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Mathematics Newsletter Volume 23. No4, March 2013

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Mathematics Newsletter Volume 23. No4, March 2013 Quaternions and Rotations in R3 Vishwambhar Pati Stat-Math Unit, Indian Statistical Institute, R.V. College P.O., Bangalore 560 059, India Abstract. In this expository note, we show that (a) rotations in 3-dimensional space are completely described by an axis of rotation and an angle of rotation about that axis. It is then natural to enquire about the axis and angle of rotation of the composite of two rotations whose axes and angles of rotation are given. We then (b) use the algebra of quaternions to answer this question. 1. Rotations Thus the jth column of the matrix [Tij ] is the image Tej of the jth basis vector ej . Clearly, with this same prescri- We recall that Euclidean space Rn comes equipped with the ption we are free to define the matrix of a linear transfor- natural dot product or inner product defined by mation with respect to any fixed chosen basis of Rn, though in n the present discussion we will mostly use the standard basis. v,w = viwi = n If v j=1 vj ej , then for the linear transformation T we i=1 have from the relation (1) above: n for two vectors v = (v1,...,vn) and w = (w1,...,wn) ∈ R . n n We will denote the ith basis vector (0,...,1 ...,0) with 1 Tv= T vj ej = vj Tej = Tij vj ei. in the ith spot and 0 elsewhere by e . The collection {e }n = = i i i=1 j j i 1 j 1 is called the standard basis of Rn.Ifv = n v e is i=1 i i In other words, if we use the common representation of the Rn = a vector in , its length or norm is defined as v / vector v as a column vector (with column entries vi), also v,v1/2 = n v2 1 2. A set of vectors {v ,...,v } is i=1 i 1 r denoted by v, then the column vector representing Tvis given called an orthonormal set if v ,v = δ for all 1 ≤ i, j ≤ r i j ij by the matrix product T.v. (Here δij is the Kronecker delta symbol defined by δij = 1if We next examine the condition on Tij for T to be an ortho- i = j, and δ = 0ifi = j). ij gonal transformation, and for it to be a rotation. If an orthonormal set is a basis, it is called an orthonormal { }n Rn basis. For example, the standard basis ei i=1 of is an Proposition 1.2 (Matrix characterisation of orthogonal orthonormal basis. transformations and rotations). The linear transformation T : Rn → Rn is an orthogonal transformation if and only if Definition 1.1 (Orthogonal transformations and rotations T t T = TTt = I, where T t denotes the transpose of T Rn Rn → Rn in ). A linear transformation T : is called an t defined by T = Tji. This is summarised by saying that the orthogonal transformation if it preserves all inner products, ij column vectors (resp. row vectors) of the matrix of T form i.e. if Tv,Tw = v,w for all v,w ∈ Rn. The set of an orthonormal set. For an orthogonal transformation T , all orthogonal transformations of Rn is denoted as O(n). det T =±1. Finally, an orthogonal transformation T is a An orthogonal transformation T is called a rotation if its deter- rotation if det T = 1. minant det T>0. The set of all rotations of Rn is denoted as SO(n). Thus SO(n) ⊂ O(n). Proof.IfT is orthogonal, we have δ = e ,e = ij i j Te,Te for all i and j. Plugging in Te = T e We note that a linear transformation T : Rn → Rn deter- i j i k ki k and Te = T e ,wefindδ = Te,Te = mines an n × n matrix with respect to the standard basis as j m mj m ij i j T T e ,e = T T δ = T t T = follows. We expand the image of the jth standard basis vector k,m ki mj k m k,m ki mj km k ik kj t t (T T)ij . Thus T is orthogonal implies T T = I. This shows ej under T in terms of the standard basis: that T t = T −1, and hence also that TTt = I. n = Conversely, if T t T = I, it follows by reversing the steps Tej Tij ei (1) = i 1 above that Tei,Tej = ei,ej for all i and j. This implies Mathematics Newsletter -257- Vol. 23 #4, March 2013 = = = ∈ ∈ ∈ that Tv,Tw i,j viwj Tei,Tej i,j viwj ei,ej S O(n), it follows that T.S O(n). If further S,T v,w for all v,w ∈ Rn. This proves the first assertion. SO(n), then det T = det S = 1. But then det(T .S) = Writing out the relation T t T = I (resp. TTt = I) shows (det T)(det S) = 1, showing that S.T ∈ SO(n) as well. This that the column (resp. row) vectors of T form an orthonormal proves the existence of a binary operation on O(n) and SO(n). set. The assertion (i) follows because multiplication of matrices Since T orthogonal implies T t T = I,wehavedet(T t T)= (or for that matter, composition of maps) is easily verified det(T t ) det T = (det T)2 = det I = 1. Hence det T =±1. to be associative. The statement (ii) is obvious, since the The second and third assertions follow. 2 identity transformation I ∈ SO(n) and in O(n) as well. The assertion (iii) follows from Proposition 1.2, where we saw The above matrix description of orthogonal transformations that T t = T −1, and also that T ∈ O(n) implies T t ∈ O(n), and rotations leads to the following important property of the because the conditions TTt = I and T t T = I are equivalent. sets O(n) and SO(n). Further T ∈ SO(n) implies det T t = det T = 1, so that Proposition 1.3 (The group structure on O(n) and SO(n)). T −1 = T t ∈ SO(n) as well. 2 The sets G = O(n) or SO(n) are groups. That is, they have a product or group operation defined by composition: 2. Rotations in R2 and R3 T.S:= T ◦ S, where (T ◦ S)v = T(Sv)for v ∈ Rn. Further- more, this operation satisfies the following axioms of a group: The following propositions give a quantitative characterisation (i) (Associativity) T .(S.R) = (T .S).R for all T,S,R ∈ G. of all the matrices that define rotations in R2 and R3. (ii) (Existence of identity) There is an identity transformation ∈ I ∈ G such that T.I = I.T = T for all T ∈ G. Proposition 2.1 (Planar Rotations). If T SO(2), then the (iii) (Existence of inverse) For each T ∈ G, there exists an matrix of T is given by: −1 ∈ −1 = −1 = element T G such that T.T T .T I. cos θ − sin θ Indeed, since SO(n) ⊂ O(n) inherits the group operation sin θ cos θ from O(n), we call SO(n) a subgroup of O(n). The group which, geometrically, describes a counterclockwise rotation in O(n) is called the orthogonal group of Rn and SO(n) the R2 by the angle θ ∈ [0, 2π). If we denote this rotation by R , special orthogonal group of Rn. θ then it is easily checked that Rθ .Rφ = Rθ+φ = Rφ+θ . (Here Proof. We first remark that the matrix of the composite trans- θ + φ means we go modulo 2π, viz. integer multiples of 2π formation T.S, by definition, is given by: have to be subtracted to get the value of θ +φ to lie in [0, 2π)). In particular, the group SO(2) is abelian, viz T.S = S.T for (T .S)e = (T .S) e . j ij i all T,S ∈ SO(2). Geometrically, we can think of the group i SO(2) as the group S1, defined as the circle consisting of However the left hand side, by definition and (1), is complex numbers in C of modulus 1 (and group operation being multiplication of complex numbers). T(Sej ) = T Skj ek = Skj (T ek) k k Proof. We know that T ∈ SO(2) implies firstly that T ∈ O( ) = Skj Tikei = TikSkj ei 2 , so that by the Proposition 1.2 above, the k i i k matrix: which implies that (T .S)ij = TikSkj . That is, if k a11 a12 we compose orthogonal transformations, the matrix of the T = a21 a22 composite transformation is the product of their respective t = 2 + 2 = matrices. satisfies the condition T T I, which implies that a11 a21 ∈ = = 2 + 2 + = Let T,S O(n). Since (T .S)v, (T .S)w T(Sv), 1 a12 a22 and a11a12 a21a22 0. The first rela- T(Sw)=Sv,Sw = v,w from the fact that T and tion implies, since aij are all real numbers, that a11 = cos θ Mathematics Newsletter -258- Vol. 23 #4, March 2013 and a21 = sin θ for some θ ∈ [0, 2π). The conditions where a,b ∈ R, and the constant term is − det T , which is −1 + = 2 + 2 = ∈ a11a12 a21a22 0 and a12 a22 1 now imply that because T SO(3). A root to this characteristic polynomial (a12,a22) =±(− sin θ,cos θ). The sign + is now determined PT is called an eigenvalue of T .
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