Figure 8.

The values of Δ = q · V represented by the sizes of the dots placed at a random distribution of quaternion points. We display the data dots at the locations of their spatial quaternion components q. (a) is the northern hemisphere of S ³, with q ₀ ≥ 0, (b) is the southern hemisphere, with q ₀ ≤ 0, and we implicitly know that the value of q ₀ is . The points in these two solid balls represent the entire space of quaternions, and it is important to note that, even though R(q) = R(−q) so each ball alone actually represents all possible unique rotation matrices, our cost function covers the entire space of quaternions, so q and −q are distinct. The spatial component of the maximal eigenvector is shown by the yellow arrow, which clearly ends in the middle of the maximum values of Δ. The small cloud at the edge of (b) is simply the rest of the complete cloud around the tip of the yellow arrow, as q ₀ passes through the ‘equator’ at q ₀ = 0, going from a small positive value at the edge of (a) to a small negative value at the edge of (b).