Abstract
We present a novel and robust algorithm for triangulating point clouds in ℝ2. It is based on a highly adaptive hexagonal subdivision scheme of the input domain. That hexagon mesh has a dual triangular mesh with the following properties:
-
any angle of any triangle lies in the range between 43.9° and 90°,
-
the aspect ratio of triangles is bound to 1.20787,
-
the triangulation has the Delaunay property,
-
the minimum triangle size is bounded by the minimum distance between input points.
The iterative character of the hexagon subdivision allows incremental addition of further input points for selectively refining certain regions. Finally we extend the algorithm to handle planar straight-line graphs (PSLG). Meshes produced by this method are suitable for all kinds of algorithms where numerical stability is affected by triangles with skinny or obtuse angles.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baker, B.S., Grosse, E., Rafferty, C.S.: Nonobtuse triangulation of polygons. Discrete Comput. Geom. 3(2), 147–168 (1988)
Bern, M., Eppstein, D., Gilbert, J.: Provably good mesh generation. J. Comput. Syst. Sci. 48(3), 384–409 (1994)
Chew, L.: Guaranteed-quality triangular meshes. Tech. Rep. TR 89-983. Cornell University (1989)
Erten, H., Üngör, A.: Triangulations with locally optimal steiner points. In: SGP 2007: Proceedings of the fifth Eurographics symposium on Geometry processing, Aire-la-Ville, Switzerland, pp. 143–152. Eurographics Association (2007)
Sußner, G., Dachsbacher, C., Greiner, G.: Hexagonal LOD for Interactive Terrain Rendering. In: Vision Modeling and Visualization, pp. 437–444 (2005)
Neugebauer, F., Diekmann, R.: Improved mesh generation: Not simple but good. In: 5th Int. Meshing roundtable, pp. 257–270. Sandia National Laboratories (1996)
Owen, S.J.: A survey of unstructured mesh generation technology. In: IMR, pp. 239–267 (1998)
Ruppert, J.: A new and simple algorithm for quality 2-dimensional mesh generation. In: SODA 1993: Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms, Philadelphia, PA, USA, pp. 83–92. Society for Industrial and Applied Mathematics (1993)
Shewchuk, J.R.: Triangle: Engineering a 2d quality mesh generator and delaunay triangulator. In: FCRC 1996/WACG 1996: Selected papers from the Workshop on Applied Computational Geormetry, Towards Geometric Engineering, London, UK, pp. 203–222. Springer, Heidelberg (1996)
Üngör, A.: Off-centers: A new type of steiner points for computing size-optimal quality-guaranteed delaunay triangulations. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 152–161. Springer, Heidelberg (2004)
Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Maurer, H.A. (ed.) New Results and New Trends in Computer Science. LNCS, vol. 555. Springer, Heidelberg (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sußner, G., Greiner, G. (2009). Hexagonal Delaunay Triangulation. In: Clark, B.W. (eds) Proceedings of the 18th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04319-2_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-04319-2_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04318-5
Online ISBN: 978-3-642-04319-2
eBook Packages: EngineeringEngineering (R0)