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

2. 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.

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 With some perturbation Or with some computational error More than four Voronoi sites are cocircular 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 NodeCuspIsolated point Singularities of cubic curves are classified into three types 図

8. Perturbation

9. Crossing at one pointCrossing at three points What wrong with robust computation? 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 Degenerate Non-Degenerate Degree three Degree two Degree one Singularity (node) Factorable (Circle x Line) Irreducible (Hyperbolic curve) Factorable (Line x Line) Never happens All AVD

12. Singularity (node) Factorable (Circle x Line) Irreducible ( Hyperbolic curve) Factorable (Line x Line) On same line On same line Same length Common endpoint Same length Parallel Same length, Parallel Diagonal lines cross vertically Same length with all endpoints in the same circle One line segment by the pair of endpoints is bisected vertically by the other Open! Degree 3Degree 2

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.