The quaternion-based spatial-coordinate and orientation-frame alignment problems

Hanson, A.J.

doi:10.1107/S2053273320002648

Figure 3
The values of $[\Delta(q) = \mathop{\rm tr}\nolimits R(q)\cdot E = q\cdot M(E)\cdot q\,]$ represented by the sizes of the dots placed randomly in the `northern' and `southern' 3D solid balls spanning the entire hypersphere S³ with (a) containing the q₀ ≥ 0 sector and (b) containing the q₀ ≤ 0 sector. We display the data dots at the locations of their spatial quaternion components q = (q₁, q₂, q₃), and we know that q₀ = ±(1 − q · q)^1/2 so the q data uniquely specify the full quaternion. Since R(q) = R(−q), the points in each ball actually represent all possible unique rotation matrices. The spatial component of the maximal eigenvector is shown by the yellow arrows, which clearly end in the middle of the maximum values of Δ(q). Note that, in the quaternion context, diametrically opposite points on the spherical surface are identical rotations, so the cluster of larger dots at the upper right of (a) is, in the entire sphere, representing the same data as the `diametrically opposite' lower-left cluster in (b), both surrounding the tips of their own yellow arrows. The smaller dots at the upper right of (b) are contiguous with the upper-right region of (a), forming a single cloud centered on $[{\bf q}_{\rm opt}]$ , and similarly for the lower left of (a) and the lower left of (b). The whole figure contains two distinct clusters of dots (related by q → −q) centered around $[\pm{\bf q}_{\rm opt}]$ .

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ISSN: 2053-2733

Volume 76| Part 4| July 2020| Pages 432-457

https://doi.org/10.1107/S2053273320002648

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