Mathematical Basics of Motion and Deformation in Computer Graphics. Ken Anjyo
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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
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
3.1 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
6.2 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
A.1 Several Versions of Rodrigues Formula
A.2 Rodrigues Type Formula for Motion Group
A.3 Proof of the Energy Formula
Preface
In the computer graphics community, many technical terms, such as Euler angle, quaternion, and affine transformation, are fundamental and quite familiar words, and have a pure mathematical background. While we usually do not have to care about the deep mathematics, the graphical meaning of such basic terminology is sometimes slightly different from the original mathematical entities. This might cause misunderstanding or misuse of the mathematical techniques. Or, if we have just a bit more curiosity about pure mathematics relevant to computer graphics, it should be easier for us to explore a new possibility of mathematics in developing a new graphics technique or tool.
This volume thus presents an intuitive introduction to several mathematical basics that are quite useful for various aspects of computer graphics, focusing on the fundamental procedures for deformation and animation of geometric objects, and curve/surface editing. The objective of this book, then, is to fill the gap between the original mathematical concepts and the practical meanings in computer graphics without assuming any prior knowledge of pure mathematics. We then restrict ourselves to the mathematics for matrices, while we know there are so many other mathematical approaches far beyond matrices in our graphics community. Though this book limits the topics to matrices, we hope you can easily understand and realize the power of mathematical approaches. In addition, this book demonstrates our ongoing work, which benefits from the mathematical formulation we develop in this book.
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