Teng and others previously showed that certain geometric graphs had small separators that could be found by lifting the graph to a sphere one dimension up and choosing a random great circle. Here we show that epsilon-cuttings and the method of conditional expectations can be used to guide a deterministic prune-and-search method for the same problem. Applications include finding the intersection graph of a collection of spheres and computing or approximating the maximum number of spheres having a common intersection.
(BibTeX -- Citations -- CiteSeer)
Speeds up a triangulation algorithm of Bern et al. ["Linear-Size Nonobtuse Triangulation of Polygons"] by finding a collection of disjoint circles which connect up the holes in a non-simple polygon. The method is to use a minimum spanning tree to find a collection of overlapping circles, then shrink them one by one to reduce the number of overlaps, using Sleator and Tarjan's dynamic tree data structure to keep track of the connectivity of the shrunken circles.
Improving on a recent breakthrough of Sharir, we use data structures from "Dynamic three-dimensional linear programming" to find two circular disks of minimum radius covering a set of points in the Euclidean plane, in randomized expected time O(n log2 n).
(BibTeX -- SODA paper -- Citations -- DREI and SODA talk slides -- CiteSeer)
We use circle-packing methods to generate quadrilateral meshes for polygonal domains, with guaranteed bounds both on the quality and the number of elements. We show that these methods can generate meshes of several types:
(BibTeX -- Citations -- CiteSeer)
We apply techniques from "Quadrilateral meshing by circle packing" to a magic trick of Houdini: fold a piece of paper so that with one straight cut, you can form your favorite polygon.
(preprint at Erik's web site -- BibTeX -- CiteSeer)
Any four mutually tangent spheres determine three coincident lines through opposite pairs of tangencies. As a consequence, we define two new triangle centers. These centers arose as part of a compass-and-straightedge construction of Soddy circles; also available are some Mathematica calculations of trilinear coordinates for the new triangle centers.
(BibTeX -- Citations -- CiteSeer)
We give linear-time quasiconvex programming algorithms for finding a Möbius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, and element size control in conformal structured mesh generation.
(BibTeX -- Citations -- CiteSeer -- WADS talk slides -- ACM DL)
We introduce the fatness parameter of a 4-dimensional polytope P, (f1+f2)/(f0+f3). It is open whether all 4-polytopes have bounded fatness. We describe a hyperbolic geometry construction that produces 4-polytopes with fatness > 5.048, as well as the first infinite family of 2-simple, 2-simplicial 4-polytopes and an improved lower bound on the average kissing number of finite sphere packings in R3. We show that fatness is not bounded for the more general class of strongly regular CW decompositions of the 3-sphere.
We describe a recursive subdivision of the plane into quadrilaterals in the form of rhombi and kites with 60, 90, and 120 degree angles. The vertices of the resulting quadrilateral mesh form the centers of a set of circles that cross orthogonally for every two adjacent vertices, and it has many other properties that are important in finite element meshing.
We show that every planar graph of maximum degree three has a planar Lombardi drawing and that some but not all 4-regular planar graphs have planar Lombardi drawings.
We characterize the graphs of two-dimensional soap bubble clusters as being exactly the bridgeless 3-regular planar graphs.
Geometry -- Publications -- David Eppstein -- Theory Group -- Inf. & Comp. Sci. -- UC Irvine
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