Degeneracy of Angular Voronoi Diagram

1. Degeneracy of Angular Voronoi Diagram Hidetoshi Muta1 and Kimikazu Kato1,2 1 Department of Computer Science, University of Tokyo 2 Nihon Unisys, Ltd.

2. Introduced by Asano et al. in ISVD06 A tool to improve a polygon of triangular meshes Definition: Angular Voronoi Diagram For given line segments, the distance to determine the dominance of the regions is defined by a visual angle.

3. Equations of angular VD For 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)

4. 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

5. Degeneracy of Euclidean VD More than four Voronoi sites are cocircular With some perturbation Or with some computational error Complex crossing structure Voronoi 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

6. 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?

7. Singular points of cubic curves Singularities of cubic curves are classified into three types Node Cusp Isolated point 図

8. Perturbation

9. What wrong with robust computation? Crossing at one point Crossing at three points The number of intersecting points can drastically change with a perturbation

10. 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.

11. Classification of degeneracy All AVD Degenerate Non-Degenerate Degree two Degree three Degree one Never happens Irreducible (Hyperbolic curve) Singularity (node) Factorable (Line x Line) Factorable (Circle x Line)

12. Degree 3 Degree 2 Singularity (node) Factorable (Circle x Line) Irreducible (Hyperbolic curve) Factorable (Line x Line) 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 other Open! On same lineSame length Common endpoint On same line

13. 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.)

14. Conclusion • We classified the types of degeneracy of an angular Voronoi diagrams • Classification of the sub-types of singular cubic 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.