656 research outputs found
Survey of two-dimensional acute triangulations
AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual
Generalizations of the Kolmogorov-Barzdin embedding estimates
We consider several ways to measure the `geometric complexity' of an
embedding from a simplicial complex into Euclidean space. One of these is a
version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove
inequalities relating the thickness and the number of simplices in the
simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved
for graphs. We also consider the distortion of knots. We give an alternate
proof of a theorem of Pardon that there are isotopy classes of knots requiring
arbitrarily large distortion. This proof is based on the expander-like
properties of arithmetic hyperbolic manifolds.Comment: 45 page
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
Applications for papers from SCG '9
Schnyder woods for higher genus triangulated surfaces, with applications to encoding
Schnyder woods are a well-known combinatorial structure for plane
triangulations, which yields a decomposition into 3 spanning trees. We extend
here definitions and algorithms for Schnyder woods to closed orientable
surfaces of arbitrary genus. In particular, we describe a method to traverse a
triangulation of genus and compute a so-called -Schnyder wood on the
way. As an application, we give a procedure to encode a triangulation of genus
and vertices in bits. This matches the worst-case
encoding rate of Edgebreaker in positive genus. All the algorithms presented
here have execution time , hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational
Geometr
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