Funzioni iperboliche della latitudine isometrica in termini delle funzioni circolari della latitudine geodetica

This paper puts forward approximate closed expressions of both hyperbolic tangent function (tanh u) and secant (1/cosh u) of the u ellipsoidal isometric latitude in terms of circular functions of the geodetic latitude and with the even powers of the first eccentricity up to the 12th grade. Expressions are deduced showing the equality of these u hyperbolic functions with the functions of the conformal ellipsoidal latitude, and enclosing them in the Taylor’s series expansion of the latter expression of the difference between the conformal latitude and the corresponding geodetic latitude. We report numerical examples in order to verify that the accuracy of the expressions is driven up to the order of double machine precision eps with which the numerical data in “Floating Point” using electronic computers and various program-ming and computing languages are represented and processed. The mixed polynomial form of expressions, with a low computational cost, will make possible, in a later work, their inversions in order to deduce approximate and accurate closed developments (up to double precision eps) in terms of geodetic latitude circular functions of the corresponding u hyperbolic functions; these inversions and further processing works last goal will be a new method to calculate Gaussian cartographic coordinates, with an adequate accuracy for high longitude in geographic time zones.