Dual Quaternion
The dual of a set R of numbers is defined as RCR" with the rules "2 D0 and "aDa" for a2R This concept effectively describes 3D rigid transformations, including rotations and translations Additionally, various equivalent representations of dual quaternions prove to be beneficial in this context.
(ii) A subalgebra ofM.2;H/, consisting of matrices of the form z w
Using Complex Numbers
A dual quaternion \( zCw \) is classified as a unit dual quaternion if the norm \( |z| = 1 \) and the inner product \( \langle z, w \rangle = 0 \) in \( \mathbb{R}^4 \) Notably, all unit quaternions qualify as unit dual quaternions, and any element represented as \( 1 + w \) with \( w \in \text{Im} \, \mathbb{H} \) is also a unit dual quaternion Additionally, each unit dual quaternion can be uniquely expressed as the product of a unit quaternion \( z \) and a unit dual quaternion of the form \( 1 + w \) where \( w \in \text{Im} \, \mathbb{H} \).
In 3D space, the action of a unit quaternion \( q = z + w \) on a point \( p = 1 + (\xi + \mathbf{j}y + \mathbf{k}z) \) facilitates rotation, while the dual quaternion \( q = 1 + w \) with \( w \in \text{Im} \mathbb{H} \) enables translation The semi-direct structure and the exponential map behavior on dual quaternions can be comprehensively understood through both the dual quaternion framework and its matrix representation, similar to the understanding of quaternions For further details, refer to Kavan (2008).
Now we give a brief comment on complex numbers Using the identification ofC withR 2 by zDxCyi$.x; y/ T , a rigid transformation in 2D can be expressed as ˛ ˇ
; (2.35) where˛De i DcosCisin, andˇ Db1Cb2i2C In other words, we have a realization of
SE.2/by a matrix with entries in complex numbers
e reflection with respect to a lineyD.tan /xis represented in complex variables by z7!e i e i zDe 2i z Dze 2i :
Note that the reflection line is also expressed asRe i
For quaternions, as well as the definitions (i)–(iii) listed in Section2.7, we have other equiv- alent definitions using complex numbers.
(iv) HDCCCj, the complex two-dimensional vector space with a multiplication rule e expression (iv) is shorter than that of (i) in Section2.7, while we note the fancy relation wjDj wforwDCDRCRi.
(v) e set of complex matrices of the form z w w z
e relation between (i)–(iii) in Section2.7and (iv), (v) is given by aCbiCcjCdkD.aCbi/C.cCdi/j DzCwj:
e multiplicativityj qq 0 j D jqj jq 0 jof norms of quaternions follows from the property of deter- minant det.AB/Ddet.A/det.B/for the corresponding matrices A and B of the form (2.37).
Dual Complex Numbers
Unit dual quaternions can represent any 3D rigid transformation, as discussed in Section 2.8, and can also be applied to 2D transformations by considering the plane within R3 To enhance the efficiency and clarity of 2D rigid transformations, we introduce anti-commutative dual complex numbers, which offer a more concise representation and accelerate computation.
For two complex numbers p0; p1 2C, the combination p0Cp1" is called an anti- commutative dual complex number (DCN, for short) and denoted byCL e multiplication of DCN is defined by
The definitions indicate that the expressions "2 D0, and p0Cp1"/.q0Cq1"/" may not equal ".q0C q1"/.p0Cp1"/ Additionally, we define the conjugate and absolute values of DCN as follows: p0Cp1" equals D Np0Cp1", and the absolute value jp0Cp1"j equals jp0j.
en DCN satisfies usual associative and distributive laws,
The computation of DCN can be performed using standard numerical methods Like unit dual quaternion numbers, unit anti-commutative complex numbers hold significant importance in mathematical applications.
C1D f Op2 LCj j Opj D1g D fe i Cp1"j2R; p1 2Cg:
is is a group with the inverse
A unit DCNpO acts onR 2 by identifyingv2CDR 2 with1Cv"2 LC:
O p.1Cv"/pOD1C.p 0 2 vC2p0p1/"; that is,vis mapped top 2 0 vC2p0p1 For example, whenp1 D0, thenpO Dp0De i mapsv2
C to p 0 2 vDe 2i v, which is the rotation around the origin of degree 2 On the other hand,
Homogeneous Expression of Rigid Transformations
The action of whenp0D1 corresponds to the translation by2p1 2CDR 2, resulting in a surjective group homomorphism from 'W LC1 to SE.2/, with a kernel of f˙1g This means that each 2D rigid transformation is associated with two unit DCN’s that have opposite signs.
The expression C3p0Cp1"7!p0Cp1j"2HCH" demonstrates compatibility with involution and conjugation while preserving the norm Additionally, when we identify vDxCiy2C with 1C.xjCyk/"D1Cvj", the resulting mapping is commutative with the action Consequently, this embedding allows DCN to be represented as a sub-ring of dual quaternion numbers.
e relations among DQN, DCN and related groups are summarized in Figure2.9 Here we add
Figure 2.9 illustrates the relationship between dual quaternions and related groups, highlighting the hierarchical structure of 2D and 3D objects The 2D object on the left is inherently contained within the 3D object on the right Vertically, the top object encompasses the bottom object, with the former derived from the latter through translation Additionally, the front-behind relationship demonstrates a two-to-one surjective mapping, where each element in the back object corresponds to two elements in the front object.
e properties of DCN and its applications are given in [Matsuda2004].
e homogeneous expression ofnD rigid transformation group is
; (2.42) and its subgroup consisting of rigid motions (i.e., non-flipnD rigid transformations) is
As for a 2D case, we have it with (2.6).
The term "homogeneous" originates from projective geometry, where in a real vector space \( R^n \), two distinct lines intersect at a single point unless they are parallel In the case of parallel lines, they are regarded as meeting at a point at infinity By incorporating these points at infinity, the concept of homogeneity is enhanced.
R n , we obtain a projective space, denoted byP n R/ is space is explicitly realized as follows: we consider a non-zero vector with.nC1/-components Œz1Wz2 W Wzn C 1:
In vector mathematics, two parallel vectors are considered identical Specifically, if there exists a non-zero scalar such that \( z_1 = k \cdot z_2 \) and so forth, we denote the vector \( [z_1, z_2, \ldots, z_{n+1}] \) as equivalent to \( [z_1', z_2', \ldots, z_n'] \) When \( z_{n+1} \neq 0 \), we can normalize the vector by taking \( D = \frac{1}{z_{n+1}} \), allowing us to treat it as a standard \( n \)-vector in \( \mathbb{R}^n \) The coordinates \( [z_1, z_2, \ldots, z_{n+1}] \) are referred to as homogeneous coordinates due to their uniformity, while classical coordinates \( (x_1, \ldots, x_n) = [x_1, \ldots, x_n, 1] \) are known as inhomogeneous coordinates Conversely, if \( z_{n+1} = 0 \), the \( (n+1) \)-vector \( [z_1, z_2, \ldots, z_{n+1}] \) represents an additional element beyond \( \mathbb{R}^n \), which is identified as a point at infinity.
The symmetry group of projective spaces is defined by projective linear transformations, which can be represented through the multiplication of a regular matrix of size (n + 1).
::: ::: ::: an1 a nn a nn C 1 an C 11 an C 1n an C 1n C 1
A projective transformation preservesR n if and only ifan C 11D Dan C 1nD0.
In such a case, sincean C 1n C 1¤0and a scalar multiple gives the same transformation, we may assume thatan C 1n C 1D1 A matrix of this form gives a homogeneous expres-
2.11 HOMOGENEOUS EXPRESSION OF RIGID TRANSFORMATIONS 21
sion of an affine transformation ofR n
In this chapter, we discuss affine transformations, i.e., matrices for deformations.
Several Classes of Transformations
The general linear group, denoted as GL, consists of invertible linear transformations on R^n, typically represented by square matrices that have non-zero determinants.
An affine transformation is a mapping in \( \mathbb{R}^n \) that transforms every line into another line, typically represented by an invertible square matrix and a vector in \( \mathbb{R}^n \) The matrix denotes the linear transformation, while the vector indicates the translation The collection of all affine transformations is denoted as Aff(n), which can also be represented within the set of invertible square matrices of size \( (n+1) \), expressed as Aff(n) GL(n+1).
is realization is called ahomogeneousexpression e composition of homogeneous ex- pressions is nothing but a multiplication of two matrices.
A negative determinant of a matrix in GL(n) or Aff(n) indicates that the associated transformation alters the orientation of objects The set of transformations that preserve orientation is denoted accordingly.
GL C n/D fA2GL.n/jdet.A/ > 0g; (3.3) Aff C n/D
We here summarize the inclusion relations of these sets of transformations in Figure3.1.
In this article, we clarify the relationships between various mathematical objects Each left-right edge indicates that the object on the left serves as an index for a subgroup of the object on the right, where the left object is connected and the right object is disconnected The vertical edges illustrate that the top object represents the semi-direct product of the bottom object with the translation group R^n Additionally, the front-behind edges denote that the object at the front is a subset of rigid transformations pertaining to the object at the back.
e motion group (or Euclidean motion group), which is denoted bySE.n/in (2.43), is the set of transformations onR n which preserve the length, angle, and the orientation Each element
Inclusions of Lie groups illustrate that the motion group can be represented as a combination of rotation and translation The congruence group, denoted as E(n), consists of transformations that maintain shape while potentially altering orientation, with reflection serving as a prime example of such transformations These transformation classes are categorized as groups, meaning that the successive composition of transformations remains within the same class, and the inverse transformation also adheres to this classification (refer to Chapter 1).
In topology, a subset of a vector space is termed compact if it is both bounded and closed For instance, the set of all rotations is considered bounded, whereas the set of all translations is unbounded Among the eight classes of groups, the special orthogonal group SO(n) and the orthogonal group O(n) are compact, while the remaining six classes do not exhibit compactness In this context, the concepts of connected and arcwise connected are equivalent, with a maximal connected subset referred to as a connected component It is important to note that continuous interpolation between two elements in different connected components is not possible; for example, one cannot interpolate between a flip and the identity.
GL(n) consists of two connected components, encompassing both the flip and the identity In contrast, SO(n) is a connected group, meaning it has a single connected component that is itself.
Note that all these eight types of groups in Figure3.1are non-commutative fornD2; 3 except forSO.2/.
The extensive array of Lie groups can be likened to the diverse software options available in computer graphics, suggesting that mathematicians appreciate complexity for its utility Just as different programming languages and platforms like C++, Maya, Python, and MATLAB serve various purposes, the variety of Lie groups is essential for addressing specific mathematical challenges This concept is rooted in Klein’s Erlangen program, which emphasizes the importance of understanding the underlying structures in mathematics.
Semidirect Product
German mathematician His research project was published in
1872 at Erlangen, and is called the Erlangen program e slogan of the Erlangen program is that “symmetry classifies geometry.”
Each geometry class is associated with specific symmetry groups, and a key objective of geometry is to describe its invariants This concept is essential for understanding Euclidean, affine, and projective geometries and their interrelations.
We see that the composition of a rotation, a translation, and the inverse rotation is another trans- lation (2.8):
In general, this fact is related to the notion ofnormal subgroupandsemi-directproducts of sub- groups.
LetGbe a group e following concepts forGare used to describe the relations among the matrix group appeared in this book:
(i) A subsetH ofGis called asubgroupifH is closed under the composition and the inversion,
(ii) A subgroupH ofGis called anormalsubgroup if the compositionghg 1 of anyh2H andg2Gbelongs toH.
For example, the setR 3 of translations is a normal subgroup ofSE.3/, while the setSO.3/of rotations centered at the identity is a subgroup ofSE.3/but not normal.
Let G be a group, H a subgroup ofG, andK a normal subgroup ofG (For example,
GDSE.3/; H DSO.3/; K DR 3 ) If the map
H K3.h; k/7!hk2G (3.6) is a bijective, thenGis thesemi-direct productofH andK, denoted byH ËK Note thathkh 1 2
In group theory, if H and K are normal subgroups of G, and the condition hk = kh holds for all h in H and k in K, then G is defined as a direct product group of H and K This relationship implies that the mapping is bijective, and it is important to note that the equality hk = kh is equivalent to hkh⁻¹ = k.
Figure 3.2: Translations are normal. and affine transformation groups are typical examples of semi-direct product groups:
Decompositions in motion can be understood as having distinct meanings; specifically, the translation component is invariant to coordinate choices and scaling, while the rotation component carries ambiguity that is influenced by the selection of the origin and the coordinate system used.
Decomposition of the Set of Matrices
Polar Decomposition
Given a matrixA2GL C n/, we haveADRS, whereRis a rotation matrix andS is a positive definite symmetric matrix e product map
SO.n/Sym C n/3.R; S /7!RS2GL C n/ (3.11) is bijective.
Note that if we reverse the order
3.3 DECOMPOSITION OF THE SET OF MATRICES 27 then it is still bijective, however it gives the different map We also note that the set Sym C n/is not a group; actually, the product of two elements in Sym C n/is not necessarily symmetric.
3.3.2 DIAGONALIZATION OF POSITIVE DEFINITE SYMMETRIC
Every positive definite symmetric matrix \( X \) can be expressed as \( X = DRDR^T \), where \( R \) is a member of the special orthogonal group \( SO(n) \) and \( D \) is a diagonal matrix with all positive diagonal entries These diagonal entries correspond to the eigenvalues of the matrix \( X \).
The mapping SO.n/Diag C.n/3.R; D/7!RDR T 2Sym C.n/(3.13) is surjective, indicating that it covers the entire target space However, it is important to note that this map is not injective If two pairs, (R; D) and (R0; D0), represent the same element X, then there exists a permutation matrix P such that D0 = D P D P T A permutation matrix is defined as a matrix with exactly one non-zero entry in each row and column, and both the inverse and the product of permutation matrices also yield permutation matrices.
If D has distinct diagonal entries, then the matrix P is unique This indicates that the expression X DRDR T is not unique; however, there is flexibility in the arrangement of the eigenvalues of X within the diagonal entries of D.
Every matrixA2GL C 2/ can be written as ADR˛ DR ˇ, whereR˛; Rˇ 2SO.2/and D is a diagonal matrix with positive diagonal entries (see Figure3.3) In general, the product map
The Singular Value Decomposition (SVD) is illustrated in Figure 3.3, where curved arrows indicate rotations and a dotted line represents directional dilation This SVD can be derived from the polar decomposition combined with the diagonalization of a positive symmetric matrix Specifically, if we denote the polar decomposition of matrix A as ADR₀ and the diagonal matrix as S = DRDRᵀ, the expression A D(R₀R) = D results in the SVD It is important to note that R₀R belongs to the special orthogonal group SO(n), allowing us to compute the SVD through polar decomposition and diagonalization Conversely, starting from the SVD representation A = ADR₀DR, the expression A D(R₀R)(RᵀDR) yields the polar decomposition of A.
The mapping O(n)/Diag C(n) is surjective, and the Singular Value Decomposition (SVD) is referred to as Cartan decomposition in mathematical literature, as noted by Helgason (1978) The compactness of SO(n) and the commutativity of Diag C(n) are important aspects in this context.
Geometric transformations serve as a fundamental mathematical framework for essential operations in computer graphics, including rotation, shear, and translation, all represented by a 4x4 homogeneous matrix While the composition of transformations is achieved through matrix multiplication, deriving a geometrically meaningful weighted sum or linear combination of transformations presents challenges This complexity has motivated graphics researchers to investigate new mathematical concepts and tools, leading to advancements in areas such as skinning, cage-based deformation, and motion analysis and compression.
In earlier sections, we explored geometric transformations using mathematical concepts linked to groups, particularly Lie groups This mathematical perspective offers a wider and more in-depth understanding of the diverse types of geometric transformations.
This chapter delves into the Lie theoretic perspective, introducing Lie algebra in relation to the Lie group of matrices as a motion group It highlights how the Lie algebra provides a linear approximation of the Lie group, enabling the application of a robust linear interpolation method for dynamic motion and deformation.
We first consider a square matrixA, which is implicitly considered an element of a Lie group of matrices e exponential ofAis then defined as exp.A/D
6A 3 C ; (4.1) whereA 0 DI is the identity matrix We’ll refer to (4.1) as the matrix exponential, for short is is motivated by Taylor expansion of the usual exponential function e x D
The series exp(A) converges rapidly for any arbitrary matrix A, similar to the conventional exponential function However, this infinite series representation is not the most efficient method for numerical computations Instead, for practical calculations involving diagonal matrices, we can utilize several beneficial properties of the exponential function.
We also see that the exponential image of a strictly upper-triangular matrix terminates into a finite sum For example, forl2R n , we have exp
Slightly more generally, for a strictly upper triangular matrixAof sizen, we haveA n DO and therefore the infinite series (4.1) terminates to a finite sum expression exp.A/DI CAC1
Significantly, we can understand Rodrigues’s rotation formula (2.15) by using the matrix exponential Every 3D rotation is expressed by
The equation ADI3Csin(juj) juj AC 1 cos(juj) juj 2 A 2 describes a transformation involving the norm of a vector u = (u1, u2, u3) in R^3, where juj is defined as the square root of the sum of the squares of its components Additionally, it is noted that juj^2 equals 1/2 times the trace of the product of matrix A and its transpose, which simplifies to 1/2 times the trace of A squared The matrix R represents a rotation around the axis defined by vector u, with an angle of juj Notably, if juj^2 is an integer, then R simplifies to the identity matrix I3.
Coming back to the general situation, we always have the exponential law exp sCt /A/Dexp.sA/exp.tA/ for all s; t 2R; A2M.n;R/: (4.8)
The generalization of the exponential law to matrices, expressed as exp(A + B) ≤ exp(A)exp(B), does not universally apply For instance, the matrices exp(A + B), exp(A)exp(B), and exp(B)exp(A) can yield distinct results, highlighting the limitations of this principle in matrix operations.
If we as- sumeAandBare commutative, i.e.,ABDBA, then we have an expected formula exp.ACB/Dexp.A/exp.B/ for A; B 2M.n;R/ with ABDBA: (4.10)
In general, the matrix exponential has the conjugate invariance property exp.P 1 AP/DP 1 exp.A/P: (4.11)
The matrix exponential is defined as an infinite sum of matrices; however, by utilizing property (4.11), we can simplify the computation of the exponential map to specific cases (4.3), (4.4), and (4.6) This approach allows us to bypass the need for calculating the infinite series when determining the exponential exp(A).
The logarithmic map, or logarithm, is defined as the inverse of the exponential map, particularly for the real scalar-valued function exp: \( \mathbb{R} \to \mathbb{R}^+ \) However, the logarithm of the complex exponential function is multi-valued, reflecting the complexity of complex numbers Euler’s formula, which connects exponential functions with trigonometric functions, illustrates that the exponential expression of a rotation is not unique, as adding \( 2\pi n \) (where \( n \in \mathbb{Z} \)) yields the same rotation This multi-valued nature complicates the logarithm of a matrix, making it similarly multi-valued Like the scalar-valued logarithm, the logarithm can be expressed through a series expansion.
If the absolute values of all eigenvalues of the matrix \( X \) are less than 1, then the function satisfies \( \exp(\log(X)) = X \) This logarithm exhibits conjugate invariance, meaning \( \log(P^{-1}XP) = P^{-1}\log(X)P \) Consequently, the computation of \( \log(X) \) can be simplified to scalar-valued functions and triangular matrices, where infinite series converge.
A; and if.A I / m DO for somem, then logAD m 1X k D 1
e setso.3/of skew-symmetric, that is, the transpose is its minus,33matrices is regarded as
Lie algebraofSO.3/(see Chapter1as well).
Figure 4.1: Lie algebra as a tangent space.
Singular Value Decomposition (SVD)
Every matrixA2GL C 2/ can be written as ADR˛ DR ˇ, whereR˛; Rˇ 2SO.2/and D is a diagonal matrix with positive diagonal entries (see Figure3.3) In general, the product map
The Singular Value Decomposition (SVD) can be understood as a combination of polar decomposition and the diagonalization of a positive symmetric matrix Specifically, if we denote the polar decomposition of matrix A as ADR₀ and express S as S = DRDRᵀ, the relationship AD(R₀R) = D results in the SVD It is important to note that R₀R belongs to the special orthogonal group SO(n) This approach allows for the computation of SVD through polar decomposition and the diagonalization of a positive symmetric matrix Conversely, when SVD is expressed as ADR₀DR, the equation AD(R₀R)(RᵀDR) leads to the polar decomposition of A.
The function O(n)/Diag C(n)/O(n)!GL(n) is surjective, and in mathematical literature, the Singular Value Decomposition (SVD) is referred to as the Cartan decomposition This perspective highlights the importance of the compactness of SO(n) and the commutativity of Diag C(n).
Geometric transformations provide a foundational mathematical framework for essential operations in computer graphics, including rotation, shear, and translation, represented through a 4x4 homogeneous matrix While the composition of these transformations is achieved through matrix multiplication, the geometric interpretations of addition and scalar products are complex To create a geometrically meaningful weighted sum of transformations, which poses significant challenges, graphics researchers have been motivated to investigate new mathematical concepts and tools This exploration has led to advancements in various areas, such as skinning, cage-based deformation, and motion analysis and compression.
In earlier chapters, we explored geometric transformations using mathematical concepts related to groups, particularly Lie groups This mathematical perspective offers a more extensive and insightful understanding of the different types of geometric transformations.
This chapter delves into the Lie theoretic perspective, introducing the concept of Lie algebra linked to the matrix Lie group as a motion group The Lie algebra provides a linear approximation of the Lie group, enabling the application of an effective linear interpolation method for dynamic motion and deformation.
We first consider a square matrixA, which is implicitly considered an element of a Lie group of matrices e exponential ofAis then defined as exp.A/D
6A 3 C ; (4.1) whereA 0 DI is the identity matrix We’ll refer to (4.1) as the matrix exponential, for short is is motivated by Taylor expansion of the usual exponential function e x D
The series exp.A converges rapidly, similar to the standard exponential function, but this infinite series is not the most efficient for numerical computations Instead, we can leverage several useful properties, particularly for diagonal matrices, to enhance computational efficiency when calculating the exponential of a matrix.
We also see that the exponential image of a strictly upper-triangular matrix terminates into a finite sum For example, forl2R n , we have exp
Slightly more generally, for a strictly upper triangular matrixAof sizen, we haveA n DO and therefore the infinite series (4.1) terminates to a finite sum expression exp.A/DI CAC1
Significantly, we can understand Rodrigues’s rotation formula (2.15) by using the matrix exponential Every 3D rotation is expressed by
The equation ADI3Csinjuj juj AC 1 cosjuj juj 2 A 2 illustrates the relationship involving the norm of a vector u, defined as juj D q u 2 1 Cu 2 2 Cu 2 3 in R³ Additionally, it is noted that juj 2 equals 1/2 tr(AAT) or 1/2 tr(A²) The matrix R represents the rotation around the axis defined by u, with an angle of juj Notably, if juj² is an integer, then R simplifies to the identity matrix I₃.
Coming back to the general situation, we always have the exponential law exp sCt /A/Dexp.sA/exp.tA/ for all s; t 2R; A2M.n;R/: (4.8)
The generalization of the exponential law to matrices, expressed as exp(A + B) ≤ exp(A) exp(B) for matrices A and B in M(n, R), does not universally apply For instance, the matrices exp(A + B), exp(A) exp(B), and exp(B) exp(A) can yield distinct results when A and B do not commute.
If we as- sumeAandBare commutative, i.e.,ABDBA, then we have an expected formula exp.ACB/Dexp.A/exp.B/ for A; B 2M.n;R/ with ABDBA: (4.10)
In general, the matrix exponential has the conjugate invariance property exp.P 1 AP/DP 1 exp.A/P: (4.11)
The matrix exponential is defined as an infinite sum of matrices; however, by utilizing property (4.11), we can simplify the computation of the exponential map to specific cases (4.3), (4.4), and (4.6), thereby eliminating the need for the infinite series in calculating exp(A).
The logarithmic map, or logarithm, serves as the inverse of the exponential map, particularly in the context of real scalar-valued functions However, when dealing with complex exponential functions, the logarithm becomes multi-valued This complexity is akin to Euler’s formula, which establishes a connection between exponential and trigonometric functions, indicating that the exponential representation of a rotation is not unique Consequently, the logarithm of a matrix also exhibits multi-valued characteristics, paralleling the behavior of the scalar-valued logarithm, which can be expressed through a series expansion.
If the absolute values of all eigenvalues of the matrix \( X \) are less than 1, then the function satisfies the relationship \( \exp(\log(X)) = X \) The logarithm exhibits conjugate invariance, allowing us to express \( \log(P^{-1}XP) = P^{-1}\log(X)P \) Consequently, the computation of \( \log(X) \) simplifies to scalar-valued functions and triangular matrices, where the infinite series converges.
A; and if.A I / m DO for somem, then logAD m 1X k D 1
e setso.3/of skew-symmetric, that is, the transpose is its minus,33matrices is regarded as
Lie algebraofSO.3/(see Chapter1as well).
Figure 4.1: Lie algebra as a tangent space.
In this book, we define a Lie group as a matrix group, where the Lie algebra serves as a linear approximation of the group at its identity Specifically, for a matrix group G in GL(n), the Lie algebra is characterized as the tangent space of G at the identity Alternatively, the Lie algebra comprises elements in M(n, R) that take a specific form.
(4.14) for any curve' WR!Gwith'.0/DI.
Norwegian mathematician He tried to control the continuous symmetry in geometry and differential equations by introducing its linearization is idea is now regarded as a core of Lie eory.
Lie groups and Lie algebras are also named after him Lie was a close friend of F Klein, and this communication led to mutual influence (see [Stubhaug2002]).
For example, let us compute the Lie algebra of the Lie groupSO.n/ For any curve 'W
R!SO.n/, the image should satisfy'.t /'.t / T DIn By differentiating with the Leibnitz rule, we obtain
The Lie algebra of the special orthogonal group SO(n) consists of matrices A that satisfy the condition A^T = -A In mathematical notation, this is often represented as "mathfrak" letters, where the Lie algebra of SO(3) is denoted as so(3) and that of the general linear group GL(n) as gl(n) This article provides multiple examples of various Lie algebras, illustrating their structures and properties.
(i) e Lie algebra ofGL.n/andGL C n/isgl.n/DM.n;R/,
(ii) e Lie algebra ofO.n/andSO.n/isso.n/D fA2M.n;R/jA T D Ag.
(iii) e Lie algebra of the group of positive real numbersR>0isR Note that the group law for
R>0is multiplicative, while that forRis additive.
(iv) e Lie algebra ofC isC,
(v) e Lie algebra ofH D fq2Hjq¤0gisH,
(vi) e Lie algebra of the groupS 3 of unit quaternions is the set ImHof imaginary quaternions.
(vii) e special linear group is defined to be the set of volume-preserving linear maps;
SL.n;R/D fA2GL.n;R/jdetAD1g Its Lie algebra is sl.n;R/D fA2M.n;R/j trAD0g, the set of traceless matrices.
(viii) e Lie algebra of the group of translations in 3D isR 3
The Lie algebra of a subgroup serves as a subspace within the larger Lie algebra framework For the direct product group \( H \times K \), the associated Lie algebra is the direct sum of the Lie algebras \( \mathfrak{h} \) and \( \mathfrak{k} \) of groups \( H \) and \( K \) In contrast, the Lie algebra of the semidirect product group \( H \rtimes K \) is represented as the direct sum of the Lie algebras \( \mathfrak{h} \) and \( \mathfrak{k} \) when viewed as a vector space This article also presents several examples illustrating the Lie algebras of both direct and semidirect product groups.
(ix) e Lie algebra of Diag C n/D fX jdiagonal matrices with positive diagonal entriesg is diag.n/D fAjdiagonal matricesg:
(x) e Lie algebra of the affine transformation groups Aff C n/ and Aff.n/, defined in (3.2 and3.4) is aff.n/D
is fact will be used in Chapter6and7.
(xi) e Lie algebra ofSE.n/, defined in (2.43) and that ofE.n/is se.n/D
is will appear in Figure5.3fornD2, and in Figure7.2fornD3of later sections. (xii) e set of invertible dual quaternion numbers
H C"HD fzCw"jz; w2H; z¤0g (4.15) is a Lie group by the multiplication As a Lie group, it is a semidirect product groupH ËH.
(xiii) e set of unit dual quaternion numbers f.z; w/jz 2S 3 ; w2H;hz; wi D0g (4.16) is a Lie group by the multiplication Its Lie algebra is ImHC"ImH.
For the Lie algebragl.n;R/, we define a binary operation, calledLie bracket, by ŒA; BDAB BA:
4.3 EXPONENTIAL MAP FROM LIE ALGEBRA 35
We can easily check that this bracket operation satisfies the following properties:
(i) e bracket ŒA; B is bilinear, i.e., linear both in A and in B To be explicit, ŒA; BDŒA; BDŒA; B, ŒACC; BDŒA; BCŒC; B and ŒA; BCC D ŒA; BCŒA; C ,
(ii) e bracket is skew-symmetric, i.e., ŒA; BD ŒB; A,
The Jacobi identity states that for all elements A, B, and C in a vector space g, the equation ŒA; ŒB; C + ŒB; ŒC; A + ŒC; ŒA; B = 0 holds true This identity, along with two other properties, defines an abstract Lie algebra Therefore, a vector space that includes a bracket operation satisfying these three properties is recognized as a Lie algebra, as referenced in [Serre1992].