Articles
You can find here the list of articles and papers we have produced.
Formalizing Screw Theory with 3D Geometric Algebra
Citation Loris Delafosse (2024) Phys. Scr. 99 056102 doi:10.1088/1402-4896/ad3787
This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler’s First and Second Laws is provided to illustrate the use of this formalism.
A New Approach to Screw Theory using Geometric Algebra
Citation Loris Delafosse (2023) hal-04177875v3
Since it was first developed by Sir Robert S. Ball at the end of the XIXth century, the Theory of Screws has known a considerable variety of reformulations, each of them underlining a different interpretation of screws: as geometrical, mechanical or algebraic objects. Beginning with an overview of the main existing formalisms, this article determines what prerequisites must be satisfied for a mathematical theory to represent screws and conciliate their geometric and algebraic aspects in a clear, elegant and pedagogical way. The mathematical framework of Geometric Algebra was precisely designed to reveal the geometrical significance of the algebraic objects of physics. A new formalism for Screw Theory is hence introduced, based on the geometric algebra G(3,0) and intended to generalize the concept of a screw (and therefore the extent of Screw Theory) and conciliate geometrical insight with algebraic efficiency. Moreover, this approach is coordinate-free and origin-independent, which makes it a powerful tool to treat affine geometry. A simple and straightforward description of finite motions appears as a natural feature of this new formulation.