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 x3+ y3= 1 , we develop an axiomatic study of what we call “Keplerian maps”, that is, functions m(κ) mapping a real interval to a planar curve, whose variable κ measures twice the signed area swept out by the O-ray when moving from 0 to κ. 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.
Keplerian trigonometry / Gambini A.; Nicoletti G.; Ritelli D.. - In: MONATSHEFTE FÜR MATHEMATIK. - ISSN 0026-9255. - ELETTRONICO. - 195:1(2021), pp. 55-72. [10.1007/s00605-021-01512-0]
Keplerian trigonometry
Gambini A.;Nicoletti G.;Ritelli D.
2021
Abstract
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 x3+ y3= 1 , we develop an axiomatic study of what we call “Keplerian maps”, that is, functions m(κ) mapping a real interval to a planar curve, whose variable κ measures twice the signed area swept out by the O-ray when moving from 0 to κ. 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.| File | Dimensione | Formato | |
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