# Properties of triangulations obtained by the longest-edge bisection

Francisco Perdomo; Ángel Plaza

Open Mathematics (2014)

- Volume: 12, Issue: 12, page 1796-1810
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topFrancisco Perdomo, and Ángel Plaza. "Properties of triangulations obtained by the longest-edge bisection." Open Mathematics 12.12 (2014): 1796-1810. <http://eudml.org/doc/268957>.

@article{FranciscoPerdomo2014,

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.},

author = {Francisco Perdomo, Ángel Plaza},

journal = {Open Mathematics},

keywords = {Triangulation; Longest-edge bisection; Mesh refinement; Mesh regularity; Finite element method; triangulation; longest-edge bisection; mesh refinement; mesh regularity; finite element method},

language = {eng},

number = {12},

pages = {1796-1810},

title = {Properties of triangulations obtained by the longest-edge bisection},

url = {http://eudml.org/doc/268957},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Francisco Perdomo

AU - Ángel Plaza

TI - Properties of triangulations obtained by the longest-edge bisection

JO - Open Mathematics

PY - 2014

VL - 12

IS - 12

SP - 1796

EP - 1810

AB - 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.

LA - eng

KW - Triangulation; Longest-edge bisection; Mesh refinement; Mesh regularity; Finite element method; triangulation; longest-edge bisection; mesh refinement; mesh regularity; finite element method

UR - http://eudml.org/doc/268957

ER -

## References

top- [1] Adler A., On the bisection method for triangles, Math. Comp., 1983, 40, 571–574 http://dx.doi.org/10.1090/S0025-5718-1983-0689473-5 Zbl0523.65033
- [2] Babuška I., Aziz A. K., On the angle condition in the finite element method. SIAM J. Numer. Anal. 1976, 13, 214–226 http://dx.doi.org/10.1137/0713021 Zbl0324.65046
- [3] Bern M., Eppstein D., Optimal Möbius transformations for information visualization and meshing, Lecture Notes in Comp. Sci., 2001, 2125, 14–25 http://dx.doi.org/10.1007/3-540-44634-6_3 Zbl0997.68536
- [4] Bookstein F.L., Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, 1991 Zbl0770.92001
- [5] Brandts J., Korotov S., KříŽek M., On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions, Comput. & Math. Appl., 2008, 55, 2227–2233 http://dx.doi.org/10.1016/j.camwa.2007.11.010 Zbl1142.65443
- [6] Ciarlet P. G., The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. Zbl0383.65058
- [7] Dryden I.L., Mardia K.V., Statistical Shape Analysis, Wiley, 1998 Zbl0901.62072
- [8] Eppstein D., Manhattan orbifolds, Topology and its Applications, 2009, 157(2), 494–507 http://dx.doi.org/10.1016/j.topol.2009.10.008 Zbl1200.68169
- [9] Eppstein D., Succinct greedy geometric routing using hyperbolic geometry, IEEE Transactions on Computing, 2011, 60(11), 1571–1580 http://dx.doi.org/10.1109/TC.2010.257
- [10] Gutiérrez C., Gutiérrez F., Rivara M.-C., Complexity of bisection method, Theor. Comp. Sci., 2007, 382, 131–138 http://dx.doi.org/10.1016/j.tcs.2007.03.004 Zbl1127.68108
- [11] Hannukainen A., Korotov S., Křížek M., On global and local mesh refinements by a generalized conforming bisection algorithm, J. Comput. Appl. Math., 2010, 235, 419–436 http://dx.doi.org/10.1016/j.cam.2010.05.046 Zbl1207.65145
- [12] Iversen B., Hyperbolic Geometry, Cambridge University Press, 1992 http://dx.doi.org/10.1017/CBO9780511569333
- [13] Korotov S., Křížek M., Kropác A., Strong regularity of a family of face-to-face partitions generated by the longestedge bisection algorithm, Comput. Math. and Math. Phys., 2008, 49, 1687–1698 http://dx.doi.org/10.1134/S0965542508090170
- [14] Křížek M., On semiregular families of triangulations and linear interpolation, Appl. Math., 1991, 36, 223–232 Zbl0728.41003
- [15] Márquez A., Moreno-González A., Plaza A., Suárez J. P., The seven-triangle longest-side partition of triangles and mesh quality improvement, Finite Elem. Anal. Des., 2008, 44, 748–758 http://dx.doi.org/10.1016/j.finel.2008.04.007
- [16] Padrón M. A., Suárez J. P., Plaza A., A comparative study between some bisection based partitions in 3D, Appl. Num. Math., 2005, 55, 357–367 http://dx.doi.org/10.1016/j.apnum.2005.04.035 Zbl1087.65615
- [17] Perdomo F., Plaza A., A new proof of the degeneracy property of the longest-edge n-section refinement scheme for triangular meshes, Appl. Math. & Compt., 2012, 219, 2342–2344 http://dx.doi.org/10.1016/j.amc.2012.08.029 Zbl1291.65353
- [18] Perdomo F., Plaza A., Proving the non-degeneracy of the longest-edge trisection by a space of triangular shapes with hyperbolic metric, Appl. Math. & Compt., 2013, 221, 424–432 http://dx.doi.org/10.1016/j.amc.2013.06.075 Zbl1332.51009
- [19] Perdomo F., Dynamics of the longest-edge partitions in a triangle space endowed with an hyperbolic metric, Ph.D. Thesis (in Spanish), Las Palmas de Gran Canaria, 2013
- [20] Plaza A., Padrón M. A., Suárez J. P., Non-degeneracy study of the 8-tetrahedra longest-edge partition, Appl. Num. Math., 2005, 55, 458–472 http://dx.doi.org/10.1016/j.apnum.2004.12.003 Zbl1086.65507
- [21] Plaza A., Suárez J. P., Padrón M. A, Falcón S., Amieiro D., Mesh quality improvement and other properties in the four-triangles longest-edge partition. Comp. Aid. Geom. Des., 2004, 21, 353–369 http://dx.doi.org/10.1016/j.cagd.2004.01.001 Zbl1069.65530
- [22] Plaza A., Suárez J. P., Carey G. F., A geometric diagram and hybrid scheme for triangle subdivision, Comp. Aid. Geom. Des., 2007, 24, 19–27 http://dx.doi.org/10.1016/j.cagd.2006.10.002 Zbl1171.65349
- [23] Rivara M.-C., Mesh refinement processes based on the generalized bisection of simplices, SIAM J. Num. Anal., 1984, 21, 604–613 http://dx.doi.org/10.1137/0721042 Zbl0574.65133
- [24] Rosenberg I. G., Stenger F., A lower bound on the angles of triangles constructed by bisecting the longest side, Math. Comp., 1975, 29, 390–395 http://dx.doi.org/10.1090/S0025-5718-1975-0375068-5 Zbl0302.65085
- [25] Stahl S., The Poincaré Half-Plane, Jones & Bartlett Learning, 1993
- [26] Stynes M., On faster convergence of the bisection method for certain triangles, Math. Comp., 1979, 33, 717–721 http://dx.doi.org/10.1090/S0025-5718-1979-0521285-4 Zbl0405.65010
- [27] Stynes M., On faster convergence of the bisection method for all triangles, Math. Comp., 1980, 35, 1195–1201 http://dx.doi.org/10.1090/S0025-5718-1980-0583497-1 Zbl0463.65005
- [28] Toth G., Glimpses of Algebra and Geometry, Springer, 2002 Zbl1027.00002

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.