We present a data structure and an algorithm for efficient and exact interference detection amongst complex models undergoing rigid motion. The algorithm is applicable to all general polygonal models. It pre-computes a hierarchical representation of models using tight-fitting oriented bounding box trees (OBBTrees). At runtime, the algorithm traverses two such trees and tests for overlaps between oriented bounding boxes based on a separating axis theorem, which takes less than 200 operations in practice. It has been implemented and we compare its performance with other hierarchical data structures. In particular, it can robustly and accurately detect all the contacts between large complex geometries composed of hundreds of thousands of polygons at interactive rates.

Adequate coverage is very important for sensor networks to fulfill the issued sensing tasks. In many working environments, it is necessary to make use of mobile sensors, which can move to the correct places to provide the required coverage. In this paper, we study the problem of placing mobile sensors to get high coverage. Based on Voronoi diagrams, we design two sets of distributed protocols for controlling the movement of sensors, one favoring communication and one favoring movement. In each set of protocols, we use Voronoi diagrams to detect coverage holes and use one of three algorithms to calculate the target locations of sensors if holes exist. Simulation results show the effectiveness of our protocols and give insight on choosing protocols and calculation algorithms under different application requirements and working conditions.

: We present novel algorithms for fast proximity queries using swept sphere volumes. The set of proximity queries includes collision detection and both exact and approximate separation distance computation. We introduce a new family of bounding volumes that correspond to a core primitive shape grown outward by some offset. The set of core primitive shapes includes a point, line, and rectangle. This family of bounding volumes provides varying tightness of fit to the underlying geometry. Furthermore, we describe efficient and accurate algorithms to perform different queries using these bounding volumes. We present a novel analysis of proximity queries that highlights the relationship between collision detection and distance computation. We also present traversal techniques for accelerating distance queries. These algorithms have been used to perform proximity queries for applications including virtual prototyping, dynamic simulation, and motion planning on complex models. As compared to ...

We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in fixed dimension into a lineartime deterministic one. The constant of proportionality is d O(d) , which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other problems (such as finding the maximum volume ellipsoid inscribed into the intersection of n halfspaces) in linear time.

The paper presents a technique for merging two genus 0 polyhedra. Merging establishes correspondences between vertices of the models as a first step in a 3D morphing process. The technique allows for the specification of scattered features to be aligned. This is accomplished by performing the following three steps: First, initial embeddings of the polyhedra on unit spheres are computed. Second, the embeddings are deformed such that user defined features (vertices) coincide on the spheres. Third, an overlay of the subdivisions is computed and the aligned vertices are fused in the merged model. Keywords. Polyhedra, Scattered Features, Morphing 1.

Standard SVM training has O(m 3) time and O(m 2) space complexities, where m is the training set size. It is thus computationally infeasible on very large data sets. By observing that practical SVM implementations only approximate the optimal solution by an iterative strategy, we scale up kernel methods by exploiting such “approximateness ” in this paper. We first show that many kernel methods can be equivalently formulated as minimum enclosing ball (MEB) problems in computational geometry. Then, by adopting an efficient approximate MEB algorithm, we obtain provably approximately optimal solutions with the idea of core sets. Our proposed Core Vector Machine (CVM) algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m. Experiments on large toy and realworld data sets demonstrate that the CVM is as accurate as existing SVM implementations, but is much faster and can handle much larger data sets than existing scale-up methods. For example, CVM with the Gaussian kernel produces superior results on the KDDCUP-99 intrusion detection data, which has about five million training patterns, in only 1.4 seconds on a 3.2GHz Pentium–4 PC.

. We use known data structures for ray-shooting and linear-programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f -face convex hull of an n-point set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 1-1/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 1-1/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2-# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in d-dimen...

The theme of this paper is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as "analyze a randomized algorithm as if it were running backwards in time, from output to input." We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others. 1 Introduction The curious phenomenon that randomness can be used profitably in the solution of computational tasks has attracted a lot of attention from researchers in recent years. The approach has proved useful in such diverse area...

This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems whichwe call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...

In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We assume that a layout of the graph is given. From this layout, in the preprocessing, we determine for each edge a geometric object containing all nodes that can be reached on a shortest path starting with that edge. Based on these geometric objects, the search space for online computation can be reduced significantly. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are traffic networks, the typical field of application for this scenario.