Abstract
Looking at the Frenet-Serret equations from the viewpoint of dynamical systems, one can prove that when the curvature of a plane curve is given as a function of the radius, the problem of reconstructing this curve is reducible to quadratures. Additionally, two different integration procedures are presented. These methods are illustrated first via the famous lemniscate of Bernoulli, which is immediately related to the Euler elastica. Relying on the new formalism, the Sturm spirals and their generalizations, the Serret curves (which have a mechanical origin) and their generalizations are parametrized explicitly. The results on the Serret curves are original, as their description up to now has been purely abstract. Finally, the same technique is applied to the Cassinian ovals, and in this way, one concludes with their alternative parameterizations.
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Mladenov, I.M., Hadzhilazova, M. (2017). Planar Curves Whose Curvature Depends Only on the Distance From a Fixed Point. In: The Many Faces of Elastica. Forum for Interdisciplinary Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-61244-7_2
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