A Structural Approach to Gudermannian Functions

The Gudermannian function relates the circular angle to the hyperbolic one when their cosines are reciprocal. Whereas both such angles are halved areas of circular and hyperbolic sectors, it is natural to develop similar considerations within the study of a class of curves images of maps with constant areal speed. After a brief exposition of some use of the Gudermannian in applied sciences, we proceed to illustrate the class of curves, called \emph{Keplerian curves}, which can be parametrised by a map $\mmm = (\cos_{\mmm}, \sin_{\mmm})$ whose areal speed is 1. In the next Sections, after a detailed study of p-circular and hyperbolic Fermat curves $\crv F_p$ and $\crv F^*_p$, we define the \emph{p-Gudermannian} as the primitive of the derivative of the p-hyperbolic sine divided by the square of the p-hyperbolic cosine: all the analogues of the classical identities are proven. Having realized that such curves correspond to each other using a homology, we extend our study to a wide class of Keplerian curves and their homologues; once again, defined the Gudermannian in an identical manner, all the analogues of classical identities subsist. Below, three examples are detailed. The last paragraph further extends this consideration, eliminating the hypothesis that the curves are parametrised by maps with areal speed 1. The Appendix illustrates integrating techniques for systems defining the Fermat curves, and the determination of the inverse of their tangent function.