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The orthogonal planes split of quaternions and its relation to quaternion geometry of rotations

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• 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 quaternions f, g ∈ H, f² = g² = −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 ∈ H is q = q_r + q_ii + q_jj + q_kk ∈ H, q_r, q_i, q_j, q_k ∈ 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 = q_r − q_ii − q_jj − q_kk, peq = qepe, which leaves the scalar part q_r

qp

unchanged. This leads to the norm of q ∈ H |q| = qqe = q_r² + q_i² + q_j² + q_k², |pq| = |p| |q| . The

12

part q = V (q) = q −q_r = (q −qe) = q_ii+q_jj +q_kk is called a pure quaternion, it squares to the negative number −(q_i² + q_j² + q_k²). Every unit quaternion ∈ S³ (i.e. |q| = 1) can be written as:

q

• 2
• 2
• 2

• b
• b
• b

q = q_r + q_ii + q_jj + q_kk = q_r + q_i + q_j + q_k q = cos α + q sin α = exp(α q), where cos α = q_r,

• q
• q

• 2
• 2

• 2
• 2
• 2
• 2
• 2
• 2

• b
• b
• b

sin α = q_i + q_j + q_k, q = q/ |q| = (q_ii + q_jj + q_kk)/ q_i + q_j + q_k, and q = −1, q ∈ S . The

2

left and right inverse of a non-zero quaternion is q⁻¹ = qe/ |q| = qe/(qqe). The scalar part of a quaternion is defined as S(q) = q_r = ¹₂ (q + qe), with symmetries ∀p, q ∈ H: S(pq) = S(qp) =

1

In memory of Hans Wondratschek, *07 Mar. 1925 in Bonn, †26 Oct. 2014 in Durlach.

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• 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

p_rq_r − p_iq_i − p_jq_j − p_kq_k, S(q) = S(qe), and linearity S(αp + βq) = α S(p) + β S(q) = αp_r + βq_r,

∀p, q ∈ H, α, β ∈ R.

The scalar part and the quaternion conjugate allow the definition of the R⁴ inner product of two quaternions p, q as p · q = S(pqe) = p_rq_r + p_iq_i + p_jq_j + p_kq_k ∈ R. Accordingly we interpret in this paper the four quaternion coefficients as coordinates in R⁴. 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 R⁴.

Definition 1.1 (Orthogonality of quaternions). Two quaternions p, q ∈ 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

R

the quaternion Fourier transform F{f}(u, v) = ₂ e^−ixuf(x, y)e^−jyvdxdy [4]. This observation³

R

(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 = q_r + q_ii + q_jj + q_kk = q_r + q_ii + q_jj + q_kij, 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

q_r, q_i, q_j, q_k ∈ R, as

• 1 ± k
• 1 ± k

q_± = {q_r ± q_k + i(q_i ∓ q_j)}

=

{q_r ± q_k + j(q_j ∓ q_i)}.

(3)

• 2
• 2

The analysis of these two components leads to the following Pythagorean modulus identity [5].

• 2
• 2
• 2

Lemma 2.1 (Modulus identity). For q ∈ H, |q| = |q₋| + |q₊| . Lemma 2.2 (Orthogonality of OPS split parts [5]). Given any two quaternions p, q ∈ 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

The notion of two-dimensional plane employed here is thus different from a two-dimensional plane in R³. 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

∼

=isomorphism Cl(3, 0)⁺ H, the notion and expression of rotations in R cannot be automatically carried over to

4

∼

rotation operators of Cl(3, 0). Only in the case when rotations are restricted to the three-dimensional subset of
=pure quaternions, then Hamilton’s original R³ 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 {i − j = i(1 + ij), 1 + ij = 1 + k}, and the q₋ plane has orthogonal basis {i + j = i(1 − ij), 1 − ij = 1 − k}. All four basis quaternions (if normed: {q₁, q₂, q₃, q₄})

{i − j, 1 + ij, i + j, 1 − ij},

(5) form an orthogonal basis of H interpreted as R⁴. 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 twodimensional 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

e^αiq_±e^βj = q_±e^(β∓α)j = e^(α∓β)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 α = π/2 or β = π/2.

−jx₂ω₂

Finally, we note the interpretation [7] of the QFT integrand e^{−ix ω} h(x) e

as a local

• 1
• 1

rotation by phase angle −(x₁ω₁ + x₂ω₂) of h₋(x) in the two-dimensional q₋ plane, spanned by {i + j, 1 − ij}, and a local rotation by phase angle −(x₁ω₁ − x₂ω₂) of h₊(x) in the twodimensional q₊ plane, spanned by {i − j, 1 + ij}. This concludes the geometric picture of the OPS of H (interpreted as R⁴) 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. The map i( )i gives

iqi = i(q_r + q_ii + q_jj + q_kk)i = −q_r − q_ii + q_jj + q_kk.

(8)
The following orthogonal planes split (OPS) with respect to the single quaternion unit i gives
1

q_± = (q ± iqi), q₊ = q_jj + q_kk = (q_j + q_ki)j, q₋ = q_r + q_ii,

(9)
2

where the q₊ plane is two-dimensional and manifestly orthogonal to the two-dimensional q₋ plane. The basis of the two planes are (if normed: {q₁, q₂}, {q₃, q₄})

• q₊-basis: {j, k},
• q₋-basis: {1, i}.

(10)
The geometric interpretation of i( )i as Coxeter half-turn is perfectly analogous to the case i( )j. This form (9) of the OPS is identical to the quaternionic simplex/perplex split applied in quaternionic signal processing, which leads in color image processing to the

luminosity/chrominance split [6].

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

Assume in the following an arbitrary pair of pure unit quaternions f, g, f² = g² = −1. The

orthogonal 2D planes split (OPS) is then defined with respect to any two pure unit quaternions

f, g as

1

q_± = (q ± fqg)

=⇒

fqg = q₊ − q₋,

(11)
2

3

• 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

q -plane

+

q -plane

+

i-j

f− g

i+j

f+g

1+ ij

1+ fg

1− ij

1− fg

180^o

180^o

q -plane

−

q -plane

−

Figure 1. Geometric pictures of the involutions i( )j and f()g as half turns. i.e. under the map f()g the q₊ part is invariant, but the q₋ part changes sign.
Both parts are two-dimensional, and span two completely orthogonal planes. For f = ±g the q₊ plane is spanned by two orthogonal quaternions {f − g, 1 + fg = −f(f − g)}, the q₋ plane is e.g. spanned by {f + g, 1 − fg = −f(f + g)}. For g = f a fully orthonormal four-dimensional basis of H is (R acts as rotation operator (rotor))

{1, f, j⁰, k⁰} = R⁻¹{1, i, j, k}R,

• R = i(i + f),
• (12)

and the two orthogonal two-dimensional planes basis:

• q₊-basis: {j⁰, k⁰},
• q₋-basis: {1, f}.
• (13)

Note the notation for normed vectors in [3] {q₁, q₂, q₃, q₄} for the resulting total orthonormal

basis of H. Lemma 3.1 (Orthogonality of two OPS planes). Given any two quaternions q, p and applying the OPS with respect to any two pure unit quaternions f, g we get zero for the scalar part of the mixed products

• Sc(p₊qe₋) = 0,
• Sc(p₋qe₊) = 0.

(14)
Note, that the two parts x_± can be represented as

• 1 ± fg
• 1 ∓ fg
• 1 ± fg
• 1 ∓ fg

x_± = x_+f

• + x_−f
• =

x_+g

+

x_−g

,

(15)

• 2
• 2
• 2
• 2

with commuting and anticommuting parts x_±f f = ±fx_±f , 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 in R⁴ determines in an