ABSTRACT
The authors need a way to represent points, vectors, lines, and planes in 3-dimensions inside the 8-dimensional vector space of dual quaternions. The quaternions are a subspace of the space of dual quaternions, so they could adopt this subspace to represent mass-points and vectors using dual quaternions. With these representations for points and planes in the space of dual quaternions, they can use the dot product to determine when a point P lies on a plane p. They shall now formalize this notion of duality in the algebra of dual quaternions by defining duality for the basis vectors and then extending to arbitrary dual quaternions by linearity. Thus, in general, if then by linearity Therefore for all dual quaternions Moreover, they have the table for duality between mass-points and planes, vectors and normal vectors, and lines and dual lines.
