Using Geometric Algebra for Navigation in Riemannian and Hard Disc Space

A “vector ” in 3D computer graphics is commonly under-stood as a triplet of three floating point numbers, eventually equipped with a set of functions operating on them. This hides the fact that there are actually different kinds of vec-tors, each of them with different algebraic properties and consequently different sets of functions. Differential Geome-try (DG) and Geometric Algebra (GA) are the appropriate mathematical theories to describe these different types of “vectors”. They consistently define the proper set of opera-tions attached to each class of “floating point triplet ” and al-low to derive what meta-information is required to uniquely identify a specific type of vector in addition to its purely numerical values. We shortly review the various types of “vectors ” in 3D computer graphics, their relations to rota-tions and quaternions, and connect these to the terminology of co-vectors and bi-vectors in DG and GA. Not only in 3D, but also in 4D, the elegant formulations of GA yield to more clarity, which will be demonstrated on behalf of the use of bi-quaternions in relativity, allowing for instance a more insightful formulation to determine the Newman-Penrose pseudo scalars from the Weyl tensor. 1