Generalised Complex Numbers in Mechanics

Abstract

Three types of generalised complex number provide concise representations for spatial points and transformations useful in geometry and mechanics. The most familiar type is the ordinary complex number, $a + ib\;(i^{ 2} = - 1)$, used to represent stretch-rotations about a point in 2D space. A second type of generalised complex number is the dual number, $a + \varepsilon b\;(\varepsilon^{ 2} = 0)$, used to represent shear transformations and inversions in 2D space. The third type of generalised complex number is the double number, $a + jb\;(j^{ 2} = + 1)$, used to represent boosts (simple Lorentz transformations) in 2D space-time. Each of the three types may be expressed in various explicit forms (Gaussian, ordered pair, matrix, parametric, exponential) for algebraic convenience, computational efficiency, and so on. Combining dual numbers with vectors and quaternions provides an efficient representation of general spatial screw displacements in 3D space.