The methods of three—dimensional vector analysis — dot and cross products, and the differential operators such as the gradient, divergence, and curl — are fundamental to modern science. It is not commonly appreciated, however, that these methods are actually a rather late development in mathematics: they devolved from a more—sophisticated theory, the algebra of the quaternions. We present here a brief introduction to quaternions and their use in describing an important set of geometrical transformations — namely, rotations in ℝ3. As we shall see in Chap. 22, this property of quaternions proves invaluable in the formulation of a sufficient—and—necessary characterization for Pythagorean hodographs in ℝ3, that is invariant under arbitrary spatial rotations.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Quaternions. In: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Geometry and Computing, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73398-0_5
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DOI: https://doi.org/10.1007/978-3-540-73398-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73397-3
Online ISBN: 978-3-540-73398-0
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