On triangulation axes of polygons

We propose the triangulation axis as an alternative skeletal structure for a simple polygon P. This axis is a straight-line tree that can be interpreted as an anisotropic medial axis of P, where inscribed disks are line segments or triangles. The underlying triangulation that specifies the anisotropy can be varied, to adapt the axis so as to reflect predominant geometrical and topological features of P. Triangulation axes typically have much fewer edges and branchings than the Euclidean medial axis or the straight skeleton of P. Still, they retain important properties, as for example the reconstructability of P from its skeleton. Triangulation axes can be computed from their defining triangulations in O ( n ) time. We investigate the effect of using several optimal triangulations for P. In particular, careful edge flipping in the constrained Delaunay triangulation leads, in O ( n log ? n ) overall time, to an axis competitive to 'high quality axes' requiring ? ( n 3 ) time for optimization via dynamic programming.

[1]  D. T. Lee,et al.  Generalized delaunay triangulation for planar graphs , 1986, Discret. Comput. Geom..

[2]  Kaleem Siddiqi,et al.  Medial Representations: Mathematics, Algorithms and Applications , 2008 .

[3]  G. Klincsek.  Minimal Triangulations of Polygonal Domains , 1980 .

[4]  Chandrajit L. Bajaj,et al.  The algebraic degree of geometric optimization problems , 1988, Discret. Comput. Geom..

[5]  Marshall W. Bern,et al.  Surface Reconstruction by Voronoi Filtering , 1998, SCG '98.

[6]  Remco C. Veltkamp,et al.  A Straight Skeleton Approximating the Medial Axis , 2004, ESA.

[7]  Tinghua Ai,et al.  GAP-Tree Extensions Based on Skeletons , 2002 .

[8]  M. Bauer,et al.  Triangulations , 1996, Discret. Math..

[9]  Tao Wang,et al.  EXTRACTION OF OPTIMAL SKELETON OF POLYGON BASED ON HIERARCHICAL ANALYSIS , 2010 .

[10]  Jonathan Richard Shewchuk,et al.  Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation , 2003, SCG '03.

[11]  Jean-Daniel Boissonnat,et al.  Stability and Computation of Medial Axes - a State-of-the-Art Report , 2009, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration.

[12]  Franz Aurenhammer,et al.  A Novel Type of Skeleton for Polygons , 1995, J. Univers. Comput. Sci..

[13]  L. Paul Chew,et al.  Constrained Delaunay triangulations , 1987, SCG '87.

[14]  Franz Aurenhammer.  Weighted skeletons and fixed-share decomposition , 2008, Comput. Geom..

[15]  Franz Aurenhammer,et al.  Medial axis computation for planar free-form shapes , 2009, Comput. Aided Des..

[16]  Robert L. Scot Drysdale,et al.  Voronoi diagrams based on convex distance functions , 1985, SCG '85.

[17]  Steven Fortune,et al.  Voronoi Diagrams and Delaunay Triangulations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[18]  Gabriella Sanniti di Baja,et al.  On Medial Representations , 2008, CIARP.