degeneracy of angular voronoi diagram
DESCRIPTION
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. Introduced by Asano et al. in ISVD06 A tool to improve a polygon of triangular meshes Definition:. Angular Voronoi Diagram. - PowerPoint PPT PresentationTRANSCRIPT
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.