Multivector

Abstract: "Introduction to Theoretical Kinematics" provides a uniform presentation of the mathematical foundations required for studying the movement of a kinematic chain that makes up robot arms, mechanical hands, walking machines, and similar mechanisms It is a concise and readable introduction that takes a more modern approach than other kinematics texts and introduces several useful derivations that are new to the literature The author employees a unique format, highlighting the similarity of the mathematical results for planar, spherical, and spatial cases by studying them all in each chapter rather than as separate topics For the first time, he applies to kinematic theory two tools of modern mathematics--the theory of multivectors and the theory of Clifford algebras--that serve to clarify the seemingly arbitrary nature of the construction of screws and dual quaternions The first two chapters formulate the matrices that represent planar, spherical, and spatial displacements and examine a continuous set of displacements which define a continuous movement of a body, introducing the "tangent operator" Chapter 3 focuses on the tangent operators of spatial motion as they are reassembled into six-dimensional vectors or screws, placing these in the modern setting of multivector algebra Clifford algebras are used in chapter 4 to unify the construction of various hypercomplex "quaternion" numbers Chapter 5 presents the elementary formulas that compute the degrees of freedom, or mobility, of kinematic chains, and chapter 6 defines the structure equations of these chains in terms of matrix transformations The last chapter computes the quaternion form ofthe structure equations for ten specific mechanisms These equations define parameterized manifolds in the Clifford algebras, or "image spaces," associated with planar, spherical, and spatial displacements McCarthy reveals a particularly interesting result by showing that these parameters can be mathematically manipulated to yield hyperboloids or intersections of hyperboloids

Abstract: We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.