A bit better: Variants of duality in geometric algebras with degenerate metrics

Multiplication by the pseudoscalar $\mathbf{I}$ has been traditionally used in geometric algebra to perform non-metric operations such as calculating coordinates and the regressive product. In algebras with degenerate metrics, such as euclidean PGA $P(\mathbb{R}^*_{3,0,1})$, this approach breaks down, leading to a search for non-metric forms of duality. The article compares the dual coordinate map $J: G \rightarrow G^*$, a double algebra duality, and Hodge duality $H: G \rightarrow G$, a single algebra duality for this purpose. While the two maps are computationally identical, only $J$ is coordinate-free and provides direct support for geometric duality, whereby every geometric primitive appears twice, once as a point-based and once as a plane-based form, an essential feature not only of projective geometry but also of euclidean kinematics and dynamics. Our analysis concludes with a proposed duality-neutral software implementation, requiring a single bit field per multi-vector.Comment: 23 pages, 5 figures, 2 table

Calculating the Rotor Between Conformal Objects

Abstract: In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects