Joseph S. B. Mitchell

Efficient collision detection using bounding volume hierarchies of k-DOPs

Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In addition to determining accurately the contacts that occur between pairs of objects, one needs also to do so at real-time rates. Applications such as haptic force feedback can require over 1000 collision queries per second. We develop and analyze a method, based on bounding-volume hierarchies, for efficient collision detection for objects moving within highly complex environments. Our choice of bounding volume is to use a discrete orientation polytope (k-DOP), a convex polytope whose facets are determined by halfspaces whose outward normals come from a small fixed set of k orientations. We compare a variety of methods for constructing hierarchies (BV-trees) of bounding k-DOPs. Further, we propose algorithms for maintaining an effective BV-tree of k-DOPs for moving objects, as they rotate, and for performing fast collision detection using BV-trees of the moving objects and of the environment. Our algorithms have been implemented and tested. We provide experimental evidence showing that our approach yields substantially faster collision detection than previous methods.

The discrete geodesic problem

We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number ofedges ofthe surface. Afterwe run our algorithm, the distance from the source to any other destination may be determined using standard techniques in time O(log n) by locating the destination in the subdivision created by the algorithm. The actual shortest path from the source to a destination can be reported in time O(k+ log n), where k is the number of faces crossed by the path. The algorithm generalizes to the case of multiple source points to build the Voronoi diagram on the surface, where n is now the maximum of the number of vertices and the number of sources.

Boundary recognition in sensor networks by topological methods

Wireless sensor networks are tightly associated with the underlying environment in which the sensors are deployed. The global topology of the network is of great importance to both sensor network applications and the implementation of networking functionalities. In this paper we study the problem of topology discovery, in particular, identifying boundaries in a sensor network. Suppose a large number of sensor nodes are scattered in a geometric region, with nearby nodes communicating with each other directly. Our goal is to find the boundary nodes by using only connectivity information. We do not assume any knowledge of the node locations or inter-distances, nor do we enforce that the communication graph follows the unit disk graph model. We propose a simple, distributed algorithm that correctly detects nodes on the boundaries and connects them into meaningful boundary cycles. We obtain as a byproduct the medial axis of the sensor field, which has applications in creating virtual coordinates for routing. We show by extensive simulation that the algorithm gives good results even for networks with low density. We also prove rigorously the correctness of the algorithm for continuous geometric domains.

The weighted region problem: finding shortest paths through a weighted planar subdivision

The problem of determining shortest paths through a weighted planar polygonal subdivision with <italic>n</italic> vertices is considered. Distances are measured according to a weighted Euclidean metric: The length of a path is defined to be the weighted sum of (Euclidean) lengths of the subpaths within each region. An algorithm that constructs a (restricted) “shortest path map” with respect to a given source point is presented. The output is a partitioning of each edge of the subdivion into intervals of ε-optimality, allowing an ε-optimal path to be traced from the source to any query point along any edge. The algorithm runs in worst-case time <italic>O</italic>(<italic>ES</italic>) and requires <italic>O</italic>(<italic>E</italic>) space, where <italic>E</italic> is the number of “events” in our algorithm and <italic>S</italic> is the time it takes to run a numerical search procedure. In the worst case, <italic>E</italic> is bounded above by <italic>O</italic>(<italic>n</italic><supscrpt>4</supscrpt>) (and we give an &OHgr;(<italic>n</><supscrpt>4</supscrpt>) lower bound), but it is likeky that <italic>E</italic> will be much smaller in practice. We also show that <italic>S</italic> is bounded by <italic>O</italic>(<italic>n</italic><supscrpt>4</supscrpt><italic>L</italic>), where <italic>L</italic> is the precision of the problem instance (including the number of bits in the user-specified tolerance ε). Again, the value of <italic>S</italic> should be smaller in practice. The algorithm applies the “continuous Dijkstra” paradigm and exploits the fact that shortest paths obey Snells Law of Refraction at region boundaries, a local optimaly property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronoi diagrams.

Approximation algorithms for lawn mowing and milling

Abstract We study the problem of finding shortest tours/paths for “lawn mowing” and “milling” problems: Given a region in the plane, and given the shape of a “cutter” (typically, a circle or a square), find a shortest tour/path for the cutter such that every point within the region is covered by the cutter at some position along the tour/path. In the milling version of the problem, the cutter is constrained to stay within the region. The milling problem arises naturally in the area of automatic tool path generation for NC pocket machining. The lawn mowing problem arises in optical inspection, spray painting, and optimal search planning. Both problems are NP-hard in general. We give efficient constant-factor approximation algorithms for both problems. In particular, we give a (3+e) -approximation algorithm for the lawn mowing problem and a 2.5-approximation algorithm for the milling problem. Furthermore, we give a simple 6 5 -approximation algorithm for the TSP problem in simple grid graphs, which leads to an 11 5 -approximation algorithm for milling simple rectilinear polygons.

Shortest paths and networks

We survey various forms of the problem, primarily in two and three dimensions, for motion of a single point, since most results have focused on these cases. We discuss shortest paths in a simple polygon (Section 31.1), shortest paths among obstacles (Section 31.2), and other metrics for length (Section 31.3). We also survey other related geometric network optimization problems (Section 31.4). Higher dimensions are discussed in Section 31.5.

APPROXIMATING POLYGONS AND SUBDIVISIONS WITH MINIMUM-LINK PATHS

We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with. no self-intersections are NP-hard.

BOXTREE: A Hierarchical Representation for Surfaces in 3D

We introduce the boxtree, a versatile data structure for representing triangulated or meshed surfaces in 3D. A boxtree is a hierarchical structure of nested boxes that supports efficient ray tracing and collision detection. It is simple and robust, and requires minimal space. In situations where storage is at a premium, boxtrees are effective alternatives to octrees and BSP trees. They are also more flexible and efficient than R‐trees, and nearly as simple to implement.

L1 shortest paths among polygonal obstacles in the plane

We present an algorithm for computingL1 shortest paths among polygonal obstacles in the plane. Our algorithm employs the “continuous Dijkstra” technique of propagating a “wavefront” and runs in timeO(E logn) and spaceO(E), wheren is the number of vertices of the obstacles andE is the number of “events.” By using bounds on the density of certain sparse binary matrices, we show thatE =O(n logn), implying that our algorithm is nearly optimal. We conjecture thatE =O(n), which would imply our algorithm to be optimal. Previous bounds for our problem were quadratic in time and space.Our algorithm generalizes to the case of fixed orientation metrics, yielding anO(nɛ−1/2 log2n) time andO(nɛ−1/2) space approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm further generalizes to the case of many sources, allowing us to compute anL1 Voronoi diagram for source points that lie among a collection of polygonal obstacles.

Voronoi Diagrams of Moving Points in the Plane

Consider a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Delaunay diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λs(n)), where λs,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites. In the special case of points moving at constant speed along straight lines, we get s = 4, implying an upper bound of O(n32α(n)), where α(n) is the extremely slowly-growing inverse of Ackermann s function. Our results are a linear-factor improvement over the naive quartic bound on the number of topological events.