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The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations

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The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations Home Search Collections Journals About Contact us My IOPscience The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Conf. Ser. 597 012042 (http://iopscience.iop.org/1742-6596/597/1/012042) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 131.169.4.70 This content was downloaded on 17/02/2016 at 22:46 Please note that terms and conditions apply. 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012042 doi:10.1088/1742-6596/597/1/012042 The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations1 Eckhard Hitzer Osawa 3-10-2, Mitaka 181-8585, International Christian University, Japan E-mail: [email protected] Abstract. Recently the general orthogonal planes split with respect to any two pure unit 2 2 quaternions f; g 2 H, f = g = −1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [7]. Applications include color image processing, where the orthogonal planes split with f = g = the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [3], that the pure quaternion units f; g and the analysis planes, which they define, play a key role in the geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections. 1. Introduction to quaternions Gauss, Rodrigues and Hamilton's four-dimensional (4D) quaternion algebra H is defined over R with three imaginary units: ij = −ji = k; jk = −kj = i; ki = −ik = j; i2 = j2 = k2 = ijk = −1: (1) The explicit form of a quaternion q 2 H is q = qr + qii + qjj + qkk 2 H, qr; qi; qj; qk 2 R. We + ∼ ∼ have the isomorphisms Cl(3; 0) = Cl(0; 2) = H, i.e. H is isomorphic to the algebra of rotation operators in Cl(3; 0). The quaternion conjugate (equivalent to Clifford conjugation in Cl(3; 0)+ and Cl(0; 2)) is defined as qe = qr − qii − qjj − qkk, pqe = qepe, which leaves the scalar part qr p q 2 2 2 2 unchanged. This leads to the norm of q 2 H jqj = qqe = qr + qi + qj + qk, jpqj = jpj jqj : The 1 part q = V (q) = q − qr = 2 (q − qe) = qii + qjj + qkk is called a pure quaternion, it squares to the 2 2 2 3 negative number −(qi + qj + qk). Every unit quaternion 2 S (i.e. jqj = 1) can be written as: q 2 2 2 q = qr + qii + qjj + qkk = qr + qi + qj + qk qb = cos α + qb sin α = exp(α qb), where cos α = qr, q 2 2 2 q 2 2 2 2 2 sin α = qi + qj + qk, qb = q= jqj = (qii + qjj + qkk)= qi + qj + qk, and qb = −1, qb 2 S . The left and right inverse of a non-zero quaternion is q−1 = q= jqj2 = q=(qq). The scalar part of a 1 e e e quaternion is defined as S(q) = qr = 2 (q + qe), with symmetries 8p; q 2 H: S(pq) = S(qp) = 1 In memory of Hans Wondratschek, *07 Mar. 1925 in Bonn, y26 Oct. 2014 in Durlach. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012042 doi:10.1088/1742-6596/597/1/012042 prqr − piqi − pjqj − pkqk, S(q) = S(qe), and linearity S(αp + βq) = α S(p) + β S(q) = αpr + βqr, 8p; q 2 H, α; β 2 R. 4 The scalar part and the quaternion conjugate allow the definition of the R inner product of two quaternions p; q as p · q = S(pqe) = prqr + piqi + pjqj + pkqk 2 R. Accordingly we interpret in 4 this paper the four quaternion coefficients as coordinates in R . In this interpretation selecting any two-dimensional plane subspace2 and its orthogonal complement two-dimensional subspace allows to split four-dimensional quaternions H into pairs of orthogonal two-dimensional planes (compare Theorem 3.5 of [7]). Dealing with rotations in this paper includes general rotations in 4 R . Definition 1.1 (Orthogonality of quaternions). Two quaternions p; q 2 H are orthogonal p ? q, if and only if S(pqe) = 0. 2. Motivation for quaternion split 2.1. Splitting quaternions and knowing what it means We deal with a split of quaternions, motivated by the consistent appearance of two terms in the quaternion Fourier transform Fffg(u; v) = R e−ixuf(x; y)e−jyvdxdy [4]. This observation3 R2 (note that in the following always i is on the left, and j is on the right) and that every quaternion can be rewritten as q = qr + qii + qjj + qkk = qr + qii + qjj + qkij, motivated the quaternion split4 with respect to the pair of orthonormal pure unit quaternions i; j 1 q = q + q ; q = (q ± iqj): (2) + − ± 2 Using (1), the detailed results of this split can be expanded in terms of real components qr; qi; qj; qk 2 R, as 1 ± k 1 ± k q = fq ± q + i(q ∓ q )g = fq ± q + j(q ∓ q )g: (3) ± r k i j 2 2 r k j i The analysis of these two components leads to the following Pythagorean modulus identity [5]. 2 2 2 Lemma 2.1 (Modulus identity). For q 2 H, jqj = jq−j + jq+j . Lemma 2.2 (Orthogonality of OPS split parts [5]). Given any two quaternions p; q 2 H and applying the OPS split of (2) the resulting two parts are orthogonal, i.e., p+ ? q− and p− ? q+, S(p+qf−) = 0; S(p−qf+) = 0: (4) Next, we discuss the map i()j, which will lead to an adapted orthogonal basis of H. We observe, that iqj = q+ − q− , i.e. under the map i()j the q+ part is invariant, but the q− part changes sign. Both parts are two-dimensional (3), and by Lemma 2.2 they span two completely orthogonal planes, therefore also the name orthogonal planes split (OPS). The q+ plane has the 2 3 The notion of two-dimensional plane employed here is thus different from a two-dimensional plane in R . The latter can be characterized by a unit bivector area element of the plane, which corresponds via the isomorphism + Cl(3; 0) =∼ H to a pure unit quaternion. This difference in interpretation means also that despite of the + 4 isomorphism Cl(3; 0) =∼ H, the notion and expression of rotations in R cannot be automatically carried over to rotation operators of Cl(3; 0). Only in the case when rotations are restricted to the three-dimensional subset of 3 pure quaternions, then Hamilton's original R interpretation of these rotations is obvious. 3 Replacing e.g. i ! j, j ! k throughout would merely change the notation, but not the fundamental observation. 4 Also called orthogonal planes split (OPS) as explained below. 2 30th International Colloquium on Group Theoretical Methods in Physics (Group30) IOP Publishing Journal of Physics: Conference Series 597 (2015) 012042 doi:10.1088/1742-6596/597/1/012042 orthogonal quaternion basis fi − j = i(1 + ij); 1 + ij = 1 + kg, and the q− plane has orthogonal basis fi + j = i(1 − ij); 1 − ij = 1 − kg. All four basis quaternions (if normed: fq1; q2; q3; q4g) fi − j; 1 + ij; i + j; 1 − ijg; (5) 4 form an orthogonal basis of H interpreted as R . Moreover, we obtain the following geometric ◦ picture on the left side of Fig. 1. The map i()j rotates the q− plane by 180 around the two- dimensional q+ axis plane. This interpretation of the map i()j is in perfect agreement with Coxeter's notion of half-turn [2]. In agreement with its geometric interpretation, the map i()j is an involution, because applying it twice leads to identity i(iqj)j = i2qj2 = (−1)2q = q: (6) We have the important exponential factor identity αi βj (β∓α)j (α∓β)i e q±e = q±e = e q±: (7) This equation should be compared with the kernel construction of the quaternion Fourier transform (QFT). The equation is also often used in our present context for values α = π=2 or β = π=2. Finally, we note the interpretation [7] of the QFT integrand e−ix1!1 h(x) e−jx2!2 as a local rotation by phase angle −(x1!1 + x2!2) of h−(x) in the two-dimensional q− plane, spanned by fi + j; 1 − ijg, and a local rotation by phase angle −(x1!1 − x2!2) of h+(x) in the two- dimensional q+ plane, spanned by fi − j; 1 + ijg. This concludes the geometric picture of the 4 OPS of H (interpreted as R ) with respect to two orthonormal pure quaternion units. 2.2. Even one pure unit quaternion can do a nice split Let us now analyze the involution i()i.
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