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 partitionsis an exclusive privilege of pre-triangulations

How to get ‘Delaunay‘ in …

View the Delaunay triangulationas follows:

S underlying set of points

f* maximal locally convex function on conv(S) such that f*(p) =< p,p> for all p in S

Here: f* is just the lower convex hull

Delaunay Minimum Complex

Restrict values of f* onlyat the corners of thedomain (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 heightsReplace 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-