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This content was downloaded from IP address 92.204.212.109 on 08/10/2023 at 07:26 The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations 1

Eckhard Hitzer

Osawa 3-10-2, Mitaka 181-8585, International Christian University, Japan

E-mail:hitzer@icu.ac.jp

Abstract.Recently the general orthogonal planes split with respect to any two pure unit quaternionsf;g2H,f2=g2=1, including the casef=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 withf=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 unitsf;g and the analysis planes, which they dene, 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 algebraHis dened overR with three imaginary units: ij=ji=k;jk=kj=i;ki=ik=j; i2=j2=k2=ijk=1:(1) The explicit form of a quaternionq2Hisq=qr+qii+qjj+qkk2H,qr;qi;qj;qk2R. We have the isomorphismsCl(3;0)+=Cl(0;2)=H, i.e.H is isomorphic to the algebra of rotation operators inCl(3;0). The quaternion conjugate (equivalent to Cliord conjugation inCl(3;0)+ andCl(0;2)) is dened aseq=qrqiiqjjqkk,epq=eqep, which leaves the scalar partqr unchanged. This leads to the norm ofq2Hjqj=pqeq= qq

2r+q2i+q2j+q2k,jpqj=jpjjqj:The

partq=V(q) =qqr=12 qeq) =qii+qjj+qkkis called apurequaternion, it squares to the negativenumber(q2i+q2j+q2k). Every unit quaternion2S3(i.e.jqj= 1) can be written as: q=qr+qii+qjj+qkk=qr+qq

2i+q2j+q2kbq= cos+bqsin= exp(bq), where cos=qr,

sin=qq

2i+q2j+q2k,bq=q=jqj= (qii+qjj+qkk)=qq

2i+q2j+q2k, andbq2=1,bq2S2. The

left and rightinverseof a non-zero quaternion isq1=eq=jqj2=eq=(qeq). Thescalar partof a quaternion is dened as S(q) =qr=12 q+eq), withsymmetries8p;q2H: S(pq) = S(qp) = 1

In memory of Hans Wondratschek, *07 Mar. 1925 in Bonn,y26 Oct. 2014 in Durlach.30th International Colloquium on Group Theoretical Methods in Physics (Group30)IOP Publishing

Journal of Physics: Conference Series597(2015) 012042 doi:10.1088/1742-6596/597/1/012042

Content from this work may be used under the terms of theCreativeCommonsAttribution 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 Ltd1

p rqrpiqipjqjpkqk, S(q) = S(eq), andlinearityS(p+q) =S(p) +S(q) =pr+qr, 8 p;q2H,;2R. The scalar part and the quaternion conjugate allow the denition of theR4inner product of two quaternionsp;qaspq= S(peq) =prqr+piqi+pjqj+pkqk2R. Accordingly we interpret in this paper the four quaternion coecients as coordinates inR4. In this interpretation selecting any two-dimensional plane subspace

2and its orthogonal complement two-dimensional subspace

allows to split four-dimensional quaternionsHinto pairs of orthogonal two-dimensional planes (compare Theorem 3.5 of [7]). Dealing with rotations in this paper includes general rotations in R 4. Denition 1.1(Orthogonality of quaternions).Two quaternionsp;q2Hareorthogonalp?q, if and only ifS(peq) = 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 thequaternion Fourier transformFffg(u;v) =R R

2eixuf(x;y)ejyvdxdy[4]. This observation3

(note that in the following alwaysiis on the left, andjis on the right) and that every quaternion can be rewritten asq=qr+qii+qjj+qkk=qr+qii+qjj+qkij, motivated the quaternion split

4with respect to the pair of orthonormal pure unit quaternionsi;j

q=q++q; q=12 qiqj):(2) Using (1), the detailed results of this split can be expanded in terms of real components q r;qi;qj;qk2R, as q =fqrqk+i(qiqj)g1k2 =1k2 f qrqk+j(qjqi)g:(3) The analysis of these two components leads to the following Pythagoreanmodulus identity[5]. Lemma 2.1(Modulus identity).Forq2H,jqj2=jqj2+jq+j2. Lemma 2.2(Orthogonality of OPS split parts [5]).Given any two quaternionsp;q2Hand applying the OPS split of (2) the resulting two parts areorthogonal, i.e.,p+?qandp?q+,

S(p+fq) = 0;S(pfq+) = 0:(4)

Next, we discuss the mapi()j, which will lead to an adapted orthogonal basis ofH. We observe, thatiqj=q+q, i.e. under the mapi()jtheq+part isinvariant, but theqpart changes sign . Both parts aretwo-dimensional(3), and by Lemma 2.2 they spantwo completely orthogonal planes , therefore also the nameorthogonal planes split (OPS). Theq+plane has the 2

The notion of two-dimensional plane employed here is thusdierentfrom a two-dimensional plane inR3. The

latter can be characterized by a unit bivector area element of the plane, which corresponds via the isomorphism

Cl(3;0)+=Hto a pure unit quaternion. This dierence in interpretation means also that despite of the

isomorphismCl(3;0)+=H, the notion and expression of rotations inR4cannot be automatically carried over to

rotation operators ofCl(3;0). Only in the case when rotations are restricted to the three-dimensional subset of

pure quaternions, then Hamilton's originalR3interpretation of these rotations is obvious.

3Replacing e.g.i!j,j!kthroughout would merely change the notation, but not the fundamental

observation.

4Also calledorthogonal planes split(OPS) as explained below.30th International Colloquium on Group Theoretical Methods in Physics (Group30)IOP Publishing

Journal of Physics: Conference Series597(2015) 012042 doi:10.1088/1742-6596/597/1/0120422 orthogonal quaternion basisfij=i(1+ij);1+ij= 1+kg, and theqplane has orthogonal basisfi+j=i(1ij);1ij= 1kg. All four basis quaternions (if normed:fq1;q2;q3;q4g) f ij;1 +ij;i+j;1ijg;(5) form an orthogonal basis ofHinterpreted asR4. Moreover, we obtain the following geometric picture on the left side of Fig. 1. The mapi()jrotatestheqplane by 180around the two- dimensionalq+axis plane. This interpretation of the mapi()jis in perfect agreement with Coxeter's notion ofhalf-turn[2]. In agreement with its geometric interpretation, the mapi()j is aninvolution, because applying it twice leads to identity i i qj)j=i2qj2= (1)2q=q:(6)

We have the important exponential factor identity

e iqej=qe()j=e()iq:(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==2 or ==2. Finally, we note the interpretation [7] of the QFT integrandeix1!1h(x)ejx2!2as alocal rotationby phase angle(x1!1+x2!2) ofh(x) in the two-dimensionalqplane, spanned byfi+j;1ijg, and alocal rotationby phase angle(x1!1x2!2) ofh+(x) in the two- dimensionalq+plane, spanned byfij;1 +ijg. This concludes the geometric picture of the OPS ofH(interpreted asR4) 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 involutioni()i. The mapi()igives i qi=i(qr+qii+qjj+qkk)i=qrqii+qjj+qkk:(8) The following orthogonal planes split (OPS) with respect to thesingle quaternion unitigives q =12 qiqi); q+=qjj+qkk= (qj+qki)j; q=qr+qii;(9) where theq+plane is two-dimensional and manifestlyorthogonalto the two-dimensionalq plane. The basis of the two planes are (if normed:fq1;q2g,fq3;q4g) q +-basis:fj;kg; q-basis:f1;ig:(10) The geometric interpretation ofi()ias Coxeterhalf-turnis perfectly analogous to the case i ()j. This form (9) of the OPS is identical to the quaternionicsimplex/perplexsplit applied in quaternionic signal processing, which leads in color image processing to the luminosity/chrominancesplit [6].

3. General orthogonal two-dimensional planes split (OPS)

Assume in the following an arbitrary pair of pure unit quaternionsf;g,f2=g2=1. The orthogonal 2D planes split (OPS)is then dened with respect toany two pure unit quaternions f;gas q =12

qfqg) =)fqg=q+q;(11)30th International Colloquium on Group Theoretical Methods in Physics (Group30)IOP Publishing

Journal of Physics: Conference Series597(2015) 012042 doi:10.1088/1742-6596/597/1/0120423 q --plane q +-plane f+g fg 1- fg 1 fg 180
180oq
--plane q +-plane f+g f - g 1 - fg 1 fg 180

180oFigure 1.Geometric pictures of the involutionsi()jandf()gas half turns.

i.e. under the mapf()gtheq+part is invariant, but theqpart changes sign. Both parts are two-dimensional, and span two completely orthogonal planes. Forf6=gthe q +plane is spanned by two orthogonal quaternionsffg;1+fg=f(fg)g, theqplane is e.g. spanned byff+g;1fg=f(f+g)g. Forg=fa fullyorthonormalfour-dimensional basis ofHis (Racts as rotation operator (rotor)) f 1 ;f;j0;k0g=R1f1;i;j;kgR; R=i(i+f);(12) and the two orthogonal two-dimensional planes basis: q +-basis:fj0;k0g; q-basis:f1;fg:(13) Note the notation for normed vectors in [3]fq1;q2;q3;q4gfor the resulting totalorthonormal basis ofH. Lemma 3.1(Orthogonality of two OPS planes).Given any two quaternionsq;pand applying the OPS with respect to any two pure unit quaternionsf;gwe get zero for the scalar part of the mixed products Sc p +eq) = 0; Sc(peq+) = 0:(14)

Note, that the two partsxcan berepresentedas

x =x+f1fg2 +xf1fg2 =1fg2 x+g+1fg2 xg;(15) with commuting and anticommuting partsxff=fxf, etc. Next we mention the possibility to perform a split along any given set of two (two-dimensional) analysis planes. It has been found, that any two-dimensional plane inR4determines in an elementary way an OPS split and vice versa, compare Theorem 3.5 of [7]. Let us turn to the geometric interpretation of the mapf()g. Itrotatestheqplane by 180 around theq+axis plane. This is in perfect agreement with Coxeter's notion ofhalf-turn[2], see the right side of Fig. 1. The followingidentitieshold e fqeg=qe()g=e()fq:(16) This leads to a straightforward geometric interpretation of the integrands of the quaternion Fourier transform (OPS-QFT) with two pure quaternionsf;g, and of the orthogonal 2D planes phase rotation Fourier transform [7]. We can further incorporate quaternion conjugation, which consequently provides a geometric interpretation of the QFT involving quaternion conjugation of the signal function. Ford=

30th International Colloquium on Group Theoretical Methods in Physics (Group30)IOP Publishing

Journal of Physics: Conference Series597(2015) 012042 doi:10.1088/1742-6596/597/1/0120424 equotesdbs_dbs35.pdfusesText_40

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