Rotors in n-dimensional Euclidean space are elements of Spin(n), the double covering group of SO(n), the rotation group. They have a natural representation as "spherical vectors" (that is, directed geodesic arcs) on the (N-1)-dimensional surface of the unit sphere in the N=n(n-1)/2-dimensional linear space of the bivectors. The rotor of a simple rotation has the form exp(B) where B is a simple bivector, and its spherical vector lies on the intersection of B with the unit sphere. The length of the spherical vector is the magnitude of B, and the act of rotating correponds to moving along the spherical vector. Any product of two rotors is given by the generally nonabelian sum of the two corresponding spherical vectors. The model has several advantages: (1) it provides probably the simplest visual representation of rotations in n-dimensions and for many purposes can replace the larger group manifold; (2) it displays the geometric relation between SO(n) and its universal double covering group Spin(n) [in the case of n=3=N, Spin(3) is isomorphic to SU(2)]; (3) the concept of distance between rotations or orientations can be formulated as the length of the corresponding difference vector; (4) interpolation between rotations is achieved simply by adding a fraction of the difference vector to the spherical vector of the original rotation; and (5) the model shows how spherical trigonometry on S^N-1 is easily calculated in the geometric algebra C(n). Several sample applications will be shown with an emphasis on the common case of n=3.

baylis@mrao.cam.ac.uk