Computer Animation. Algorithms and Techniques. z< MORGAN KAUFMANN PUBLISHERS. Rick Parent Ohio State University AN IMPRINT OF ELSEVIER SCIENCE

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1 Computer Animation Algorithms and Techniques Rick Parent Ohio State University z< MORGAN KAUFMANN PUBLISHERS AN IMPRINT OF ELSEVIER SCIENCE AMSTERDAM BOSTON LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO

2 Foreword xv Preface xvii Color Insert following page 298 Chapter 1 Introduction Perception The Heri tage of Animation Early Devices The Early Days of "Conventional" Animation Disney Contributions of Others Other Media for Animation Principles of Computer Animation Animation Production Computer Animation Production Tasks 15 Vll

3 viii Digital Editing Digital Video A Brief History of Computer Animation Early Activity TheMiddleYears Animation Comes ofage Chapter Summary 29 References 29 Chapter 2 Technical Background Spaces and Transformations The Display Pipeline Homogeneous Coordinates and the Transformation Matrix Compounding Transformations: Multiplying Transformation Matrices Basic Transformations Representing an Arbitrary Orientation Extracting Transformations from a Matrix Description of Transformations in the Display Pipeline Round-off Error Considerations Orientation Representation Fixed Angle Representation Euler Angle Representation Angle and Axis Quaternions Chapter Summary 61 References 61 Chapter 3 Interpolation and Basic Techniques Interpolation The Appropriate Function Controlling the Motion Along a Curve 68

4 ix Computing Are Length Speed Control Ease-in/Ease-out Constant Acceleration: Parabolic Ease-In/ Ease-Out General Distance-Time Functions Curve Fitting to Position-Time Pairs Interpolation of Rotations Represented by Quaternions Path Following Orientation along a Path Smoothing a Path Determining a Path along a Surface Key-Frame Systems Animation Languages Artist-Oriented Animation Languages Articulation Variables Graphical Languages Actor-Based Animation Languages Deforming Objects Warping an Object Coordinate Grid Deformation Morphing(2D) Coordinate Grid Approach Feature-Based Morphing D Shape Interpolation Matching Topology Star-Shaped Polyhedra Axial Slices Map to Sphere Recursive Subdivision Summary Chapter Summary 169 References 170

5 Chapter4 Advanced Algorithms Automatic Camera Control Hierarchical Kinematic Modeling Representing Hierarchical Models Forward Kinematics 184 v Local Coordinate Frames 185 > Inverse Kinematics Summary Rigid Body Simulation Bodies in Free Fall Bodies in Contact Enforcing Soft and Hard Constraints Flexible Objects Virtual Springs Energy Minimization Space-Time Constraints Controlling Groups of Objects Particle Systems Flocking Behavior Autonomous Behavior Implicit Surfaces Basic Implicit Surface Formulation Animation Using Implicitly Defined Objects Collision Detection Deforming the Implicit Surface as a Result of Collision Summary Chapter Summary 267 References 268 Chapter 5 Natural Phenomena 271 with contributions by David S. Ebert 5.1 Plants A Little Bit of Botany L-Systems 275

6 XI Animating Plant Growth Summary Water Still Waters and Small-Amplitude Waves The Anatomy of Waves Modeling Ocean Waves Finding Its Way Downhill Summary Gaseous Phenomena General Approaches to Modeling Gas Computational Fluid Dynamics Clouds Fire Summary Chapter Summary 312 References 312 Chapter 6 Modeling and Animating Articulated Figures 317 with contributions by Meg Geroch, Scott King, Matt Lewis, and Doug Roble 6.1 Reaching and Grasping Modeling the Arm The Shoulder Joint The Hand Coordinated Movement Reaching Around Obstacles Strength Walking The Mechanics of Locomotion The Kinematics of the Walk Using Dynamics to Help Produce Realistic Motion Forward Dynamic Control Summary Facial Animation Types of Facial Models Creating the Model 342

7 xii Textures Approaches to Animating the Face Overview of Virtual Human Representation Representing Body Geometry Geometry Data Acquisition Geometry Deformation Clothing Hair Surface Detail Layered Approach to Human Figure Modeling Cloth and Clothing Simple Draping Getting into Clothes Motion Capture Processing the Images Camera Calibration D Position Reconstruction Fitting to the Skeleton Modifying Motion Capture Summary Chapter Summary 379 References 379 Appendix A Rendering issues 385 A.1 Double Buffering 385 A.2 Compositing 386 A.2.1 Compositing without Pixel Depth Information 389 A.2.2 Compositing with Pixel Depth Information 395 A.3 Dispiaying Moving Objects: Motion Blur 398 A.4 Drop Shadows 401 A.5 Summary 406 References 406

8 xm Appendix B Background Information and Techniques 409 B.l Vectors and Matrices 409 B. 1.1 Inverse Matrix and Solving Linear Systems 411 B.1.2 Singular Value Decomposition 419 B.2 Geometrie Computations 420 B.2.1 Components of a Vector 420 B.2.2 Lengthofa Vector 420 B.2.3 Dot Product of Two Vectors 420 B.2.4 Cross Product of Two Vectors 422 B.2.5 Vector and Matrix Routines 424 B.2.6 Closest Point between Two Lines in Three-Space 427 B.2.7 Area Calculations 428 B.2.8 The Cosine Rule 429 B.2.9 Barycentric Coordinates 430 B.2.10 Computing Bounding Shapes 431 B.3 Transformations 447 B.3.1 Transforming a Point Using Vector-Matrix Multiplication 447 B.3.2 Transforming a Vector Using Vector-Matrix Multiplication 448 B.3.3 Axis-Angle Rotations 449 B.3.4 Quaternions 450 B.4 Interpolating and Approximating Curves 453 B.4.1 Equations: Some Basic Terms 453 B.4.2 Simple Linear Interpolation: Geometrie and Algebraic Forms 454 B.4.3 Parameterization by Are Length 456 B.4.4 Computing Derivatives 456 B.4.5 Hermite Interpolation 457 B.4.6 Catmull-Rom Spline 458 B.4.7 Four-Point Form 461 B.4.8 Blended Parabolas 462 B.4.9 Bezier Interpolation/Approximation 463 B.4.10 De Casteljau Construction of Bezier Curves 464 B.4.11 Tension, Continuity, and Bias Control 465 B.4.12 B-Splines 467

9 B.4.13 Fitting Curves to a Given Set of Points Randomness 470 B.5.1 Noise 471 B.5.2 Turbulence 474 B.5.3 Random Number Generator Physics Primer 476 B.6.1 Position, Velocity, and Acceleration 476 B.6.2 Circular Motion 477 B.6.3 Newtons Laws of Motion 478 B.6.4 Inertia and Inertial Reference Frames 479 B.6.5 Center of Mass 479 B.6.6 Torque 480 B.6.7 Equilibrium: Balancing Forces 480 B.6.8 Gravity 480 B.6.9 Centripetal force 481 B.6.10 Contact Forces 481 B.6.11 Centrifugal Force 483 B.6.12 Work and Potential Energy 484 B.6.13 Kinetic Energy 484 B.6.14 Conservation of Energy 484 B.6.15 Conservation of Momentum 484 B.6.16 Oscillatory Motion 485 B.6.17 Damping 486 B.6.18 Angular Momentum 486 B.6.19 Inertia Tensors Numerical Integration Techniques 488 B.7.1 Function Integration 488 B.7.2 Integrating Ordinary Differential Equations Standards for Moving Pictures 493 B.8.1 In the Beginning, There Was Analog 493 B.8.2 In the Digital World Camera Calibration 502 References 507 Index 509 About the Author 529