Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0710.0925

Degeneracy of Angular Voronoi Diagram

Hidetoshi Muta·Kimikazu Kato
View original source ↗Read source document ↗Export metadata ↓

Abstract

Angular Voronoi diagram was introduced by Asano et al. as fundamental research for a mesh generation. In an angular Voronoi diagram, the edges are curves of degree three. From view of computational robustness we need to treat the curves carefully, because they might have a singularity. We enumerate all the possible types of curves that appear as an edge of an angular Voronoi diagram, which tells us what kind of degeneracy is possible and tells us necessity of considering a singularity for computational robustness.

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hidetoshi Muta, Kimikazu Kato. 2007-10-04. Degeneracy of Angular Voronoi Diagram. https://arxiv.org/abs/0710.0925

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Simultaneous Embedding of Two Paths on the Grid

Simultaneous geometric embedding of graphs on an integer grid is a classical research topic in Graph Drawing. When the graphs are two paths, one can additionally require that one path is x-monotone and the other is y-monotone (the monotone setting). A simple and elegant algorithm is known to simultaneously embed two paths on an integer grid in the monotone setting. The resulting embedding is, however, not optimal in terms of grid size or edge lengths. In this paper, we study some quality measures of simultaneous embeddings of two paths on an integer grid, in both the monotone and in the unconstrained setting. In the monotone setting, we present almost-linear time algorithms to minimize either the perimeter of the bounding box or the sum of the edge lengths in the $L_1$-norm. Both results are based on transforming the problem into that of finding a minimum vertex cover (weighted for the $L_1$-norm problem) on a related bipartite graph derived from the two paths. As for the unconstrained setting, we show that both minimizing the length of the longest edge and minimizing the sum of the edge lengths of a simultaneous embedding of two paths on an integer grid is NP-hard.

cs.CG↗
arXiv2026

Using Persistent Homology to Analyze Access to Heterogeneous-Quality Resources and Heterogeneous-Severity Nuisances

We develop a framework to use multiparameter persistent homology (PH) to examine access to heterogeneous-quality resources and exposure to heterogeneous-severity nuisances in a geographic region. Persistent homology, which is a type of topological data analysis {(TDA)}, has been employed previously to examine resource coverage. Unlike prior approaches, which used one-parameter PH to study resource coverage and nuisance exposure, our method accounts for heterogeneous-quality resources. Our framework, which employs a computationally-efficient approximation of multiparameter PH, allows one to study access to any resource ({or} exposure of any nuisance) using any notion of quality (or severity). Using the city of Chicago as an example region, we employ our framework to detect clusters of poor access to public parks, overexposure to landfills, and both underexposure and overexposure to pubs and bars.

cs.CG↗
arXiv2026

Symbolic Constraints in Polyhedral Enclosure and Tetrahedral Decomposition in Genus-0 Polyhedra

I present an exploratory, coordinate-free framework for organizing two related problems in polyhedral combinatorics: screening polygonal face multisets for genus-zero enclosure and examining tetrahedral decompositions that preserve the original boundary vertex set. The boundary is summarized using a face-degree excess measure that records how far the surface departs from full triangulation, while standard Euler and incidence relations provide exact constraints and parity-dependent bounds on the external structure. For tetrahedral decompositions, standard incidence relations for a triangulated 3-ball are separated from the additional patterns generated by a restricted boundary-driven decomposition procedure. Within the manually constructed examples, bipyramidal configurations provide an extremal reference class, and departures from bipyramidal structure are associated with changes in attainable tetrahedral states. The examples also suggest a contraction of the realized internal state space as boundary face-degree excess increases, motivating the notions of S-flexibility and incidence-gated transitions. These patterns are presented as empirical structural observations and conjectural directions rather than general classification theorems. The resulting framework provides a symbolic pre-screening workflow that can reject some impossible polygonal face multisets before explicit graph construction or geometric realization, and is implemented in the open-source R package polyenclose.

cs.CG↗