Abstract
Plane-based geometric algebra (PGA) offers a way to represent Euclidean motions that is directly built on the primitives of affine geometry, and thus provides a seamless framework for objects and their movement. We show how this universal treatment includes the actual physical motions of objects with mass under forces and torques. PGA unifies the linear and angular aspects compactly, and in a coordinate-free manner; inertia maps become simply additive (without displacement terms). We demonstrate the simple equations and straightforward numerical code that result. We show explicitly how to embed the vector-based concepts of the usual classical Newtonian mechanics into the 3D PGA framework, and why it is advantageous to do so.
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Notes
- 1.
In the dimensionality of PGA, with d Euclidean dimensions and one ‘extra’ dimension, one may recognize the principles of homogeneous coordinates; but the metric aspects are new and essential. The usual projective interpretation of homogeneous coordinates uses no metric—to its detriment.
- 2.
Using distribution of dot over wedge: \(a \cdot (B \wedge C) = (a\cdot B) \wedge C + \widehat{B} \wedge (a \cdot C)\), the hat denoting grade involution, supplying a minus sign to odd-grade elements.
- 3.
Whether an element becomes ‘purely Euclidean’ on its basis (as opposed to requiring an ideal element with factor \(\boldsymbol{\epsilon }\)) depends fully on one’s choice of basis, notably the choice of origin. Being origin-attached, it is a non-geometrical term, only employed to show how the null-elements act algebraically and/or in an implementation. We will encounter this non-geometrical concept ‘purely Euclidean’ again, when reverting to the classical vector-based treatment; but it is otherwise to be avoided.
- 4.
While it is clear in Eq. (17) from its form that the term \(\textbf{p}\cdot C\) is a join line (like Eq. (3.3)) and therefor a dual bivector, the angular part \(\boldsymbol{\epsilon }\wedge \boldsymbol{\ell }\) perhaps appears to be a meet line. However, \(\boldsymbol{\ell }\equiv {I}_c{\boldsymbol{{\omega }}}\) is an axial vector, the Euclidean dual of a Euclidean bivector \(\textbf{L}\equiv \boldsymbol{\ell }\textbf{I}_3\), so that \(\boldsymbol{\epsilon }\wedge \boldsymbol{\ell }= \boldsymbol{\epsilon }\textbf{L}/ \textbf{I}_3 = -\textbf{L}\boldsymbol{\epsilon }\textbf{I}_e = -\textbf{L}{{\mathcal {I}}}= \star {\textbf{L}}\), and therefore indeed a proper dual bivector.
- 5.
The same dual distinctions occur in other 6D frameworks: screws and coscrews (for rate aspects) and wrenches and co-wrenches (for forque aspects) in Screw Theory; displacements and forces in Spatial Vector Algebra, see Sect. 9.
- 6.
In a treatment that places the dual elements in a dual space of k-forms rather than k-vectors, one can evaluate the dual bivector (a 2-form) on the bivector (a 2-vector) to produce a scalar; this is essentially what Gunn does with his J-map in [15].
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Dorst, L., De Keninck, S. (2024). Physical Geometry by Plane-Based Geometric Algebra. In: Araujo Da Silva, D.W.H., Hildenbrand, D., Hitzer, E. (eds) Advanced Computational Applications of Geometric Algebra. ICACGA 2022. Springer Proceedings in Mathematics & Statistics, vol 445. Springer, Cham. https://doi.org/10.1007/978-3-031-55985-3_2
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