Degeneracy of Angular Voronoi Diagram

Hidetoshi Muta1 and Kimikazu Kato1,2

1 Department of Computer Science, University of Tokyo2 Nihon Unisys, Ltd.

Angular Voronoi Diagram Introduced by Asano et

al. in ISVD06 A tool to improve a

polygon of triangular meshes

Definition:For given line segments, the distance to determine the dominance of the regions is defined by a visual angle.

Equations of angular VDFor given two line segments, as a boundary, there appear two equations which are the flip side of each other.

Both equations are cubic (of degree three)

Why interested in the degeneracy of angular VD? It has a much more complicated structure

than Euclidean VD It gives a hint for an extension of the existing

complexity analysis for a general VD which regards its sites are in a general position

It provides a good case study for computational robustness of a general VD

Degeneracy of Euclidean VD

With some perturbation

Or with some computational error

More than four Voronoi sites are cocircular

Complex crossing structureVoronoi edges meet at one point

In theoretical context, they tend to avoid analysis of degeneracy, saying “assume the sites are in a general position”

However, degeneracy takes special care in actual computation to achieve robustness

Computational complexity of algebraic VD Computational complexity of two dimensional

VD whose boundaries are algebraic curves is shown to be [Halperin-Sharir 1994]

It is proved by analyzing the structure of algebraic surfaces whose lower envelope is the VD

Here again, it is assumed that the surfaces are “in a general position.”

What happens in special cases?

Singular points of cubic curves

Node Cusp Isolated point

Singularities of cubic curves are classified into three types

図

Perturbation

Crossing at one point Crossing at three points

What wrong with robust computation?

The number of intersecting points can drastically change with a perturbation

Degeneracy of angular VD For Euclidean VD, degeneracy is a concept

of a position of multiple edges. However, for an angular VD, degeneracy is

defined for a single edge. Degeneracy is defined as a curve which will

change a topological position with a perturbation.

Classification of degeneracy

DegenerateNon-Degenerate

Degree three Degree two Degree one

Singularity

(node)

Factorable

(Circle x Line)

Irreducible

(Hyperbolic curve)

Factorable

(Line x Line)

Never happens

All AVD

Singularity(node)

Factorable(Circle x Line)

Irreducible

(Hyperbolic curve)

Factorable(Line x Line)

On same lineOn same lineSame length

Common endpoint

Same lengthParallel

Same length, ParallelDiagonal lines cross vertically

Same lengthwith all endpoints in the same circle

One line segment by the pair of endpoints is bisected vertically by the otherOpen!

Degree 3 Degree 2

Singularity of cubic curve It is proved that a node

appears as a singularity of the boundary of an angular VD

Whether other types of singularities (cusp and isolated point) appear or not is still open. (With some observation, we conjecture that they do not appear.)

Conclusion We classified the types of degeneracy of an a

ngular Voronoi diagrams Classification of the sub-types of singular cub

ic curve case, i.e. whether a node is an only possible type of singularity, is still open.

Our research shed light on degeneracy problem of a general Voronoi diagram w.r.t. an arbitrary distance.