Attitude and orbit error in n-dimensional spaces

Abstract

This paper focuses on the theoretical aspects associated with the definition of error for Special Orthogonal and General Linear transformations, in any dimensional space, real or complex, including both Euclidean and Riemannian spaces. In particular, we show that the orbit error can be described by a complex number with phase representing the orbit orientation error and modulus describing the orbit shape error. We also show that the angle between two quaternions has a specific geometrical meaning. More specifically, the QR decomposition allows us to see a General Linear transformation as a combination of two subsequent effects: a dilation and a rigid rotation. This decomposition is also applied to the Lorentz transformations, highlighting the relativity effects of length change and coordinate axes bending.