Martin Held

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.

Pocket machining based on contour-parallel tool paths generated by means of proximity maps

Abstract A fundamental NC-machining problem is the clearing of areas within specified boundaries from material. The paper presents an efficient algorithm for generating largely optimal tool paths to solve this pocket-machining problem for multiply connected planar pocket areas bounded by curvilinear boundaries. Using certain concepts of computational geometry, i.e. Voronoi diagrams and monotonic pouches, offsets of a pocket boundary are efficiently generated. Further, it is explained how the tool path can be optimized with respect to several criteria arising from technological requirements. The concepts presented have been implemented, and they form the basis of the pocketing package lark .

Voronoi diagrams and offset curves of curvilinear polygons

This paper studies the practical generation of Voronoi diagrams and offset curves of simply-connected planar areas bounded by straight lines and circular arcs. We present and analyze a wavefront-propagation algorithm for the generation of Voronoi diagrams and compare it experimentally to a tuned version of Lees divide-and-conquer algorithm. Key performance parameters of these two algorithms are compared based on machine-generated test data. We conclude that the wavefront-propagation algorithm can be expected to outperform the divide-and-conquer algorithm for all, but pathological test data. In particular, its practical running time seems to grow only linearly. We also used our implementation in order to gather statistics on the CPU-consumption of offsetting based on Voronoi diagrams. All tests clearly showed the practical suitability of using Voronoi diagrams for the offsetting of curvilinear polygons. The CPU-time consumptions recorded also compare very favorably with other published codes for computing Voronoi diagrams.

A geometry-based investigation of the tool path generation for zigzag pocket machining

We present a detailed description of a zigzag algorithm for pocket machining. The algorithm is capable of computing correct zigzag tool paths for multiply-connected planar areas (“pockets”) bounded by a wide class of curves. It features a number of optimizations with respect to geometrical and technological objectives. In particular, a near-optimum inclination of the tool path is automatically determined. The underlying geometric principles are simple enough to allow the algorithm to be included in a numerical control computer.

Fast and effective stripification of polygonal surface models

A fundamental algorithmic problem in computer graphics is t hat of computing a succinct encoding of a triangulation of a polygo nal surface model in order to be able to transmit and render it efficie ntly. The goal is to take a given polygonal surface model, whose fac ts are given by (possibly multiply-connected) polygons, tria ngulate its facets, and then decompose the triangulation into a small nu mber of “tristrips,” each of which has its connectivity stored im plicitly in the ordering of the data points. We develop methods that ar e effective in solving the stripification problem, both in the ory (provably good encodings) and in practice. Our methods are based o n carefully constructed search trees in the dual graph, follo wed by algorithms to decompose dual trees into tristrips. One decomp osition algorithm is provably optimal (based on dynamic programmin g), allowing us a sound basis of comparison among our other (heur istic) algorithms. We demonstrate the speed and effectivenes s of our algorithms through a battery of experiments. In comparison with the recently released STRIPE system for stripification, we find that our stripifier,FTSG, produces comparable or better quality encodings, while requiring significantly less computing time on a large variety of datasets. Further, FTSG is carefully engineered and implemented to be robust, even in the face of highly degenerate and corrupted real-world data. CR Categories: I.3.3 [Computer Graphics]: Picture/Image Generation—displaying algorithms; I.3.5 [Computer Graph ics]: Computational Geometry and Object Modeling—Geometric alg orithm, languages, and systems.

Optimization problems related to zigzag pocket machining

Abstract. A fundamental problem of manufacturing is to produce mechanical parts from billets by clearing areas within specified boundaries from the material. Based on a graph-theoretical formulation, the algorithmic handling of one particular machining problem—``zigzag pocket machining—is investigated. We present a linear-time algorithm that ensures that every region of the pocket is machined exactly once, while attempting to minimize the number of tool retractions required. This problem is shown to be n

Collision detection for fly-throughs in virtual environments

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PVD: A Stable Implementation for Computing Voronoi Diagrams of Polygonal Pockets

-hard for pockets with holes. Our algorithm is provably good in the sense that the machining path generated for a pocket with h holes requires at most 5 . . . OPT + 6 . . . h retractions, where OPT is the (unknown) minimum number of retractions required by any algorithm. The algorithm has been implemented, and practical tests for pockets without holes suggest that one can expect an approximation factor of about 1.5 for practical examples, rather than the factor 5 as proved by our analysis.

Recognizing polygonal parts from width measurements

Real-time collision detection is of criticaJ importance in physical simulations and in interactive use of virtual environments. For example, the goal of one application in the aircraft industry is to let mechanics verify the serviceability of an aircraft in a VR environment (CAD model of the aircraft), rather than on a full-scale physical mock-up. Wearing a sensor-laden suit, they should be able to test and train routine maintenance tasks such as the inspection and replacement of parts. High-speed collision detection within the virtual environment is essential in order to provide interactive feedback about the feasibility of the mechanics’ actions, Haptic force-feedback can require about 1000 simulation updates per second. We have been investigating various approaches to collision detection in a polyhedral environment 2 (whose boundary representation is given), in which one or more polyhedral “flying objects” move. The assumption is that the environment t may be huge (millions of facets), but that it is largely static and can be preprocessed to support efficient intersection queries with the flying objects, which are assumed to be relatively small in number and moderate in complexity (thousands of facets).

Hamiltonian triangulations for fast rendering

Voronoi diagrams of pockets, i.e. polygons with holes, have a variety of important applications but are particularly challenging to compute robustly. We report on an implementation of a simple algorithm which does not rely on exact arithmetic to achieve robustness; rather, it achieves its robustness through carefully engineered handling of geometric predicates. Although we do not give theoretical guarantees for robustness or accuracy, the software has sustained extensive experimentation (on real and simulated data) and day-to-day usage on real-world data. The algorithm is shown experimentally to compare favorably in running time with prior methods.