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Minimum Distance Queries for Haptic Rendering

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Minimum Distance Queries for Haptic Rendering MINIMUM DISTANCE QUERIES FOR HAPTIC RENDERING by David Edward Johnson A dissertation submitted to the faculty of the University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science School of Computing The University of Utah May 2005 Copyright © David Edward Johnson 2005 All Rights Reserved THE UNIVERSITY OF UTAH GRADUATE SCHOOL SUPERVISORY COMMITTEE APPROVAL of a dissertation submitted by David E. Johnson This dissertation has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory. _________________ ______________________________ Chair: Elaine Cohen _________________ ______________________________ Richard Riesenfeld _________________ ______________________________ Samuel Drake _________________ ______________________________ William Thompson _________________ ______________________________ John Hughes THE UNIVERSITY OF UTAH GRADUATE SCHOOL FINAL READING APPROVAL To the Graduate Council of the University of Utah: I have read the dissertation of David E. Johnson in its final form and have found that (1) its format, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the supervisory committee and is ready for submission to The Graduate School. _________________ ______________________________ Date Elaine Cohen Chair: Supervisory Committee Approved for the Major Department ______________________________ Christopher Johnson Chair Approved for the Graduate Council ______________________________________ David S. Chapman Dean of The Graduate School ABSTRACT Finding the closest points between two modeled objects is a fundamental operation in robotics, computer graphics, and computational geometry. This dissertation is motivated by the use of distance functions in haptic interfaces for virtual prototyping, where distance measures provide the basis for forces that are applied to a human user. The requirements for haptic interfaces mean that these distances must be computed both quickly and robustly. This dissertation begins by exploring the robustness of simple numerical methods for finding the minimum distance between a point and a curve. A geometric analysis of the convergence conditions yields an algorithm for precomputing a set of starting values with robustness guarantees. Embedding this simple local method within a geometric convergence test then provides some guarantees of global convergence. The requirements of haptic interfaces motivate another approach, based on normal cones, for global search of local minima. This technique extends to surfaces and mixed with numerical methods allows a haptic rendering system for NURBS models. Finally, the normal cone approach is applied to polygonal models, which provides the basis for a general 6DOF haptic interface for virtual prototyping. These methods provide significant performance and reliability benefits over existing haptic rendering techniques. For Susan and Lucy TABLE OF CONTENTS 1. ABSTRACT................................................................................................................. iv 2. ACKNOWLEDGMENTS ........................................................................................... ix 3. INTRODUCTION ........................................................................................................ 1 Virtual Prototyping .............................................................................................. 2 Haptic Interfaces .................................................................................................. 3 Distance Measures ............................................................................................... 4 Summary.............................................................................................................. 5 4. BACKGROUND .......................................................................................................... 6 Distance................................................................................................................ 6 Modeling.............................................................................................................. 7 Distance Queries for Polygonal Models .............................................................. 7 Parametric Models ............................................................................................... 8 Other Representations.......................................................................................... 9 Haptic Rendering ............................................................................................... 10 Discussion.......................................................................................................... 15 5. DISTANCE TO PARAMETRIC MODELS .............................................................. 17 Distance from a Point to a Curve....................................................................... 17 Extremal Distance.............................................................................................. 20 Distance from a Point to a Surface .................................................................... 22 Distance Between Two Surfaces ....................................................................... 24 Extremal Distance Formulation ......................................................................... 24 Discussion.......................................................................................................... 26 6. THE SCALED EVOLUTE BOUND FOR RELIABLE CONVERGENCE OF POINT-CURVE MINIMUM DISTANCE QUERIES .............................................. 27 Newton’s Method for Minimum Distance Queries ........................................... 28 Computing Seed Points...................................................................................... 42 Discussion.......................................................................................................... 54 7. LOWER BOUND PRUNING FOR GLOBAL MINIMUM DISTANCE QUERIES BETWEEN A POINT AND A CURVE.................................................................... 55 Lower Bounds.................................................................................................... 56 Results................................................................................................................ 60 Discussion.......................................................................................................... 60 8. GLOBAL SEARCH FOR POINT-CURVE DISTANCE MINIMA USING NORMAL CONES ..................................................................................................... 63 Normal Cone Approach ..................................................................................... 63 Bounding the Range of Normals with Normal Cones ....................................... 64 Checking the Collinearity Condition ................................................................. 66 Computing an Exact Solution ............................................................................ 66 Results................................................................................................................ 68 9. MINIMUM DISTANCE QUERIES BETWEEN A POINT AND A SURFACE .... 69 Multidimensional Newton’s Method ................................................................. 69 Degeneracy Conditions...................................................................................... 71 Degeneracy Along the Normal .......................................................................... 73 Additional Examples of the Degeneracy Quadric ............................................. 75 Discussion.......................................................................................................... 77 10. GLOBAL SEARCH FOR POINT-SURFACE DISTANCE EXTREMA USING NORMAL CONES ..................................................................................................... 78 Coherency with the Dynamic Subdivision Tree ................................................ 82 Discussion.......................................................................................................... 82 11. HAPTIC RENDERING OF NURBS SURFACES .................................................... 84 Finding Local Minima ....................................................................................... 84 Local Update...................................................................................................... 85 Results................................................................................................................ 86 12. GLOBAL SEARCH FOR LOCAL DISTANCE MINIMA BETWEEN POLYGONAL MODELS........................................................................................... 88 The Spatialized Normal Cone Hierarchy........................................................... 89 Constructing the Hierarchy................................................................................ 89 Minimum Distance Computations with a SNCH............................................... 90 Results................................................................................................................ 95 Discussion.......................................................................................................... 97 13. SIX DEGREE-OF-FREEDOM HAPTIC RENDERING OF COMPLEX POLYGONAL MODELS..........................................................................................
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