Interior Parameters, Exterior Parameters, and a Cayley-Like Transform

The direct relationship between a particular three-parameter orientation set, the modified Rodrigues parameters (MRP), and the orientation matrix was investigated. Extracting the modified Rodrigues parameters directly from the orientation matrix leads to a new categorization that is different from the usual modified Rodrigues parameters and their shadow set. The new categorization is composed of a useful set of parameters, the interior parameters, which possess a jump discontinuity at 180 deg, and the exterior parameters, which possess multiple orientation singularities in addition to a jump discontinuity. This approach leads to a new algorithm for calculating the interior parameters from the orientation matrix without using the Euler parameters, which are typically used. The approach is general and can be applied to any orientation parameter set. The new relationship between the interior parameters and the orientation matrix can be concisely written as a Cayley-like transform.