Dual quaternion representation of points, lines and planes

Background. The bulk of the work on dual quaternions is devoted to their application to describe helical motion. Little attention is paid to the representation of points, lines, and planes (primitives) using them. Purpose. It is necessary to consistently present the dual quaternion theory of the representation of primitives and refine the mathematical formalism. Method. It uses the algebra of dual numbers, quaternions and dual quaternions, as well as elements of the theory of screws and sliding vectors. Results. Formulas have been obtained and systematized that use exclusively dual quaternionic operations and notation to solve standard problems of three-dimensional geometry. Conclusions. Dual quaternions can serve as a full-fledged formalism for the algebraic representation of a three-dimensional projective space.