EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

IP1: ADVANCED VORONOI AND DELAUNAY STRUCTURES

Franz Aurenhammer

IGI TU Graz Austria

2EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

PLANNED TOPICS

(1) Shape Delaunay structures

(1), (3), and (4) are related; start work there. (2) is of a different flavor, and maybe more tough.

(2) Zone diagrams

(3) Straight skeletons

(4) Generalized medial axes

3EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

SHAPE DELAUNAY STRUCTURES

Convex shape C (with center o), instead of empty circle. Edge inclusion, empty shape property.

Voronoi diagram for convex distance function, take the dual (gives shape Delaunay for reflected shape)

Diagram does not change combinatorially, when center moves within C

4EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

SHAPE DELAUNAY STRUCTURES

Not a full triangulation of the convex hull if C is not smooth Support hull

Direction-sensitivity to shape

O(n logn) algorithms exist (D & C)

Flipping works, too (criterion, termination)

5EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

SHAPE DELAUNAY STRUCTURES

Flipping: Exclude certain edges

Flipping criterion: Angles don‘t work any more Radii (Scaling factors of C): max(r,s) < max(t,u) can be shown min(r,s) < min(t,u) not true, i.g.

If I contains points no empty shape for edge exists, excluded.

No point in II can give triangles with edge, as they cannot be covered by shape.

6EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

SHAPE DELAUNAY STRUCTURES

Minimum spanning tree? d(C) is not a metric, unless C is symmetric d(C) depends on choice of center

MST(C*) is in DT(C*) and the same as the MST w.r.t. m(C).

The latter tree is part of DT(C) (as C* is union of shapes C)

Define symmetric metric m(C) = d(C*) (C* = Minkowski sum of C and C‘)

7EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

SHAPE DELAUNAY STRUCTURES

Open questions:

-- More optimization properties (Flip: r+s < t+u ?)

-- Number of flips, given C ?

-- Deciding whether T is some shape Delaunay (find C) Characterization of DT(C)...

-- Is the number of shape Delaunays polynomial for every point set?

8EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Quite new, challenging concept

Set S of n points in the plane. Zone of p in S: Domain closer to p than to any other zone [Asano et al.]

Implicit definition, neutral zone, existence, uniqueness?

`Kingdom interpretation´

9EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Some properties

Zones are convex, and contained in Voronoi region.

Zones can expand, when site is added.

Trisector of two sites, algebraic? ∩ with line...

10EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Approximation

Family of subsets A = (A1,...,An), consider B = (B1,...,Bn) with Bi = dom(pi, UAj)

Operator OP, B = OP(A)

Start with family Z0 = (p1,...,pn)Iterate Zi+1 = OP(Zi)

Z1 = (reg(p1),...,reg(pn))Even (odd) Zi gives inner (outer) approximation of the zone diagramFixed point of OP is the zone diagram of p1,...,pn

11EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

`Mollified version´: Territory diagrams

Zone diagram: Z(pi) = dom(pi, UZ(pj) Territory diagram: T(pi) contained in dom(pi, UT(pj) (Equality is replaced by inclusion)

Trivial cases ...Territories can be larger than zones, and even nonconvexMaximal territory diagrams (cannot be expanded) Zone diagrams are not the only instance...

12EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Polynomial root finding

Set of n points in the plane...interpreted as the roots of a complex polynomial of degree n

Root finding: Initial value z0, iterative method.Convergence to some root zi = point location, yielding site pi

Basin of attraction for each site

Seqence of iteration functions Bm [Kalantari]B2 = Newton‘s method, B3 = Halley‘s method Fractal behavior...Uniform approximation of Voronoi diagram, as m grows.

13EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Similarity to zone diagram

`Save´ convex areas inside immediate basins of attraction

14EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

ZONE DIAGRAMS

Open questions

More insight into the structure of zone diagrams

Construction algorithms, approximation...

Is there a link to basins of attraction?

15EuroGIGA CRP “Spatial Decompositions and Graphs” (VORONOI)

Thank you