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Let G be a set associated with an operation “·”. If the pair (G, ·) satisfies the following properties, then it is called a group. Or we would call G itself a group:
1. For any a, b ∊ G, the result of the operation, denoted by a · b, also belongs to G.
2. For any a, b and c ∊ G, we have a · (b · c) = (a · b) · c.
3. There exists an element e ∊ G, such that e · a = a · e = a, for any element a ∊ G. (The element is then called the identity of G).
4. For each a ∊ G, there exists an element b ∊ G such that a · b = b · a = e, where e is the identity. (The element b is then called the inverse of a.)
As usual, R and C denote the set of all real numbers and the set of all complex numbers, respectively. R or C is then a group with addition (i.e., the operation “·” simply means +), and called commutative, since a + b = b + a holds for any element a, b of R or C. In the following sections, we’ll see many groups of matrices. For example, the set of all invertible square matrices constitutes a group with composition as its group operation. The group consisting of the invertible matrices with size n is called the general linear group of order n, and will be denoted by GL(n, R) or GL(n, C).
LIE GROUP AND LIE ALGEBRA
A Lie group is defined to be a smooth manifold with a group structure. But we never mind what is a manifold (i.e., locally it is diffeomorphic to n-dimensional open disk). In applications, a matrix group, that is, a group consisting of matrices, like GL(n, R) for instance, are enough to be considered as a Lie group. The totality of quaternions of unit length constitutes another Lie group. Although there is a general definition of Lie algebra, in this book we restrict ourselves to consider the Lie algebra associated with a Lie group. We then define the Lie algebra as a tangent space at the identity of the Lie group. In this sense, the Lie algebra can be considered as a linear approximation of the Lie group, which will be more explicitly described for the matrix groups in the following chapters.
QUATERNION
The original definition of quaternion by William Hamilton seems a bit different from the one we use in graphics. In 1835 he justified calculation for complex numbers x + iy as those for ordered pairs of two real numbers (x, y). As is well known, complex numbers can express 2D rotations. This motivates many mathematicians to find a generalization of numbers which can describe 3D rotations. In 1843 he finally discovered it, referring to the totality of those numbers as quaternions. In this book, the set of quaternions is denoted by H, and expressed as H = R + Ri + Rj + Rk, where we introduce the three numbers i, j and k satisfying the following rules:
H is then called an algebra or field (see [Ebbinghaus1991] for more details). We also note that, as shown in the above rules, it is not commutative. A few more alternative definitions of quaternions will also be given later for our graphics applications. In particular we’ll see how 3D rotations can be represented with quaternions of unit length.
DUAL QUATERNION
In 1873, as a further generalization of quaternions, William K. Clifford obtained the concept called biquaternions, which is now known as a Clifford algebra. The concept of dual quaternions, which is another Clifford algebra, was also introduced in the late 19th century. A dual quaternion can be represented with q = q0 + qεε, where q0, qε ∊ H and ε is the dual unit (i.e., ε commutes with every element of the algebra, while satisfying ε2 = 0). We’ll see later how rigid transformations in 3D space can be represented with dual quaternions of unit length.
CHAPTER 2
Rigid Transformation
In physics, a rigid body means as an object which preserves the distances between any two points of it with or without external forces over time. So describing rigid transformation (or rigid motion) means finding the non-flip congruence transformations parametrized over time. For a rigid body X, an animation (or a motion) X(t) indexed by a time parameter t can be described by a series of rigid transformations S(t) with X(t) = S(t)X(0), instead of dealing with the positions of all the particles consisting of X. In the following sections, a non-flip congruence transformation may also be called a rigid transformation. The totality of the non-flip rigid transformations constitutes a group, which will be denoted by SE(n), where n is the dimension of the world where rigid bodies live (n = 2 or 3). So let’s start with 2D translation, a typical rigid transformation in R2.
2.1 2D TRANSLATION
A translation Tb by a vector b ∊ R2 gives a rigid transformation in 2D. The composition of two translations and the inverse of a translation, which is denoted by
This can be rephrased as the totality of translations forms a group (recall Chapter 1). Moreover, they satisfy also
This means that the totality of translations forms a commutative group. This property is illustrated in Figure 2.1. A commutative group is also called abelian group named after Niels Abel. The totality of 2D translations are denoted by R2, as is the two-dimensional vector space.
Figure 2.1: Example of groups—commutative group.
Niels Henrik Abel (1802–1829)
Norway mathematician. In his 28-year life, he gave a lot of important insight, which now has become a mathematical notion, such as Abelian groups, Abelian integral, Abelian functions, named after him. Also, the Abel prize was founded in 2002, the two-hundredth anniversary of Abel’s birth. This prize is awarded to one or few outstanding mathematicians each year with six million kroner (approx. one million dollars).
2.2 2D ROTATION
A
