Abstract
There is a requirement to be able to represent three-dimensional objects and their transforms in many applications, including computer graphics and mechanism and machine design. A geometric algebra is constructed which can model three-dimensional geometry and rigid-body transforms. The representation is exact since the square of one of the basis vectors is treated symbolically as being infinite. The non-zero, even-grade elements of the algebra represent precisely all rigid-body transforms. By allowing the transform to vary, smooth motions are obtained. This can be achieved using Bézier and B-spline combinations of even-grade elements.
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Notes
- 1.
Editorial note: However, this representation is explored in detail in Chap. 15 (this volume).
- 2.
Editorial note: In its basic definitions of geometric algebra, this chapter repeats some of the elementary constructions given in given in the tutorial (Chap. 21) in this volume. Since the anomalous element e 0 changes some of the details crucially, we kept this re-explanation.
- 3.
Editorial note: This somewhat unusual construction may find its motivation in a limiting procedure from curved spaces to flat Euclidean space, see Chap. 18.
- 4.
Editorial note: This deviates from the usage of the term “pseudoscalar” in an n-D algebra elsewhere in this book, where it is restricted to pure n-blades.
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Acknowledgements
The work reported in the paper was carried within the Innovative Design and Manufacturing Research Centre at the University of Bath, and the second author is funded by a studentship provided via the Centre. The Centre is funded by the Engineering and Physical Sciences Research Council (EPSRC), and this support is gratefully acknowledged.
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Mullineux, G., Simpson, L. (2011). Rigid-Body Transforms Using Symbolic Infinitesimals. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_17
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