Symmetry detection of rational space curves

Abstract

We present a novel, deterministic, and efficient method to detect whether a given rational space curve C is symmetric. The method combines two ideas. On one hand in a similar way to [1], [2], if C is symmetric then the symmetry provides a second parametrization of the curve; furthermore, whenever the first parametrization is proper, i.e. injective except for finitely many parameter values, the latter is also proper and both are related by means of a Mobius transformation [3] that completely determines the symmetry. On the other hand, if C is symmetric then the curvature and torsion of C at corresponding points must coincide. By putting together these two ideas we can give an algorithm to directly find the Mobius transformations defining the symmetries of the curve. From here we can compute these symmetries and its characteristic elements (symmetry axes, symmetry planes, etc.) This completes and improves on an earlier method addressing a similar problem [3]. Keywords Symmetry Detection, Space Curves, Rational Curves References [1] Alcazar J.G. (2014), Efficient detection of symmetries of polynomially parametrized curves, Journal of Computational and Applied Mathematics vol. 255, pp. 715–724. [2] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Similarity of Plane Rational Plane Curves, Journal of Computational and Applied Mathematics, Vol. 269, pp. 1–13. [3] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Symmetries of Rational Plane and Space Curves, to appear in Computer Aided Geometric Design.