On triangulation axes of polygons

Wolfgang Aigner, Franz Aurenhammer, Bert Jüttler

Abstract: 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)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> 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(nlog⁡n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mi mathvariant="normal" is="true">log</mi><mo is="true">⁡</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> overall time, to an axis competitive to ‘high quality axes’ requiring Θ(n3)<math><mi is="true">Θ</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> time for optimization via dynamic programming.