A new paradigm for rigid body simulation is presented and analyzed. Current techniques for rigid body simulation run slowly on scenes with many bodies in close proximity. Each time two bodies collide or make or break a static contact, the simulator must interrupt the numerical integration of velocities and accelerations. Even for simple scenes, the number of discontinuities per frame time can rise to the millions. An efficient optimization-based animation (OBA) algorithm is presented which can simulate scenes with many convex threedimensional bodies settling into stacks and other “crowded” arrangements. This algorithm simulates Newtonian (second order) physics and Coulomb friction, and it uses quadratic programming (QP) to calculate new positions, momenta, and accelerations strictly at frame times. The extremely small integration steps inherent to traditional simulation techniques are avoided. Contact points are synchronized at the end of each frame. Resolving contacts with friction is known to be a difficult problem. Analytic force calculation can have ambiguous or non-existing solutions. Purely impulsive techniques avoid these ambiguous cases, but still require an excessive and computationally expensive number of updates in the case of

We develop a combinatorial algorithm for determining a minimum volume simplex enclosing a set of points in R 3 . If the convex hull of the points has n vertices, then our algorithm takes (n 4 ) time. Combining our exact but slow algorithm with a simple but crude approximation technique, we also develop an "-approximation algorithm. The algorithm computes in O(n + 1=" 6 ) time a simplex whose volume is within (1 + ") factor of the optimal, for any " > 0. 1 Introduction Approximating a geometric body by a combinatorially simpler shape is a problem with many applications. In computer graphics and robotics, for instance, checking for collision between complex geometric models is frequently a computational bottleneck. Therefore, collision detection packages commonly use simple bounding objects, such as axis-aligned bounding boxes [4, 14, 16], discrete oriented polytopes [9, 13], or spheres [10], to quickly eliminate pairs whose bounding objects are collision-free. Since interse...

Detecting whether two geometric objects intersect and computing the region of intersection are fundamental problems in computational geometry. Geometric intersection problems arise naturally in a number of applications. Examples include geometric packing and covering, wire and component layout in VLSI, map overlay

Copyright c ○ 2000 by James F. O’BrieniiAcknowledgments I would like to thank my family, friends, and colleagues who provided me with their friendship and support during my time in graduate school. My parents and family were, and continue to be, a constant source of encouragement. They inspired me to set ambitious goals for myself, and they heartened me when those goals seemed distant. The members of the GVU Center, especially the other students in the Animation Lab and Geometry Group, created the camaraderie that made long hours in the lab often seem more like play than work. They were there to lend a hand when deadlines loomed all too near, and to lend an ear when deadlines seemed all too many. I am deeply grateful to my advisor, Jessica Hodgins, who’s excellent advice and example were invaluable; and to the members of my committee, Greg Turk, Irfan Essa, Jarek Rossignac, and Demetri Terzopoulos, who provided insightful criticism and direction. I am also thankful to Jim Foley and Norberto Ezquerra who helped to set me on the right track during my first years of graduate school.