Chapter 18 - Implementation Issues

Abstract

An efficient geometric algebra implementation uses symbolic manipulations only to bootstrap the implementation, and not during actual run-time computations. All multivectors can be decomposed as a sum of basis blades. A k-blade only contains elements of grade k, and therefore has many zero entries in its coefficient vector, making the representation by 2n elements rather wasteful. This effect becomes stronger in higher dimensions. By taking the sum of basis blade implementation as the core approach, a geometric algebra implementation naturally splits into four levels. The levels include selecting the basis blades and implementing the basic operations on them, implementing linear operations for multivectors, implementing nonlinear operations on multivectors, and the application level. Geometric algebra contains some important operations that are elementary but not linear or distributive. As a consequence, the approach used in the middle-level implementation cannot be used here. These operations are implemented using specialized algorithms, which are actually largely independent of the middle-level implementation. The linearity and associativity of geometric algebra suggests representing the whole algebra with its geometric product by single matrix algebra: all elements become represented as 2n × 2n matrices, and the geometric product is represented as the matrix product.