Mathematical Basics of Motion and Deformation in Computer Graphics. Ken Anjyo

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the set of imaginary quaternions, 14 |q| the absolute value of a quaternion, 14 S³ the set of unit quaternions, 15 exp exponential map, 16, 29 slerp(q₀, q₁, t) spherical linear interpolation, 16 ε dual number, 16 M(2, H) the set of square matrices of size 2 with entries in H, 16 the set of anti-commutative dual complex numbers (DCN), 18 E(n) rigid transformation group, 19 SE(n) n-dimensional motion group, 20 GL(n) general linear group, 23 Aff(n) affine transformations group, 23 GL⁺(n) general linear group with positive determinants, 23 Aff⁺(n) the set of orientation-preserving affine transformations, 23 ⋉ semi-direct product, 25 Sym⁺(n) the set of positive definite symmetric matrices, 26 Diag⁺(n) the set of diagonal matrices with positive diagonal entries, 27 SVD Singular Value Decomposition, 28 exp(A) exponential of a square matrix, 29 C^× the set of non-zero complex numbers, 31 gl(n) Lie algebra of GL(n), 33 so(n) Lie algebra of SO(n), 33 sl(n) Lie algebra of SL(n), 33 aff(n) Lie algebra of Aff(n), 34 se(n) Lie algebra of SE(n), 34 [A, B] Lie bracket, 34 Jx, Jy, Jz basis of so(3), 37 log logarithmic map (logarithm), 31, 42 AL, AP, AE interpolant, 42 EP, EF, ES, ER error functions, 49 ||·||F Frobenius norm of a matrix, 49 se(3) Lie algebra of SE(3), 53 sym(3) the set of symmetric matrices of size three, 53 ϕ, ψ map between Aff⁺(3) and a vector space, 53 ι embedding M(3, R) → M(4, R), 54 R̂, X̂ element of SE(3) and se(3), 54 ∇ gradient, 58 ∆ Laplacian, 58 div divergence, 58 ∂Ω boundary, 58

      CHAPTER 1

       Introduction

       ORGANIZATION

      In the latter half of this chapter we give a very rough sketch of several mathematical concepts that will reappear throughout this book.

      In Chapters 2 and 3, we describe rigid and non-rigid transformations, while explaining the basic definitions regarding the matrix group. We thereafter show that Lie theoretic framework gives us comprehensive understanding of affine transformations, quaternions, and dual quaternions in Chapters 4 and 5. The Lie theoretic approach is also successfully applied to parametrization issues in Chapters 6 and 7, where we provide several useful recipes for rigid motion description and global deformation, along with our recent work. Finally in Chapter 8, we show a list of further readings, suggesting the power of mathematical approaches in graphics far beyond the present volume.

      Here are a few additional notes that make this book easy to read and more enjoyable. First there are several colored columns in this book, which give brief, interesting stories of mathematicians or deeper explanations of the mathematical concepts in the body text. You may skip them at the first reading, but they will give you good guidance for your further study. Second, in this book, a point in Euclidean space is given as a row vector, whereas many geometric transformations are described with matrices. The action of a matrix to a vector then means multiplication from the left. As you may know, OpenGL takes the same manner of matrix multiplication, whereas DirectX does not.

       A FEW MATHEMATICAL CONCEPTS

      In this section, we therefore take a brief look at the original mathematical concepts related with matrices. These will be useful when we reuse or extend the basic ideas behind those concepts that are usually not well described in the computer graphics literature.

      However, except the concept of group, we won’t mention their rigorous definitions in mathematics. Rather we would like to describe the crude introduction of the mathematical concepts that are important even in computer graphics. A bit more precise definitions of them may also be given in later chapters. It would, however, be more important to think of why those mathematical concepts are useful in our graphics context, rather than learning deeply their rigorous mathematical entities.

       GROUP

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