LE-bisection of a triangle ∆ is obtained by joining the midpoint on the longest edge of the triangle to the opposite vertex.…”

Section: Triangle Normalizationmentioning

confidence: 99%

Properties of triangulations obtained by the longest-edge bisection

Perdomo

Plaza

2014

Open Mathematics

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Abstract:The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle of the initial angle. The novelty of the proofs is the use of an hyperbolic metric in a shape space for triangles. MSC:51M09, 65M50, 65N50

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A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes

Cited by 6 publications

References 8 publications

Corrigendum to “A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes” [Applied Mathematics and Computation 219 (4) (2012) 2342-2344]

Corrigendum to “A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes” [Applied Mathematics and Computation 219 (4) (2012) 2342-2344]

Asymptotically optimal feedback planning using a numerical Hamilton-Jacobi-Bellman solver and an adaptive mesh refinement

Properties of triangulations obtained by the longest-edge bisection

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