GENERALIZED QUATERNIONS AND ROTATION IN 3-SPACE E 3 αβ

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1 TWMS J. Pure Appl. Math. V.6, N., 015, pp.4-3 GENERALIZED QUATERNIONS AND ROTATION IN 3-SPACE E 3 αβ MEHDI JAFARI 1, YUSUF YAYLI Abstract. The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional E 3 αβ space. We review of some algebraic properties of generalized quaternions and operations between them and then show their relation with the rotation matrix. Keywords: generalized quaternion, quasi-orthogonal matrix, rotation. AMS Subject Classification: 15A Introduction The quaternions algebra was invented by W.R. Hamilton as an extension to the complex numbers. He was able to find connections between this new algebra and spatial rotations. The unit quaternions form a group that is isomorphic to the group SU() and is a double cover of SO(3), the group of 3-dimensional rotations. Under these isomorphisms the quaternion multiplication operation corresponds to the composition operation of rotations [18]. Kula and Yaylı [13] showed that unit split quaternions in H determined a rotation in semi-euclidean space E 4. In [15], is demonstrated how timelike split quaternions are used to perform rotations in the Minkowski 3-space E1 3. Rotations in a complex 3-dimensional space are considered in [0] and applied to the treatment of the Lorentz transformation in special relativity. A brief introduction of the generalized quaternions is provided in [16]. Also, this subject have investigated in algebra [19]. Recently, we studied the generalized quaternions, and gave some of their algebraic properties [7]. It is shown that the set of all unit generalized quaternions with the group operation of quaternion multiplication is a Lie group of 3-dimension. Their Lie algebra and properties of the bracket multiplication are looked for. Also, a matrix corresponding to Hamilton operators that is defined for generalized quaternions is determined a Homothetic motion in Eαβ 4 [9]. Furthermore, we showed how these operators can be used to describe rotation in Eαβ 4 [10]. In this paper, after developing some elementary properties of generalized quaternions, it is shown how unit generalized quaternions can be used to described rotation in 3-dimensional space Eαβ 3.. Preliminaries Quaternions are hypercomplex numbers used to represent spatial rotations in three dimensions. The basic definition of a real quaternion given in [3, 4] as follows: Definition.1. A real quaternion is defined as q = a 0 + a 1 i + a j + a 3 k 1 Department of Mathematics, University College of Science and Technology Elm o Fan, Urmia, Iran Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey mj msc@yahoo.com, yayli@science.ankara.edu.tr Manuscript received June

2 M. JAFARI, Y. YAYLI: GENERALIZED QUATERNIONS AND ROTATION... 5 where a 0, a 1, a and a 3 are real numbers and 1, i, j, k of q may be interpreted as the four basic vectors of Cartesian set of coordinates; and they satisfy the non-commutative multiplication rules i = j = k = 1 ij = k = ji, jk = i = kj and ki = j = ik, ijk = 1. The quaternion algebra H is the even subalgebra of the Clifford algebra of the 3-dimensional Euclidean space. The Clifford algebra Cl(Ep n ) = Cl n p,p for the n-dimensional non-degenerate vector space Ep n having an orthonormal base {e 1, e,..., e n } with the signature (p, n p) is the associative algebra generated by 1 and {e i } with satisfying the relations e i e j + e j e i = 0 for i j and e 1, if i=1,,...,p i = { 1, if i=p+1,...,n The Clifford algebra Cl n p,p has the basis{e i1, e i,..., e ik : 1 i 1 < i <... < i k < n} that is the division algebra of quaternions H is isomorphic with the even subalgebra Cl 3,0 + of the Clifford algebra Cl 3,0 such that Cl 3,0 + has the basis {1, e 1e i, e e 3 j, e 1 e 3 k}. The conjugate of the quaternion q = S q +v q is denoted by q, and defined as q = S q v q. The norm of a quaternion q = (a 0, a 1, a, a 3 ) is defined by qq = qq = a 0 + a 1 + a + a 3 and is denoted by N q and say that q 0 = q N q is unit quaternion where q 0. Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions[0]. One can represent a quaternion q = a 0 + a 1 i + a j + a 3 k by a complex matrix (with i being the usual complex imaginary); [ a0 + i a A = 1 i ] a 1 + a i a 1 a a 0 i a 3 or by a 4 4 real matrix a 0 a 1 a a 3 A = a 1 a 0 a 3 a a a 3 a 0 a 1 a 3 a a 1 a 0 The Euler s and De-Moivre s formulae for the matrix A are studied in [8]. It is shown that as the De Moivre s formula implies, there are uncountably many matrices of unit quaternion A n = I 4 satisfying for n >. In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. There are at least eight methods used commonly to represent rotation, including: i) orthogonal matrices, ii) axis and angle, iii) Euler angles, iv) Gibbs vector, v) Pauli spin matrices, vi) Cayley-Klein parameters, vii) Euler or Rodrigues parameters, and viii) Hamilton s quaternions [6]. But to use the unit quaternions is a more useful, natural, and elegant way to perceive rotations compared to other methods. Theorem.1. All the rotation about lines through the origin in ordinary space form a group, homomorphic to the group of all unit quaternions []. If a simple rotation is only in the three space dimensions, i.e. about a plane that is entirely in space, then this rotation is the same as a spatial rotation in three dimensions. But a simple rotation about a plane spanned by a space dimension and a time dimension is a boost, a

3 6 TWMS J. PURE APPL. MATH., V.6, N., 015 transformation between two different reference frames, which together with other properties of spacetime determines the relativistic relationship between the frames. The set of these rotations forms the Lorentz group [1]. With Cartesian point coordinates in 3-space, a rotation in 3-space about the origin can be represented by the orthogonal matrix R = r 11 r 1 r 13 r 1 r r 3 r 31 r 3 r 33 where RR T = I 3 and det R = 1. It is known that unit quaternions can represent rotations about the origin. Wittenburg [1] gives the following conversion formulae. For any unit quaternion q the entries of the rotation matrix are r 11 = (a 0 + a 1) 1, r 1 = (a 1 a + a 0 a 3 ), r 31 = (a 1 a 3 a 0 a ) r 1 = (a 1 a a 0 a 3 ), r = (a 0 + a ) 1, r 3 = (a a 3 a 0 a ) r 13 = (a 1 a 3 a 0 a ), r 3 = (a a 3 a 0 a ), r 33 = (a 0 + a ) 1 [5]. Example.1. For the unit real quaternion q = (1, 1, 0), the rotation matrix is R q = the axis of this rotation is spanned by the vector (1, 1, 0) and the angle of rotation is φ = π. Definition.. Let u = (u 1, u, u 3 ), v = (v 1, v, v 3 ) R 3. If α, β R +, the generalized inner product of u and v is defined by It could be written,, u, v = αu 1 v 1 + βu v + αβu 3 v 3. u, v = u T α β αβ Also, if α > 0, β < 0, u, v is called the generalized semi-euclidean inner product. We put Eαβ 3 = ( R 3,, ). The vector product in Eαβ 3 is defined by u v = βi αj k u 1 u u 3 v 1 v v 3 where i j = k, j k = βi, k i = αj. v. = β(u v 3 u 3 v )i + α(u 3 v 1 u 1 v 3 )j + (u 1 v u v 1 )k, Special cases: 1. If α = β = 1, then E 3 αβ is an Euclidean 3-space E3.. If α = 1, β = 1, then E 3 αβ is a semi-euclidean 3-space E3. Definition.3. A generalized quaternion is an expression of form q = a 0 + a 1 i + a j + a 3 k

4 M. JAFARI, Y. YAYLI: GENERALIZED QUATERNIONS AND ROTATION... 7 where a 0, a 1, a and a 3 are real numbers and i, j, k are quaternionic units which satisfy the equalities and i = α, j = β, k = αβ ij = k = ji, jk = βi = kj ki = αj = ik, α, β R. The set of all generalized quaternions are denoted by H αβ. A generalized quaternion q is a sum of a scalar and a vector, called scalar part, S q = a 0, and vector part V q = a 1 i++a j +a 3 k R 3 αβ. Therefore, H αβ forms a 4-dimensional real space which contains the real axis R and a 3- dimensional real linear space E 3 αβ, so that, H αβ = R E 3 αβ. Special cases: 1) If α = β = 1 is considered, then H αβ is the algebra of real quaternions [4]. ) If α = 1, β = 1 is considered, then H αβ is the algebra of split quaternions [13]. 3) If α = 1, β = 0 is considered, then H αβ is the algebra of semi-quaternions [14]. 4) If α = 1, β = 0 is considered, then H αβ is the algebra of split semi-quaternions [1]. 5) If α = 0, β = 0 is considered, then H αβ is the algebra of 1 4 quaternions [11,17]. The multiplication rule for generalized quaternions is defined as qp = S p S q V q, V p + S q V p + S p V q + V p V q, where It could be expressed as S p = a 0, S q = b 0, V q, V p = αa 1 b 1 + βa b + αβa 3 b 3, V p V q = β(a b 3 a 3 b )i + α(a 3 b 1 a 1 b 3 )j + (a 1 b a b 1 )k. qp = a 0 αa 1 βa αβa 3 a 1 a 0 βa 3 βa a αa 3 a 0 αa 1 a 3 a a 1 a 0 Obviously, quaternion multiplication is an associative and distributive with respect to addition and subtraction, but the commutative law does not hold in general. Corollary.1. H αβ with addition and multiplication has all the properties of a number field expect commutativity of the multiplication. It is therefore called the skew field of quaternions. The conjugate of the quaternion q = S q + V q is denoted by q and defined as q = S q V q. The norm of a quaternion q = (a 0, a 1, a, a 3 ) is defined by N q = qq = qq = a 0 + αa 1 + βa + αβa 3 and say that q 0 = q N q is a unit generalized quaternion where q 0. The set of unit generalized quaternions, G, with the group operation of quaternion multiplication is a Lie group of 3-dimension. The inverse of q is defined as q 1 = q N q, N q 0. The scalar product of two generalized quaternions q = S q + V q and p = S p + V p is defined as q, p s = S p S q + V q, V p = S pq. b 0 b 1 b b 3

5 8 TWMS J. PURE APPL. MATH., V.6, N., 015 Also, using the scalar product we can defined an angle λ between two quaternions q, p to be such S pq cos λ =. Np Nq Definition.4. A matrix A 3 3 is called a quasi-orthogonal matrix if A T εa = ε and det A = 1 where α 0 0 ε = 0 β 0, 0 0 αβ and α, β R {0}.The set of all quasi-orthogonal matrices with the operation of matrix multiplication is called rotation group in 3-space E 3 αβ. Definition.5. A matrix S 3 3 = 0 βs 3 βs αs 3 0 αs 1 s s 1 0 is called a generalized skew-symmetric matrix if S T ε = εs where α 0 0 ε = 0 β 0, 0 0 αβ E 3 αβ and α, β R {0}. 3. Relationship of generalized quaternion to rotation In this section, we show that a unit generalized quaternion represents a rotation in 3-space. Let q be a unit generalized quaternion. The map ϕ acting on a pure quaternion ω: ϕ : E 3 αβ E3 αβ ϕ(ω) = qωq 1 is a 3D vector, a length-preserving function of 3D vectors, a linear transformation and does not have a reflection component. Since E 3 αβ = span {i, j, k} and if q = a 0 + a 1 i + a j + a 3 k G then ϕ(i) = (a 0 + αa 1 βa αβa 3)i + α(a 1 a + a 0 α 3 ) j + (αa 1 α 3 a 0 a )k, ϕ(j) = β(a 1 a a 0 a 3 )i + (a 0 αa 1 + βa αβa 3) j + (βa α 3 + a 0 a 1 )k, ϕ(k) = β(a 0 a + αa 1 a 3 )i + α(βa α 3 a 0 a 1 ) j + (a 0 αa 1 βa + αβa 3)k. So that the matrix representation of the map ϕ is a 0 + αa 1 βa αβa 3 β(a 1 a a 0 a 3 ) β(a 0 a + αa 1 a 3 ) M = α(a 1 a + a 0 α 3 ) a 0 αa 1 + βa αβa 3 α(βa α 3 a 0 a 1 ) αa 1 α 3 a 0 a βa α 3 + a 0 a 1 a 0 αa 1 βa + αβa 3 We investigate matrix M in two different cases. Case 1: Let α, β are positive numbers. Theorem 3.1. Let q be a unit generalized quaternion, then the matrix M can be written as M (φ,s) = I 3 + sin φ S + (1 cos φ)s, (1) where ϕ is a elliptic, such that cos ϕ = a 0, sin ϕ = αa 1 + βa + αβa 3, and S is a generalized skew-symmetric matrix. The formula (1.1) known as Rodrigues rotation formula.

6 M. JAFARI, Y. YAYLI: GENERALIZED QUATERNIONS AND ROTATION... 9 Proof. Every unit generalized quaternion q = a 0 + a 1 i + a j + a 3 k can be written in polar form then the matrix M can be written as M = q = cos φ + S sin φ, cos φ + (αs 1 βs αβs 3 ) sin φ β(s 1 s sin φ s 3 cos φ sin φ ) β(αs 1s sin φ s cos φ sin φ ) α(s 1 s sin φ + s 3 cos φ sin φ ) cos φ + ( αs 1 + βs αβs 3 ) sin φ α(βs s 3 sin φ s 1 cos φ sin φ ) (αs 1 s 3 sin φ s cos φ sin φ ) (βs s 3 sin φ + s 1 cos φ sin φ ) cos φ + ( αs 1 βs + αβs 3 ) sin φ = I 3 + (αs 1 βs αβs 3 1) sin φ βs 1 s sin φ βs 3 sin φ βαs 1 s sin φ βs cos φ sin φ αs 1 s sin φ + αs 3 sin φ ( αs 1 + βs αβs 3 1) sin φ αβs s 3 sin φ + αs 1 sin φ αs 1 s 3 sin φ s sin φ βs s 3 sin φ + s 1 sin φ ( αs 1 βs + αβs 3 + 1) sin φ with used of sin φ = 1 cos φ and αs 1 + βs + αβs 3 = 1, we have 0 βs 3 βs M = I 3 +sin φ αs 3 0 αs 1 +(1 cos φ) s s 1 0 so, proof is complete., αβs 3 βs βs 1 s αβs 1 s 3 αs 1 s αs 1 αβs 3 αβs s 3 αs 1 s 3 βs s 3 αs 1 βs Special case: For the case α = β = 1, we have M = (M H ) for real quaternion H. M H is a orthogonal matrix, then the map ϕ corresponds to a rotation in E 3. If we take the rotation axis to be S = (s 1, s, s 3 ), then the Rodrigues rotation formula is where S is a skew-symmetric matrix, M H = I 3 + S sin θ + (1 cos θ)s S = 0 s 3 s s 3 0 s 1 s s 1 0 Theorem 3.. Let q be a unit generalized quaternion. Then M is a quasi-orthogonal matrix, i.e. M T εm = ε and det M = 1 where α 0 0 ε = 0 β αβ Corollary 3.1. Let q = cos φ + S sin φ be a unit generalized quaternion. The linear map ϕ(ω) = qωq 1 represent a rotation of the original vector ω by an angle φ around the axis S in 3-space Eαβ 3. Example 3.1. Let q = ( 1 1 α, β, 0) be a unit generalized quaternion. The rotation matrix is M = 1 β β α α 1 β α 1 1 β α 0 Case. Let α be a positive number and β be a negative number.,

7 30 TWMS J. PURE APPL. MATH., V.6, N., 015 Theorem 3.3. Let q be a unit generalized quaternion and the norm of its vector part is negative. i.e. αs 1 + βs + αβs 3 < 0, then the matrix M can be written as M (φ,c) = I 3 + sinh φc + ( 1 + cosh φ)c, where is a hyperbolic angle, such that cosh φ = a 0, sinh φ = αs 1 βs αβs 3 generalized skew-symmetric matrix. and C is a Proof. Every unit generalized quaternion q = a 0 + a 1 i + a j + a 3 k can be written in polar form then the matrix M can be written as M = q = cosh φ + C sinh φ, cosh φ + (αc 1 βc αβc 3 ) sinh φ β(c 1 c sinh φ c 3 cosh φ sinh φ ) β(αc 1c 3 sinh φ + c cosh φ sinh φ ) α(c 1 c sinh φ + c 3 cosh φ sinh φ ) cosh φ + ( αc 1 + βc αβc 3 ) sinh φ α(βc c 3 sinh φ + c 1 cosh φ sinh φ ) (αc 1 c 3 sinh φ c cosh φ sinh φ ) (βc c 3 sinh φ + c 1 cosh φ sinh φ ) cosh φ + ( αc 1 βc + αβc 3 ) sinh φ = I 3 + (αc 1 βc αβc 3 ) sinh φ βc 1 c sinh φ βc 3 sinh φ αβc 1 c 3 sinh φ + βc sinh φ αc 1 c sinh φ + αc 3 sinh φ ( αc 1 + βc αβc 3 + 1) sinh φ αβc c 3 sinh φ + αc 1 sinh φ αc 1 c 3 sinh φ c sinh φ βc c 3 sinh φ + c 1 sinh φ ( αc 1 βc + αβc 3 + 1) sinh φ with used of sinh φ = 1 + cosh φ and αc 1 + βc + αβc 3 = 1, we have 0 βc 3 βc M = I 3 + sinh φ αc 3 0 αc 1 + c c 1 0 αβc 3 βc βc 1 c αβc 1 c 3 +( 1 + cosh φ) αc 1 c αc 1 αβc 3 αβc c 3, αc 1 c 3 βc c 3 αc 1 βc so, proof is complete. Corollary 3.. Let q = cosh φ + C sinh φ be a unit generalized quaternion. The linear map ϕ(ω) represents a rotation of the original vector ω by an angle φ around the axis C in 3-space Eαβ 3. Example 3.. Let q = + 1 (0, 1, 1) be a unit generalized quaternion and α = 1, β =. The rotation matrix is 3 M = 1 1 M is a quasi-orthogonal matrix, i.e., M T εm = ε and det M = 1, where ε = and therefore it represents a rotation in 3-space E1, 3. The axis of this rotation is spanned by the vector (0, 1, 1 ) and the hyperbolic angle of rotation is φ, such taht cosh φ = and sinh φ = 1. Theorem 3.4. Let q be a unit generalized quaternion and the norm of its vector part is positive, i.e. αa 1 + βa + αβa 3 > 0, then the matrix M can be written as M (φ,s) = I 3 + sin φs + (1 cos φ)s,

8 M. JAFARI, Y. YAYLI: GENERALIZED QUATERNIONS AND ROTATION where φ is a hyperbolic angle, such taht cos φ = a 0, s i sin φ = αa 1 + βa αβa 3, and S is a skew-symmetric matrix. Proof. The proof is similar to the proof of Theorem 3.1. Special case: For the case α = 1, β = 1, we have M(= M H ) for split quaternion H. M H is a semi-orthogonal matrix, then the map ϕ corresponds to a rotation in E 3 1. The Rodrigues rotation formula is as follows: i. If the rotation axis C = (c, c, c) is timelike, ii. If the rotation axis C = (c, c, c) is spacelike, M H = I 3 + sin θc + (1 cos θ)c, M H = I 3 + (sinh γ)c + ( 1 + cosh γ)c, where C is a skew-symmetric matrix, i.e. C T = χcχ and 0 c 3 c C = c 3 0 c 1, χ = c c That is, kind of rotation matrix in the Minkowski space depends on the rotation axis. Example 3.3. Let q = + (1, 0, 1 ) be a unit generalized quaternion and α = 1, β =. The rotation matrix is χ M = M is a quasi-orthogonal matrix, i.e., M T εm = ε and det M = 1 where ε = and therefore it represents a rotation in 3-space E1, 3. The unit quaternion q represents rotation ( ) through an angle 90 about the axis S =, 0, 1. References [1] Brannon, R.M., (00), A review of useful theorems involving proper orthogonal matrices referenced to three-dimensional physical space, Albuquerque: Sandia National Laboratories. [] Coxeter, H.S.M., (1946), Quaternions and reflections, American Math. Mon., 53(3), pp [3] Girard, P. R., (007), Quaternions, Clifford Algebras and Relativistic Physics, Birkhauser. Verlag, Basel, p.179. [4] Hacisalihoglu, H.H., (1971), Hareket Geometrisi ve Kuaterniyonlar Teorisi, Published by Gazi University, Ankara, Turkey. [5] Hoffmann, C.M. Yang, W., (003), Compliant Motion Constraints, Proceedings of the Sixth Asian Symposium, Beijing, China, April. [6] Horn Berthold, K.P., (001), Some Notes on Unit Quaternions and Rotation, Copyright. [7] Jafari, M., (01), Generalized Hamilton Operators and Lie groups, Ph.D. thesis, Ankara University, Ankara, Turkey. [8] Jafari, M., Mortazaasl, H., Yayli, Y., (011), De Moivre s Formula for Matrices of Quaternions, JP J. of Algebra, Number Theory and Appl., Vol. 1(1), pp [9] Jafari, M. & Yayli, Y., (013), Rotation in four dimensions via generalized Hamilton operators, Kuwait journal of Science, 40(1), pp [10] Jafari, M. & Yayli, Y., (010), Homothetic motions at E 4 αβ, International Journal Contemp. of Mathematics Sciences, 5(47), pp

9 3 TWMS J. PURE APPL. MATH., V.6, N., 015 [11] Jafari, M., (014), On properties of Quasi-quaternions algebra, Communication Faculty of Sciences University of Ankara Series A1, 63(1), pp [1] Jafari, M., (015), Split semi-quaternions algebra in semi-euclidean 4-Space, Cumhuriyet Science Journal, 36(1), pp [13] Kula, L. & Yayli, Y., (007), Split quaternions and rotations in semi-euclidean space E 4, Journal of Korean Math. Soc., 44(6) pp [14] Mortazaasl, H., Jafari M., (013), A study on semi-quaternions algebra in semi-euclidean 4-Space, Mathematical sciences and applications E-notes, 1(), pp.0-7. [15] Ozdemir, M., Ergin, A.A., (006), Rotations with unit timelike quaternions in Minkowski 3-space, J. of Geometry and Physics 56, pp [16] Pottman, H., Wallner, J., (000), Computational Line Geometry. Springer-Verlag Berlin Heidelberg, New York. [17] Rosenfeld, B.A., (1997), Geometry of Lie Groups, Kluwer Academic Publishers, Dordrecht. [18] Stahlke, D., (009), Quaternions in Classical Mechanics, PHYS 61, Stahlke.org. [19] Unger, T., Markin, N., (008), Quadratic Forms and space-time Block Codes from GeneralizedQuaternion and Biquaternion Algebras arxiv: v1. [0] Ward, J.P., (1997), Quaternions and Cayley Numbers Algebra and Applications, Kluwer Academic Publishers, London. [1] Wittenburg, J., (1977), Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgart, Germany. Mehdi Jafari received his B.Sc. degree in Mathematics in 003 from Urmia University and his M.Sc. degree in pure mathematics in 005 from I.A University, Science and Research Branch, Tehran. He got his Ph.D. degree from Ankara University, Ankara, Turkey in 01. His research interests include quaternion theory, kinematics. Yusuf Yaylı received his B.Sc. degree from Inönü University, Ankara, Turkey in 1983, M.S. and Ph.D. degrees in geometry from Ankara University, Ankara, Turkey in 1985 and 1988 respectively. He received associate professorship in Since 001 he has been a professor at the Mathematics Department, Ankara University, Ankara, Turkey. His research is focused on Motion geometry, Lorentzian geometry, curves and surfaces theory.