giving the coordinates of points on a rhumb line with course through .

Inverting this relation, to find the geodetic latitude given
the longitude , can most readily be done iteratively, using:

(9) |

Combining equations (6) and (7)
with constant gives us a differential equation for the
arc-length along the rhumb line:

(10) |

(12) |

Formally, can be expressed in terms of the elliptic
integral of the second kind by[1]

(13) |

Expanding to , is approximately given
by[4]:

(14) |

Along a parallel, which is an E-W rhumb line, Eqns. (8)
and (11) diverge,
but since is constant, we have from Eqn. (6):

(15) |

A map with longitude as the x-axis and as the y-axis has a Mercator[4] projection (with the equator as the standard parallel) on which rhumb lines plot as straight lines with the correct azimuth.

2002-03-21