Quadrilateral Decomposition by Two-Ear Property Resulting in CAD Segmentation
The objective is to split a simply connected polygon
into a set of convex quadrilaterals without inserting new
boundary nodes. The presented approach consists in repeatedly
removing quadrilaterals from the polygon. Theoretical results
pertaining to quadrangulation of simply connected polygons are
derived from the usual 2-ear theorem. It produces a quadrangulation
technique with O(n) number of quadrilaterals. The
theoretical methodology is supplemented by practical results
and CAD surface segmentation.
[1] M. Bern and D. Eppstein, "Quadrilateral Meshing by circle packing",
Int. J. Comput. Geom. Appl., vol. 10, no. 4, pp. 347-360, 2000.
[2] D. Bremner, F. Hurtado, S. Ramaswami and V. Sacristan, Small convex
quadrangulations of point sets, in: Proc. 12th international symposium,
ISAAC 2001, Christchurch, New Zealand, 2001, pp. 623-635.
[3] G. Brunnett, "Geometric design with trimmed surfaces", Computing
Supplementum, vol. 10, pp. 101-115, 1995.
[4] C. Lee and S. Lo, "A new scheme for the generation of a graded
quadrilateral mesh", Comput. Struct., vol. 52, no. 5, pp. 847-857, 1994.
[5] G. Meister, "Polygons have ears", Amer. Math. Mon., vol. 82, pp. 648-
651, 1975.
[6] S. Owen, "Non-simplicial unstructured mesh generation". Ph.D. dissertation,
Dept. Civil Envir. Engin., Carnegie Mellon University, Pennsylvania,
1999.
[7] S. Ramaswami, P. Ramos and G. Toussaint, "Converting triangulations
to quadrangulations", Comput. Geom., vol. 9, no. 4, pp. 257-276, 1998.
[8] M. Randrianarivony, "Geometric processing of CAD data and meshes
as input of integral equation solvers". Ph.D. dissertation, Dept. Comput.
Science, Chemnitz University of Technology, Chemnitz, Germany, 2006.
[9] M. Randrianarivony and G. Brunnett, "Molecular surface decomposition
using geometric techniques", in Proc. Conf. Bildverarbeitung f¨ur die
Medizine, Berlin, 2008, pp. 197-201.
[10] M. Randrianarivony and G. Brunnett, "Preparation of CAD and Molecular
Surfaces for Meshfree Solvers", in Proc. Int. Workshop Meshfree
Methods for PDE, Bonn, 2007, pp. 231-245.
[1] M. Bern and D. Eppstein, "Quadrilateral Meshing by circle packing",
Int. J. Comput. Geom. Appl., vol. 10, no. 4, pp. 347-360, 2000.
[2] D. Bremner, F. Hurtado, S. Ramaswami and V. Sacristan, Small convex
quadrangulations of point sets, in: Proc. 12th international symposium,
ISAAC 2001, Christchurch, New Zealand, 2001, pp. 623-635.
[3] G. Brunnett, "Geometric design with trimmed surfaces", Computing
Supplementum, vol. 10, pp. 101-115, 1995.
[4] C. Lee and S. Lo, "A new scheme for the generation of a graded
quadrilateral mesh", Comput. Struct., vol. 52, no. 5, pp. 847-857, 1994.
[5] G. Meister, "Polygons have ears", Amer. Math. Mon., vol. 82, pp. 648-
651, 1975.
[6] S. Owen, "Non-simplicial unstructured mesh generation". Ph.D. dissertation,
Dept. Civil Envir. Engin., Carnegie Mellon University, Pennsylvania,
1999.
[7] S. Ramaswami, P. Ramos and G. Toussaint, "Converting triangulations
to quadrangulations", Comput. Geom., vol. 9, no. 4, pp. 257-276, 1998.
[8] M. Randrianarivony, "Geometric processing of CAD data and meshes
as input of integral equation solvers". Ph.D. dissertation, Dept. Comput.
Science, Chemnitz University of Technology, Chemnitz, Germany, 2006.
[9] M. Randrianarivony and G. Brunnett, "Molecular surface decomposition
using geometric techniques", in Proc. Conf. Bildverarbeitung f¨ur die
Medizine, Berlin, 2008, pp. 197-201.
[10] M. Randrianarivony and G. Brunnett, "Preparation of CAD and Molecular
Surfaces for Meshfree Solvers", in Proc. Int. Workshop Meshfree
Methods for PDE, Bonn, 2007, pp. 231-245.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51004", author = "Maharavo Randrianarivony", title = "Quadrilateral Decomposition by Two-Ear Property Resulting in CAD Segmentation", abstract = "The objective is to split a simply connected polygon
into a set of convex quadrilaterals without inserting new
boundary nodes. The presented approach consists in repeatedly
removing quadrilaterals from the polygon. Theoretical results
pertaining to quadrangulation of simply connected polygons are
derived from the usual 2-ear theorem. It produces a quadrangulation
technique with O(n) number of quadrilaterals. The
theoretical methodology is supplemented by practical results
and CAD surface segmentation.", keywords = "Quadrangulation, simply connected, two-ear theorem.", volume = "2", number = "8", pages = "523-7", }