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.

We present a general technique for approximating various descriptors of the extent of a set P of n points in R . For a given extent measure and a parameter " > 0, it computes in time O(n + 1=" ) a subset Q P of size 1=" , with the property that (1 ")(P ) (Q) (P ). The speci c applications of our technique include "-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of P , (ii) maintaining all the previous measures for a set of moving points, and (iii) tting spheres and cylinders through a point set P . Our algorithms are considerably simpler, and faster in many cases, than the known algorithms.

We present an efficient O(n + 1/ε^4.5)-time algorithm for computing a (1 + 1/ε)-approximation of the minimum-volume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online [Har00].

We speed up previous (1 + ")-factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimum-radius enclosing cylinder, minimum-width annulus, minimum-volume bounding box, minimum-width cylindrical shell, etc.

Layout and packing are NP-hard geometric optimization problems of practical importance for which finding a globally optimal solution is intractable if P!=NP. Such problems appear in industries such as aerospace, ship building, apparel and shoe manufacturing, furniture production, and steel construction. At their core, layout and packing problems have the common geometric feasibility problem of containment: find a way of placing a set of items into a container. In this thesis, we focus on containment and its applications to layout and packing problems. We demonstrate that, although containment is NP-hard, it is fruitful to: 1) develop algorithms for containment, as opposed to heuristics, 2) design containment algorithms so that they say "no" almost as fast as they say "yes", 3) use geometric techniques, not just mathematical programming techniques, and 4) maximize the number of items for which the algorithms are practical. Our approach to containment is based on a new restrict/evaluate...

They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap. – The Hunting of the Snark,

The hierarchical radiosity algorithm solves for the global transfer of diffuse illumination in a scene. While its potential algorithmic complexity is superior to both previous radiosity methods and distributed ray tracing, for scenes containing detailed polygonal models, or highly tessellated curved surfaces, its time performance and memory consumption are less than ideal. Also, the density and orientation of the polygons in the input scene unduly affect the output of the method. The aim of this thesis will be to show that by using flexible surface hierarchies similar to those in the surface simplification literature, the use of regular refinement and to a large extent isotropic volume clusters can be avoided, increasing both the speed and the quality of the basic algorithm. I will develop a radiosity system incorporating these ideas, and show that its performance is superior to existing hierarchical radiosity algorithms, in the domain of scenes containing complex models. The underlying goal of my thesis work is to make high-quality radiosity possible with such scenes. 1

is concerned with finding the smallest shape of some type that encloses all the points of P . Well-known instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in R , find the smallest shape in question that contains at least k points or a certain quantile of the data. This type of problem is known as a k-enclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time.

In this paper, a novel model-based optical tracking and model estimation system for composite interaction devices is presented. Devices consist of a set of linked segments, where each segment can have combinations of translational and rotational degrees of freedom (DOFs) relative to a parent segment. The system automatically constructs the geometric skeleton structure, DOF relations, and DOF constraints between segments. Pre-defined models are not required. The system supports segments with only a single marker, so that interaction devices can be small with a low number of markers. The model is computed in an offline procedure. The tracking method uses the obtained device model to recognize the device and reconstruct all DOF parameters describing the pose of each segment. The tracking method can handle partial occlusion. Results show the proposed techniques are efficient and robust.

: 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. CR Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling Additional Key Words and Phrases: hierarchical data structure, collision detection, shape approximation, contac...