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Firstly, we introduce a new frame and a new curvature function for a fixed parametrization r of a plane curve C. This new frame is called Jacobi since it involves the rotation with the first two Jacobi elliptic functions of the usual Frenet frame. The Jacobi-curvature involves only the third Jacobi elliptic function w and is computed for some remarkable examples; the inequalities satisfied by w imply inequalities for the Jacobi-curvature. Secondly, we introduce a whole family of new parametrizations \(r_{\rho }\) for C with \(r=r_{\rho =0}\) . The expression of \(r_{\rho }\) involves an integral containing the curvature function k of r, and all \(r_{\rho }\) have the same curvature.

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The Jacobi Geometry of Plane Parametrized Curves and Associated Inequalities

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  • 发表日期：

    2026年

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  • Mircea Crasmareanu

摘要

Firstly, we introduce a new frame and a new curvature function for a fixed parametrization r of a plane curve C. This new frame is called Jacobi since it involves the rotation with the first two Jacobi elliptic functions of the usual Frenet frame. The Jacobi-curvature involves only the third Jacobi elliptic function w and is computed for some remarkable examples; the inequalities satisfied by w imply inequalities for the Jacobi-curvature. Secondly, we introduce a whole family of new parametrizations \(r_{\rho }\) for C with \(r=r_{\rho =0}\) . The expression of \(r_{\rho }\) involves an integral containing the curvature function k of r, and all \(r_{\rho }\) have the same curvature.