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

Hanson, A.J.

doi:10.1107/S2053273320002648

Figure 7
(a) Projecting the geodesic versus chord distances from the origin to sampled points in a set of frame-displacement data $[t_{{k}} = r_{{k}}\star\bar{p}_{{k}}\to{\widetilde{t}}_{{k}}]$ . Since the q spatial quaternion paths project to a straight line from the origin, we use the (q₀, q_x, q_y) coordinates instead of our standard q coordinates to expose the curvature in the arc-length distances to the origin. (b) Histogram of the chord-length distances to the origin (in blue) compared to the histogram of the geodesic arc-length distances (in yellow), sampled using a uniform distribution of random quaternions over a portion of S³. If there were no errors, all the points would have the same distance from the origin located at q = (1, 0, 0, 0) [red axis in (a)], and there would be one blue spike, appearing at a slightly smaller position than the yellow spike because arc length is always longer than chord length. The arc-length method has a different distribution, as expected, and produces a very slightly better barycenter. However, the optimal quaternions for the arc-length versus chord-length measure for this simulated data set differ by only a fraction of a degree, so drawing the positions of the two distinct optimal quaternions would not reveal any noticeable difference in image (a).

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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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