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Title: Refining a triangulation of a planar straight-line graph to eliminate large angles

Abstract

Triangulations without large angles have a number of applications in numerical analysis and computer graphics. In particular, the convergence of a finite element calculation depends on the largest angle of the triangulation. Also, the running time of a finite element calculation is dependent on the triangulation size, so having a triangulation with few Steiner points is also important. Bern, Dobkin and Eppstein pose as an open problem the existence of an algorithm to triangulate a planar straight-line graph (PSLG) without large angles using a polynomial number of Steiner points. We solve this problem by showing that any PSLG with {upsilon} vertices can be triangulated with no angle larger than 7{pi}/8 by adding O({upsilon}{sup 2}log {upsilon}) Steiner points in O({upsilon}{sup 2} log{sup 2} {upsilon}) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. Some PSLGs require {Omega}({upsilon}{sup 2}) Steiner points in any triangulation achieving any largest angle bound less than {pi}. Hence the number of Steiner points added by our algorithm is within a log {upsilon} factor of worst case optimal. We note that our refinement algorithm works on arbitrary triangulations: Given any triangulation, we show how tomore » refine it so that no angle is larger than 7{pi}/8. Our construction adds O(nm+nplog m) vertices and runs in time O(nm+nplog m) log(m+ p)), where n is the number of edges, m is one plus the number of obtuse angles, and p is one plus the number of holes and interior vertices in the original triangulation. A previously considered problem is refining a constrained triangulation of a simple polygon, where p = 1. For this problem we add O({upsilon}{sup 2}) Steiner points, which is within a constant factor of worst case optimal.« less

Authors:
Publication Date:
Research Org.:
Sandia National Labs., Albuquerque, NM (United States)
Sponsoring Org.:
USDOE, Washington, DC (United States)
OSTI Identifier:
10165881
Report Number(s):
SAND-93-1392C; CONF-931167-1-Extd.Abst.
ON: DE93016510
DOE Contract Number:  
AC04-76DP00789
Resource Type:
Conference
Resource Relation:
Conference: 34. meeting of the Institute of Electrical and Electronics Engineers Foundation of Computer Science,Palo Alto, CA (United States),3-5 Nov 1993; Other Information: PBD: 13 May 1993
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; FINITE ELEMENT METHOD; TRIANGULAR CONFIGURATION; NUMERICAL ANALYSIS; CONVERGENCE; DIAGRAMS; COMPUTER GRAPHICS; MESH GENERATION; 990200; MATHEMATICS AND COMPUTERS

Citation Formats

Mitchell, S A. Refining a triangulation of a planar straight-line graph to eliminate large angles. United States: N. p., 1993. Web.
Mitchell, S A. Refining a triangulation of a planar straight-line graph to eliminate large angles. United States.
Mitchell, S A. 1993. "Refining a triangulation of a planar straight-line graph to eliminate large angles". United States. https://www.osti.gov/servlets/purl/10165881.
@article{osti_10165881,
title = {Refining a triangulation of a planar straight-line graph to eliminate large angles},
author = {Mitchell, S A},
abstractNote = {Triangulations without large angles have a number of applications in numerical analysis and computer graphics. In particular, the convergence of a finite element calculation depends on the largest angle of the triangulation. Also, the running time of a finite element calculation is dependent on the triangulation size, so having a triangulation with few Steiner points is also important. Bern, Dobkin and Eppstein pose as an open problem the existence of an algorithm to triangulate a planar straight-line graph (PSLG) without large angles using a polynomial number of Steiner points. We solve this problem by showing that any PSLG with {upsilon} vertices can be triangulated with no angle larger than 7{pi}/8 by adding O({upsilon}{sup 2}log {upsilon}) Steiner points in O({upsilon}{sup 2} log{sup 2} {upsilon}) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. Some PSLGs require {Omega}({upsilon}{sup 2}) Steiner points in any triangulation achieving any largest angle bound less than {pi}. Hence the number of Steiner points added by our algorithm is within a log {upsilon} factor of worst case optimal. We note that our refinement algorithm works on arbitrary triangulations: Given any triangulation, we show how to refine it so that no angle is larger than 7{pi}/8. Our construction adds O(nm+nplog m) vertices and runs in time O(nm+nplog m) log(m+ p)), where n is the number of edges, m is one plus the number of obtuse angles, and p is one plus the number of holes and interior vertices in the original triangulation. A previously considered problem is refining a constrained triangulation of a simple polygon, where p = 1. For this problem we add O({upsilon}{sup 2}) Steiner points, which is within a constant factor of worst case optimal.},
doi = {},
url = {https://www.osti.gov/biblio/10165881}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Thu May 13 00:00:00 EDT 1993},
month = {Thu May 13 00:00:00 EDT 1993}
}

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