10 – The vector space model: The algebra of directions

Publisher Summary When geometric algebra was developed in the first part of this book, it illustrated the principles with pictures in which vectors are represented as arrows at the origin, bi-vectors as area elements at the origin, and so on. This is the purest way to show the geometric properties corresponding to the algebra. The examples show this algebra of the mathematical vector space Rn can already be used to model useful aspects of Euclidean geometry, for it is the algebra of directions of n-dimensional Euclidean space. Some more properties of this model are explored in this chapter, with special emphasis on computations with directions in 2-D and 3-D. Most topics are illustrated with programming exercises at the end of the chapter. First, this chapter shows how the vector space model can be used to derive fundamental results in the mathematics of angular relationships. It gives the basic laws of trigonometry in the plane and in space, and shows how rotors can be used to label and classify the crystallographic point groups. Then it computes with 3-D rotations in their rotor representation, establishing some straightforward techniques to construct a rotor from a given geometrical situation, either deterministically or in an optimal estimation procedure. The logarithm of a 3-D rotor enables it to interpolate rotations. Finally, external camera calibration is given an application to show how the vector space model can mix directional and locational aspects.