Using geometric algebra for 3D linear transformations

Different methods exist for deriving a 3D linear transformation, including vector algebra, geometric algebra, and matrix algebra (Artin, 1991; Hestenes, 1986, Foley et al., 2002). In this paper, we introduce Ron Goldman's unified derivation method in vector algebra (Goldman, 2003) combined with our developments in geometric algebra (Hestenes, 1986; Dorst and Mann, 2002; Mann and Dorst, 2002). Our work includes using a unified method to derive various geometrical transformations, which are different from the current ray-tracing and transformation methods implemented in graphics algorithms and hardware (Foley et al., 2002). We hope that these derivations extend the geometric algebra research and applications to the computer graphics field