The barycentric ratio as invariant of projective geometry

Geometric invariants help to focus on the main aspects of a geometric problem by hiding irrelevant information. The fundamental invariant of projective geometry is the cross-ratio, an invariant of four collinear points. It is of particular importance for the understanding and conception of the projective plane. All other projective invariants can be expressed in terms of cross-ratios but this does not depricate other invariants as they are still beneficial for the understanding of the projective geometry. This article provides a generalisation of the cross-ratio to point configurations in higher dimensional projective spaces. The central insight is that the ratio of barycentric coordinates is projectively invariant. This article discusses the properties of the invariant and shows applications to the estimation of the camera position and 3D point reconstruction.