616 research outputs found
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Piecewise-Linear Farthest-Site Voronoi Diagrams
Voronoi diagrams induced by distance functions whose unit balls are convex polyhedra are piecewise-linear structures. Nevertheless, analyzing their combinatorial and algorithmic properties in dimensions three and higher is an intriguing problem. The situation turns easier when the farthest-site variants of such Voronoi diagrams are considered, where each site gets assigned the region of all points in space farthest from (rather than closest to) it.
We give asymptotically tight upper and lower worst-case bounds on the combinatorial size of farthest-site Voronoi diagrams for convex polyhedral distance functions in general dimensions, and propose an optimal construction algorithm. Our approach is uniform in the sense that (1) it can be extended from point sites to sites that are convex polyhedra, (2) it covers the case where the distance function is additively and/or multiplicatively weighted, and (3) it allows an anisotropic scenario where each site gets allotted its particular convex distance polytope
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
Constructing L∞ Voronoi diagrams in 2D and 3D
Voronoi diagrams and their computation are well known in the Euclidean L2 space. They are easy to sample and render in generalized Lp spaces but nontrivial to construct geometrically. Especially the limit of this norm with p → ∞ lends itself to many quad- and hex-meshing related applications as the level-set in this space is a hypercube. Many application scenarios circumvent the actual computation of L∞ diagrams altogether as known concepts for these diagrams are limited to 2D, uniformly weighted and axis-aligned sites. Our novel algorithm allows for the construction of generalized L∞ Voronoi diagrams. Although parts of the developed concept theoretically extend to higher dimensions it is herein presented and evaluated for the 2D and 3D case. It further supports individually oriented sites and allows for generating weighted diagrams with anisotropic weight vectors for individual sites. The algorithm is designed around individual sites, and initializes their cells with a simple meshed representation of a site's level-set. Hyperplanes between adjacent cells cut the initialization geometry into convex polyhedra. Non-cell geometry is filtered out based on the L∞ Voronoi criterion, leaving only the non-convex cell geometry. Eventually we conclude with discussions on the algorithms complexity, numerical precision and analyze the applicability of our generalized L∞ diagrams for the construction of Centroidal Voronoi Tessellations (CVT) using Lloyd's algorithm
Power to the points: validating data memberships in clusterings
pre-printIn this paper, we present a method to attach affinity scores to the implicit labels of individual points in a clustering. The affinity scores capture the confidence level of the cluster that claims to "own" the point. We demonstrate that these scores accurately capture the quality of the label assigned to the point. We also show further applications of these scores to estimate global measures of clustering quality, as well as accelerate clustering algorithms by orders of magnitude using active selection based on affinity. This method is very general and applies to clusterings derived from any geometric source. It lends itself to easy visualization and can prove useful as part of an interactive visual analytics framework. It is also efficient: assigning an affinity score to a point depends only polynomially on the number of clusters and is independent both of the size and dimensionality of the data. It is based on techniques from the theory of interpolation, coupled with sampling and estimation algorithms from high dimensional computational geometry
Shapes of polyhedra and triangulations of the sphere
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm
IST Austria Thesis
The main objects considered in the present work are simplicial and CW-complexes with vertices forming a random point cloud. In particular, we consider a Poisson point process in R^n and study Delaunay and Voronoi complexes of the first and higher orders and weighted Delaunay complexes obtained as sections of Delaunay complexes, as well as the ÄŒech complex. Further, we examine theDelaunay complex of a Poisson point process on the sphere S^n, as well as of a uniform point cloud, which is equivalent to the convex hull, providing a connection to the theory of random polytopes. Each of the complexes in question can be endowed with a radius function, which maps its cells to the radii of appropriately chosen circumspheres, called the radius of the cell. Applying and developing discrete Morse theory for these functions, joining it together with probabilistic and sometimes analytic machinery, and developing several integral geometric tools, we aim at getting the distributions of circumradii of typical cells. For all considered complexes, we are able to generalize and obtain up to constants the distribution of radii of typical intervals of all types. In low dimensions the constants can be computed explicitly, thus providing the explicit expressions for the expected numbers of cells. In particular, it allows to find the expected density of simplices of every dimension for a Poisson point process in R^4, whereas the result for R^3 was known already in 1970's
- …