Loxodrome

loxodrome

A loxodrome.

Loxodrome (shown as a tube, Frenet frame)

A loxodrome (also called a rhumb line) is a route that a boat
would take if it kept a constant compass heading (so that on a
Mercator projection it is simply a straight line).. To be more formal,
a loxodrome is a path that lies on the unit sphere in R^3 and that
makes a constant angle with the great circles of longitude. Thus the
loxodromes are analogous to the logarithmic spirals in the complex
plane, which make a constant angle with the rays through the
origin. In fact, since sterographic projection from the complex plane
to the unit sphere is conformal (and in particular angle-preserving)
and since the stereographic projection of the radial lines in the
plane are the circles of longitude, it follows that the loxodromes are
given by stereographically projecting the logarithmic spirals. On the
other hand, since the exponential map of the complex plane to itself
is conformal and maps the lines parallel to the real axis to the
radial lines, it follows that the logarithmic spirals are just the
images under the exponential map of straight lines, i.e., the images
of the maps t ---> aa*t + i t .

 Hence we can define the loxodromes parametrically by

       t --->  StereographicProjection(Exponential(aa*t + i t))