Chapter 21 - Fundamental Algorithms for Nonlinear Products

Abstract

This chapter discusses the implementation of nonlinear geometric algebra operations. The exponentials of general bivectors generate the continuous motions in geometry, and only in fewer than four dimensions are bivectors always 2-blades. In the important conformal model, rigid body motions are exponentials of bivectors that are not 2-blades. At times, it may be useful to have an algorithm that classifies a multivector as either a blade, a versor, or a nonversor such as classification is useful as a check of interactive input or to verify whether and how a result could be displayed geometrically. Yet testing whether a multivector is a versor or a blade is nontrivial in the additive representation. A blade test requiring that the multivector is of a single grade is insufficient. The classification algorithm performs this test correctly. It is one algorithm that can be used either for the versor test or for the blade test. It is natural that these tests should structurally be very similar, for all invertible blades are also versors. The determination of whether a multivector V is a versor clearly depends on the precise properties of the metric. So for a versor test, one has to run the algorithm in the actual metric of the algebra of the versor.