Helmstetter Formula and Rigid Motions with CLIFFORD

Abstract

CLIFFORD is a Maple package for symbolic computations in Clifford algebras Cl(B) of an arbitrary symbolic or numeric bilinear form B. The purpose of this paper is to show usability and power of CLIFFORD when performing computer-based proofs and explorations of mathematical aspects of Clifford algebras and their applications. It is intended as an invitation to engineers, computer scientists, and robotics to use Clifford algebra methods as opposed to coordinate/matrix methods. CLIFFORD has been designed as a tool to promote and facilitate explorative mathematics among non Clifford-algebra specialists. As an example of the power of CLIFFORD, we restate a formula due to Helmstetter which relates the product in Cl(g), the Clifford algebra of the symmetric part of B, to the product in Cl(B). Then, with CLIFFORD, we prove it in dimension 3. Clifford algebras of a degenerate quadratic form provide a convenient tool with which to study groups of rigid motions in ℝ3. Using CLIFFORD we will actually explicitly describe all elements of Pin(3) and Spin(3). Rotations in ℝ3 can then be generated by unit quaternions realized as even elements in Cl 0,3 + Simple computations using quaternions are then performed with CLIFFORD. Throughout this paper we illustrate actual CLIFFORD commands and steps undertaken to solve the problems.