Keplerian trigonometry

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