Hyeon-Suk Na

Voronoi diagrams on the sphere

A warning panel trailer in which the warning panel, with the plurality of discrete electrical image display means, is raised to various operative positions along a column support and lowered to an inoperative position in an elongated seat on top of a platform of the trailer. The seat extends between the front and back of the trailer so that the travelling trailer stores the warning panel with an edge facing the direction of travel and the plane of the panel facing away from such direction of travel. The warning panel is lowered and raised by a winched cable which turns around a lower pulley assembly adjacent the platform of the trailer and an upper, pivotally mounted pulley assembly which rotates with the warning panel when selectively positioning the warning panel.

Approximate shortest paths in anisotropic regions

Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with <i>n</i> vertices. Let ρ ≥ 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For all ε ∈ (0, 1), and for any two points <i>v<inf>s</inf></i> and <i>v<inf>d</inf></i>, we give an algorithm that finds a path from <i>v<inf>s</inf></i> to <i>v<inf>d</inf></i> whose cost is at most (1 + ε) times the minimum cost. Our algorithm runs in <i>O</i> (ρ₂logρ/ε²n³ log (ρn/ε)) time. This bound does not depend on any other parameters; in particular, it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in [1, ρ] ∪ {∞}, the time bound of our algorithm improves to <i>O</i> (ρ₂logρ/εn³ log (ρn/ε)).

On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes

It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n2). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n).

The Expected Number of 3D Visibility Events Is Linear

In this paper, we show that, amongst

Geometric permutations of disjoint unit spheres

n

Constructing optimal highways

uniformly distributed unit balls in

Lines and Free Line Segments Tangent to Arbitrary Three-Dimensional Convex Polyhedra

\mathbb{R}^3

The probabilistic complexity of the Voronoi diagram of points on a polyhedron

, the expected number of maximal nonoccluded line segments tangent to four balls is linear. Using our techniques we show a linear bound on the expected size of the visibility complex, a data structure encoding the visibility information of a scene, providing evidence that the storage requirement for this data structure is not necessarily prohibitive. These results significantly improve the best previously known bounds of

Querying approximate shortest paths in anisotropic regions

O(n^{8/3})

THE ALIGNED K-CENTER PROBLEM

[F. Durand, G. Drettakis, and C. Puech, {ACM Transactions on Graphics}, 21 (2002), pp. 176--206]. Our results generalize in various directions. We show that the linear bound on the expected number of maximal nonoccluded line segments that are not too close to the boundary of the scene and tangent to four unit balls extends to balls of various but bounded radii, to polyhedra of bounded aspect ratio, and even to nonfat three-dimensional objects such as polygons of bounded aspect ratio. We also prove that our results extend to other distributions such as the Poisson distribution. Finally, we indicate how our probabilistic analysis provides new insight on the expected size of other global visibility data structures, notably the aspect graph.