Abstract
This chapter introduces basic concepts on the geometry of plane curves, including arc length, curvature and change of parameter. It also contains a presentation of implicit and non-local representations, and discusses special cases of invariant parametrizations, relative to specific groups of transformations.
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- 1.
The general expression of the area of a triangle (A, B, C) is \(|\det (AB, AC)|/2\), half the area of the parallelogram formed by the two vectors.
- 2.
This would give \(Q(z_1) = Q(e_1) = \text {const}\) and \(Q(\lambda _1 z_1) = \lambda _1 Q(z_1) = Q(z_1)\) for all \(\lambda _1>0\), yielding \(Q=0\).
- 3.
To complete the argument, one needs to check that the required conditions are satisfied for the obtained Q; this is indeed the case, although we skip the computation.
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© 2019 Springer-Verlag GmbH Germany, part of Springer Nature
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Younes, L. (2019). Parametrized Plane Curves. In: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol 171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-58496-5_1
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DOI: https://doi.org/10.1007/978-3-662-58496-5_1
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-58495-8
Online ISBN: 978-3-662-58496-5
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