Abstract
We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.
Highlights
We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori
We prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut, and we generate it by means of a composite rotor in conformal geometric algebra in Sect
In [5] we showed how 3D conformal motions can be generated as the orbit of a rotor that is the exponential of the sum of two commuting 2-blades of CGA
Summary
By Consa [3] and Hestenes [8], models of the electron are proposed that involve it moving rapidly internally around a torus. The parametrization of that motion in both papers is done by the two uniformly increasing angles evolving from the usual Euclidean image of generating a torus as a circle moving around a circle. A second physical situation in which orbits on a torus occur is in the knotted solutions to Maxwell’s equation in vacuum, in which the Poynting vector of the electromagnetic field lies on a torus knot [2]. This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECCUNICAMP, Campinas, Brazil, edited by Sebastia Xambo-Descamps and Carlile Lavor. ∗Corresponding author
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