LITMIR.BIZ

       ABSTRACT

      This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation.

      This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.

       KEYWORDS

      motion, deformation, quaternion, Lie group, Lie algebra

       Contents

       Preface

       Preface to the Second Edition

       Symbols and Notations

       1 Introduction

       2 Rigid Transformation

       2.1 2D Translation

       2.2 2D Rotation

       2.3 2D Rigid Transformation

       2.4 2D Reflection

       2.5 3D Rotation: Axis-angle

       2.6 3D Rotation: Euler Angle

       2.7 3D Rotation: Quaternion

       2.8 Dual Quaternion

       2.9 Using Complex Numbers

       2.10 Dual Complex Numbers

       2.11 Homogeneous Expression of Rigid Transformations

       3 Affine Transformation

       3.1 Several Classes of Transformations

       3.2 Semidirect Product

       3.3 Decomposition of the Set of Matrices

       3.3.1 Polar Decomposition

       3.3.2 Diagonalization of Positive Definite Symmetric Matrix

       3.3.3 Singular Value Decomposition (SVD)

       4 Exponential and Logarithm of Matrices

       4.1 Definitions and Basic Properties

       4.2 Lie Algebra

       4.3 Exponential Map from Lie Algebra

       4.4 Another Definition of Lie Algebra

       4.5 Lie Algebra and Decomposition

       4.6 Loss of Continuity: Singularities of the Exponential Map

       4.7 The Field of Blending

       5 2D Affine Transformation between Two Triangles

       5.1 Triangles and an Affine Transformation

       5.2 Comparison of Three Interpolation Methods

       6 Global 2D Shape Interpolation

       6.1 Local to Global

       6.2 Formulation

       6.3 Error Function for Global Interpolation

       6.4 Examples of Local Error Functions

       6.5 Examples of Constraint Functions

       7 Parametrizing 3D Positive Affine Transformations

       7.1 The Parametrization Map and its Inverse

       7.2 Deformer Applications

       7.3 Integrating with Poisson Mesh Editing

       7.3.1 The Poisson Edits

       7.3.2 Harmonic Guidance

       7.3.3 The Parametrization Map for Poisson Mesh Editing

       8 Further Readings

       A Formula Derivation

       A.1 Several Versions of Rodrigues Formula

       A.2 Rodrigues Type Formula for Motion Group

       A.3 Proof of the Energy Formula

       Bibliography

       Authors’ Biographies