A metric for spatial displacement using biquaternions on SO(4)

In this paper we use the fact that 4×4 homogeneous transforms can be viewed as limiting cases of rotations in four dimensional Euclidean space, E⁴, to construct a metric for spatial displacements. For each spatial displacement, we compute an associated four-dimensional rotation and determine the associated biquaternion representation. We then use the standard Euclidean metric for these eight-dimensional vectors, in order to obtain a bi-invariant metric on SO(4). The result is an induced metric on SE(3) that is bi-invariant to a specific degree of approximation. As examples we determine the distance between two specified displacements, and find the reference frame “equidistant” between the two given frames for various positions of the global reference frame