Differential Equation of the Loxodrome on a Rotational Surface

The loxodrome is a curve fairly often used in marine surface navigation primarily because it does not require a change of course.On the Mercator projection (standard marine nautical chart) the loxodrome is shown as a straight line , on Earth conceived as a sphere it has the form of a spiral curve , while on the Earth shaped as a rotational elipsoid it takes the form of an irregular curve.In this paper the rotational surface (S) is given by parametrisation and the differential equation of the loxodrome on the rotational surface (S) determined in this way is derived by the application of differential geometry thereby defining at the same time the metric of such rotational surface (S) , what is very convinient for use in marine applications.