Taking the hint from usual parametrization of circle and hyperbola, and inspired by the pathwork initiated by Cayley and Dixon for the parametrization of the “Fermat” elliptic curve \(x^3+y^3=1\), we develop an axiomatic study of what we call “Keplerian maps”, that is, functions \({{\,\mathrm{{\mathbf {m}}}\,}}(\kappa )\) mapping a real interval to a planar curve, whose variable \(\kappa \) measures twice the signed area swept out by the O-ray when moving from 0 to \(\kappa \). Then, given a characterization of k-curves, the images of such maps, we show how to recover the k-map of a given parametric or algebraic k-curve, by means of suitable differential problems.

借鉴圆和双曲线的常规参数化的提示，并受Cayley和Dixon发起的“费马”椭圆曲线\（x ^ 3 + y ^ 3 = 1 \）的参数化的启发，我们进行了公理研究所谓的“ Keplerian映射”，即函数\（{{\\ mathrm {{\ mathbf {m}}} \，}}（\ kappa} \）将实际区间映射到平面曲线，其变量\（\ kappa \）在从0移到\（\ kappa \）时，测量的是O射线扫出的有符号区域的两倍。然后，给定k曲线的特征，这些图的图像，我们展示了如何通过适当的微分问题来恢复给定参数或代数k曲线的k图。