«431 432 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 The group of rotations SO(2) is isomorphic to the group U(1) of complex ...»
The Quaternions and the Spaces S 3,
SU(2), SO(3), and RP3
The Algebra H of Quaternions
In this chapter, we discuss the representation of rotations
of R3 and R4 in terms of quaternions.
Such a representation is not only concise and elegant, it
also yields a very eﬃcient way of handling composition of
It also tends to be numerically more stable than the rep-
resentation in terms of orthogonal matrices.
432 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 The group of rotations SO(2) is isomorphic to the group U(1) of complex numbers eiθ = cos θ + i sin θ of unit length. This follows imediately from the fact that the map cos θ − sin θ eiθ → sin θ cos θ is a group isomorphism.
Geometrically, observe that U(1) is the unit circle S 1.
We can identify the plane R2 with the complex plane C, letting z = x + iy ∈ C represent (x, y) ∈ R2.
Then, every plane rotation ρθ by an angle θ is represented by multiplication by the complex number eiθ ∈ U(1), in the sense that for all z, z ∈ C, z = ρθ (z) iﬀ z = eiθ z.
10.1. THE ALGEBRA H OF QUATERNIONS 433 In some sense, the quaternions generalize the complex numbers in such a way that rotations of R3 are repre- sented by multiplication by quaternions of unit length.
This is basically true with some twists.
For instance, quaternion multiplication is not commuta- tive, and a rotation in SO(3) requires conjugation with a (unit) quaternion for its representation.
Instead of the unit circle S 1, we need to consider the sphere S 3 in R4, and U(1) is replaced by SU(2).
Recall that the 3-sphere S 3 is the set of points (x, y, z, t) ∈ R4 such that x2 + y 2 + z 2 + t2 = 1, and that the real projective space RP3 is the quotient of S 3 modulo the equivalence relation that identiﬁes antipo- dal points (where (x, y, z, t) and (−x, −y, −z, −t) are antipodal points).
434 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 The group SO(3) of rotations of R3 is intimately related to the 3-sphere S 3 and to the real projective space RP3.
The key to this relationship is the fact that rotations can be represented by quaternions, discovered by Hamilton in 1843.
Historically, the quaternions were the ﬁrst instance of a noncommutative ﬁeld. As we shall see, quaternions rep- resent rotations in R3 very concisely.
It will be convenient to deﬁne the quaternions as certain 2 × 2 complex matrices.
10.1. THE ALGEBRA H OF QUATERNIONS 435 We write a complex number z as z = a + ib, where a, b ∈ R, and the conjugate z of z is z = a − ib.
Let 1, i, j, and k be the following matrices:
i0 1 0 1= i=
where x = a + ib and y = c + id. The matrices in H are called quaternions.
436 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 The null quaternion is denoted as 0 (or 0, if confusions arise).
Quaternions of the form bi + cj + dk are called pure quaternions. The set of pure quaternions is denoted as Hp.
Note that the rows (and columns) of such matrices are vectors in C2 that are orthogonal with respect to the Hermitian inner product of C2 given by
Using these identities, it can be veriﬁed that H is a ring (with multiplicative identity 1) and a real vector space of dimension 4 with basis (1, i, j, k).
The quaternions H are often deﬁned as the real algebra generated by the four elements 1, i, j, k, and satisfying the identities just stated above.
The problem with such a deﬁnition is that it is not obvious that the algebraic structure H actually exists.
A rigorous justiﬁcation requires the notions of freely generated algebra and of quotient of an algebra by an ideal.
Our deﬁnition in terms of matrices makes the existence of H trivial (but requires showing that the identities hold, which is an easy matter).
10.1. THE ALGEBRA H OF QUATERNIONS 439 Given any two quaternions X = a1 + bi + cj + dk and Y = a 1 + b i + c j + d k, it can be veriﬁed that
It is worth noting that these formulae were discovered independently by Olinde Rodrigues in 1840, a few years before Hamilton (Veblen and Young [?]).
However, Rodrigues was working with a diﬀerent formalism, homogeneous transformations, and he did not discover the quaternions.
440 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 The map from R to H deﬁned such that a → a1 is an injection which allows us to view R as a subring R1 (in fact, a ﬁeld) of H.
Similarly, the map from R3 to H deﬁned such that (b, c, d) → bi + cj + dk is an injection which allows us to view R3 as a subspace of H, in fact, the hyperplane Hp.
Given a quaternion X = a1 + bi + cj + dk, we deﬁne its conjugate X as
Clearly, X is nonnull iﬀ N (X) = 0, in which case X/N (X) is the multiplicative inverse of X.
Thus, H is a noncommutative ﬁeld.
Since X + X = 2a1, we also call 2a the reduced trace of X, and we denote it as T r(X).
A quaternion X is a pure quaternion iﬀ X = −X iﬀ T r(X) = 0. The following identities can be shown (see
Berger [?], Dieudonn´ [?], Bertin [?]):
whenever Z = 0.
442 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 If X = bi + cj + dk and Y = b i + c j + d k, are pure quaternions, identifying X and Y with the corresponding vectors in R3, the inner product X · Y and the crossproduct X ×Y make sense, and letting [0, X ×Y ] denote the quaternion whose ﬁrst component is 0 and whose last three components are those of X × Y, we have the remarkable identity
The above formula for quaternion multiplication allows us to show the following fact.
Let Z ∈ H, and assume that ZX = XZ for all X ∈ H.
Then, the pure part of Z is null, i.e., Z = a1 for some a ∈ R.
Remark : It is easy to check that for arbitrary quaternions X = [a, U ] and Y = [a, U ],
Since quaternion multiplication is bilinear, for a given X, the map Y → XY is linear, and similarly for a given Y, the map X → XY is linear. If the matrix of the ﬁrst map is LX and the matrix of the second map is RY, then
Observe that the columns (and the rows) of the above matrices are orthogonal.
Thus, when X and Y are unit quaternions, both LX and RY are orthogonal matrices. Furthermore, it is obvious that LX = LX, the transpose of LX, and similarly RY = RY.
It is easily shown that
This shows that when X is a unit quaternion, LX is a rotation matrix, and similarly when Y is a unit quaternion, RY is a rotation matrix (see Veblen and Young [?]).
It is easily veriﬁed that ϕ is bilinear, symmetric, and deﬁnite positive. Thus, the quaternions form a Euclidean space under the inner product deﬁned by ϕ (see Berger [?], Dieudonn´ [?], Bertin [?]).
e It is immediate that under this inner product, the norm of a quaternion X is just N (X).
It is also immediate that the set of pure quaternions is orthogonal to the space of “real quaternions” R1.
As a Euclidean space, H is isomorphic to E4.
The subspace Hp of pure quaternions inherits a Euclidean structure, and this subspace is isomorphic to the Euclidean space E3.
10.1. THE ALGEBRA H OF QUATERNIONS 447 Since H and E4 are isomorphic Euclidean spaces, their groups of rotations SO(H) and SO(4) are isomorphic, and we will identify them.
Similarly, we will identify SO(Hp) and SO(3).
448 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3
10.2 Quaternions and Rotations in SO(3) We just observed that for any nonnull quaternion X, both maps Y → XY and Y → Y X (where Y ∈ H) are linear maps, and that when N (X) = 1, these linear maps are in SO(4).
This suggests looking at maps ρY,Z : H → H of the form X → Y XZ, where Y, Z ∈ H are any two ﬁxed nonnull quaternions such that N (Y )N (Z) = 1.
In view of the identity N (U V ) = N (U )N (V ) for all U, V ∈ H, we see that ρY,Z is an isometry.
10.2. QUATERNIONS AND ROTATIONS IN SO(3) 449
We will prove that every rotation in SO(4) arises in this fashion.
Also, observe that when Z = Y −1, the map ρY,Y −1, denoted more simply as ρY, is the identity on 1R, and maps Hp into itself.
Thus, ρZ ∈ SO(3), i.e., ρZ is a rotation of E3.
We will prove that every rotation in SO(3) arises in this fashion.
The quaternions of norm 1, also called unit quaternions, are in bijection with points of the real 3-sphere S 3.
450 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3 It is easy to verify that the unit quaternions form a subgroup of the multiplicative group H∗ of nonnull quaternions. In terms of complex matrices, the unit quaternions correspond to the group of unitary complex 2×2 matrices of determinant 1 (i.e., xx + yy = 1)
with respect to the Hermitian inner product in C2.
This group is denoted as SU(2).
The obvious bijection between SU(2) and S 3 is in fact a homeomorphism, and it can be used to transfer the group structure on SU(2) to S 3, which becomes a topological group isomorphic to the topological group SU(2) of unit quaternions.
It should also be noted that the fact that the shere S 3 has a group structure is quite exceptional.
10.2. QUATERNIONS AND ROTATIONS IN SO(3) 451 As a matter of fact, the only spheres for which a continuous group structure is deﬁnable are S 1 and S 3.
One of the most important properties of the quaternions is that they can be used to represent rotations of R3, as stated in the following lemma.
1 For every quaternion Z = 0, the map ρZ : X → ZXZ −1 (where X ∈ H) is a rotation in SO(H) = SO(4) whose restriction to the space Hp of pure quaternions is a rotation in SO(Hp) = SO(3). Conversely, every rotation in SO(3) is of the form ρZ : X → ZXZ −1, for some quaternion Z = 0, and for all X ∈ Hp.
Furthermore, if two nonnull quaternions Z and Z represent the same rotation, then Z = λZ for some λ = 0 in R.
452 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3
Since SU(2) and S 3 are homeomorphic as topological spaces, this shows that SO(3) is homeomorphic to the quotient of the sphere S 3 modulo the antipodal map.
But the real projective space RP3 is deﬁned precisely this way in terms of the antipodal map π: S 3 → RP3, and thus SO(3) and RP3 are homeomorphic.
10.2. QUATERNIONS AND ROTATIONS IN SO(3) 453 This homeomorphism can then be used to transfer the group structure on SO(3) to RP3 which becomes a topological group.
Moreover, it can be shown that SO(3) and RP3 are diffeomorphic manifolds (see Marsden and Ratiu [?]).
Thus, SO(3) and RP3 are at the same time, groups, topological spaces, and manifolds, and in fact they are Lie groups (see Marsden and Ratiu [?] or Bryant [?]).
The axis and the angle of a rotation can also be extracted from a quaternion representing that rotation.
454 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3
Also note that V V = −1, and thus, formally, every unit quaternion looks like a complex number cos ϕ + i sin ϕ, except that i is replaced by a unit vector, and multiplication is quaternion multiplication.
In order to explain the homomorphism ρ: SU(2) → SO(3) more concretely, we now derive the formula for the rotation matrix of a rotation ρ whose axis D is determined by the nonnull vector w and whose angle of rotation is θ.
For simplicity, we may assume that w is a unit vector.
Letting W = (b, c, d) be the column vector representing w and H be the plane orthogonal to w, recall that the matrices representing the projections pD and pH are
with A and B as above, but (b, c, d) not necessarily a unit vector. We will study exponential maps later on.
10.2. QUATERNIONS AND ROTATIONS IN SO(3) 461
But since every pure quaternion X is a vector whose ﬁrst component is 0, we see that the rotation matrix R(Z) associated with the quaternion Z is
This expression for a rotation matrix is due to Euler (see Veblen and Young [?]).
10.2. QUATERNIONS AND ROTATIONS IN SO(3) 463 It is remarkable that this matrix only contains quadratic polynomials in a, b, c, d. This makes it possible to compute easily a quaternion from a rotation matrix.
From a computational point of view, it is worth noting that computing the composition of two rotations ρY and ρZ speciﬁed by two quaternions Y, Z using quaternion multiplication (i.e. ρY ◦ ρZ = ρY Z ) is cheaper than using rotation matrices and matrix multiplication.
On the other hand, computing the image of a point X under a rotation ρZ is more expensive in terms of quaternions (it requires computing ZXZ −1 ) than it is in terms of rotation matrices (where only AX needs to be computed, where A is a rotation matrix).
Thus, if many points need to be rotated and the rotation is speciﬁed by a quaternion, it is advantageous to precompute the Euler matrix.
464 CHAPTER 10. THE QUATERNIONS, THE SPACES S 3, SU(2), SO(3), AND RP3
10.3 Quaternions and Rotations in SO(4) For every nonnull quaternion Z, the map X → ZXZ −1 (where X is a pure quaternion) deﬁnes a rotation of Hp, and conversely every rotation of Hp is of the above form.
Remarkably, it turns out that we get all the rotations of H.
10.3. QUATERNIONS AND ROTATIONS IN SO(4) 465 Lemma 10.3.1 For every pair (Y, Z) of quaternions such that N (Y )N (Z) = 1, the map ρY,Z : X → Y XZ (where X ∈ H) is a rotation in SO(H) = SO(4).