Abstract
We discuss a fully covariant Lagrangian-based description of 3D rigid body motion, employing spinors in 5D conformal space. The use of this space enables the translational and rotational degrees of freedom of the body to be expressed via a unified rotor structure, and the equations of motion in terms of a generalised ‘moment of inertia tensor’ are given. The development includes the effects of external forces and torques on the body. To illustrate its practical applications, we give a brief overview of a prototype multi-rigid-body physics engine implemented using 5D conformal objects as the variables.
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Notes
- 1.
Editorial note: This notation is used in [3]. Compared to the notation in the tutorial appendix (Chap. 21), Eq. (21.1), by setting λ=1, we find that n=n ∞ and \(\bar{n}= -2 n_{o}\). Correspondingly, \(e =\frac{1}{2}(n+\bar{n}) = -n_{o}+ \frac{1}{2}n_{\infty}= -\sigma_{+}\) and \(\bar{e}= \frac{1}{2}(n-\bar{n}) =n_{o}+ \frac{1}{2}n_{\infty}= \sigma_{-}\). Note that \(\bar{n}\cdot n = 2\) corresponds to n o ⋅n ∞=−1. As a compromise between the notation in this book and [3] that avoids awkward factors, in this chapter we will use \(\bar{n}\) but replace n by n ∞.
- 2.
See also [2], which contains further details.
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© 2011 Springer-Verlag London Limited
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Lasenby, A., Lasenby, R., Doran, C. (2011). Rigid Body Dynamics and Conformal Geometric Algebra. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_1
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DOI: https://doi.org/10.1007/978-0-85729-811-9_1
Publisher Name: Springer, London
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