Minimal rotations in arbitrary dimensions with applications to hypothesis testing and parameter estimation

Abstract

The rotation of a vector around the origin and in a plane constitutes a minimal rotation. Such a rotation is of vital importance in many applications. Examples are the re-orientation of spacecraft or antennas with minimal effort, the smooth interpolation between sensor poses, and the drawing of circular arcs in 2D and 3D. In numerical linear algebra, minimal rotations in different planes are used to manipulate matrices, e.g., to compute the QR decomposition of a matrix. This review compiles the concepts and formulas for minimal rotations in arbitrary dimensions for easy reference and provides a summary of the mathematical background necessary to understand the techniques described in this paper. The discussed concepts are accompanied by important examples in the context of photogrammetric image analysis. Hypothesis testing and parameter estimation for uncertain geometric entities are described in detail. In both applications, minimal rotations play an important role.