Figure 7.

(a) Projecting the geodesic versus chord distances from the origin to sampled points in a set of frame-displacement data . Since the q spatial quaternion paths project to a straight line from the origin, we use the (q ₀, 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 S ³. 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).