PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips Franz Aurenhammer Institute for Theoretical Computer Science Graz University of Technology,

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Presentation transcript:

PRE-TRIANGULATIONS Generalized Delaunay Triangulations and Flips Franz Aurenhammer Institute for Theoretical Computer Science Graz University of Technology, Austria

Why do we like Voronoi diagrams?

What do we do when a (nice) structure does not exactly fit our purposes? Generalize - Shape of sites - Distance function - Underlying space

Hand in hand with Voronoi diagrams goes the Delaunay triangulation

Surprisingly, duals of generalized Voronoi diagrams play a minor role

Why are Delaunay triangulations harder to generalize than Voronoi diagrams? Voronoi diagram: Fix properties (mainly distance function), study the shape of regions Delaunay triangulation: Fix the shape of regions (triangles), study resulting (combinatorial) properties.  Generalize the Delaunay triangulation independently!

What are Delaunay triangulations special for? - Unique structure - Local ‘Delaunayhood‘ - Flippability of edges - Liftability to a surface in 3D  When generalizing the Delaunay triangulation, we want to keep these properties.

How to generalize a triangulation, anyway? Triangle: Exactly 3 vertices without reflex angle  Pseudo-triangle, Pre-triangle

Pseudo-triangulations Pre-triangulations Data Structure: Visibility, collision detection Graph: Rigidity properties Fairly new concept Robust liftability of polygonal partitions is an exclusive privilege of pre-triangulations

How to get ‘ Delaunay‘ in … View the Delaunay triangulation as follows: S underlying set of points f* maximal locally convex function on conv(S) such that f*(p) = for all p in S Here: f* is just the lower convex hull

Delaunay Minimum Complex Restrict values of f* only at the corners of the domain (no reflex angle)  Pseudo-triangulation Unique, liftable, and locally Delaunay (convex) ….not to be confused with the constrained Delaunay triangulation

Delaunay Minimum Complex  Pre-triangulation Complex of smallest combinatorial size with the desired Delaunay properties!

… and Flippability ? - We should be able to flip any given pre-triangulation into the Delaunay minimum complex - And flips should be consistent with existing flips for triangulations and pseudo-triangulations

A General Flipping Scheme FLIP(edge) Choose domain Give heights Replace by f*

Flipping Domain ok no pre-triangulation!

Implications - Canonical Delaunay pre-triangulation (or pseudo-triangulation) for polygonal regions exists - Can be reached by improving flips (convexifying flips) from every pre-triangulation - Extends the well-known properties of Delaunay triangulations Can we obtain similar results for 3-space?

‘Delaunay‘ for a Nonconvex Polytope Pseudo-tetrahedra (4 corners)

Bistellar Flip for Tetrahedra  Generalizes for pseudo-tetrahedra!

Exhibition of (Pseudo)-Delaunay Art

-simple removing flip-

-simple exchanging flip-

-Splitting off a secondary cell-

-Inserting flip-

-large exchanging flip-

-tunnel flip-

-Thank you-