T Modeling 3d Euclidean Geometry 2

he space we live in is well described as 3D Euclidean geometry for most computer graphics applications. Although it would seem straightforward to directly implement this for realistic-image generation and object simulation including their properties , most computer graphics programmers find a more indirect method attractive. That is, we prefer constructing a computational model of the 3D geometry to implement. This often improves programs in structure and efficiency. For example, the widespread use of homogeneous coordinates, which uses a 4D linear algebra to perform some of the 3D Euclidean geometry. But the vectors from 3D linear algebra also have their uses, as do quaternions— which appear to live in a 4D complex algebra—and even Plücker coordinates—which describe 3D lines using an unfamiliar 6D space. In fact, the choice of models is getting confusing. Explanatory papers often suggest different algebras for different aspects of geometry. 1-4 Our programs typically reflect this approach. To many, the recently discovered geometric algebra appears to be just one more possibility, but there is another way of looking at this scheme. 5-7 Instead, in geometric algebra we finally have a framework containing all these modeling options and approaches in an organized manner. This approach streamlines applications by assigning various tricks such as quaternions and Plücker coordinates to a proper geometric algebra of appropriate real, interpretable vector spaces. This article compares five models of 3D Euclidean geometry—not theoretically, but by demonstrating how to implement a simple recursive ray tracer in each of them. It's meant as a tangible case study of the profitability of choosing an appropriate model, discussing the trade-offs between elegance and performance for this particular application. This article acts as a practical sequel to two tutorials on geometric algebra previously published in IEEE CG&A, and we frequently reference those tutorials. The models we compare are ■ 3D linear algebra (3D LA); ■ 3D geometric algebra (3D GA), which naturally absorbs the quaternions into 3D real geometry; ■ 4D linear algebra (4D LA), that is, the familiar homogeneous coordinates; ■ 4D geometric algebra (4D GA), implements the homogeneous model, which naturally absorbs Plück-er coordinates of lines and planes into homogeneous computations; and ■ 5D geometric algebra (5D GA), which implements the conformal model. This model gives coordinates to circles and spheres and provides the most compact expressions for 3D Euclidean computations known to date. We picked both 3D LA and 4D LA because we wanted …