Let us go back to the properties of the Mercator projection:
latitudes and longitudes go to an orthogonal grid, and the
projection is conformal. Conformality means that at any point
the vertical and horizontal stretching are the same. If we
use longitude *L* and height *h* (measured in the same units as
longitude) as coordinates on the map, then the equator is
not stretched at all: it and its image have length
.
But a circle of latitude at latitude has length
on the sphere, and the same length
as the equator in the Mercator projection. It has been stretched
by a factor
For conformality,
the meridians must be stretching by
as they pass through latitude ,
more and more in
higher and higher latitudes. In terms of calculus,
.

In order to apply calculus to the problem it is most
convenient to work entirely in radian measure, and convert
to degrees at the end. In radians, we have two points
on the sphere, one at latitude
,
the other at latitude
,
and
separated by
in longitude.

The height *h* on the map corresponding to latitude is the integral
.
This integral, the bane of
generations of Freshmen, has a useful application! As they learned,
.
So latitude
corresponds to height *h*=.711, and
to *h*=.649.

The function
is invertible; in fact
the formula can be explicitly inverted. First write
as
,
and note that the function is
positive on the sphere so the absolute value signs can
be discarded. Then solve for *x* to get
.
Call this function *G*(*h*).

On the Mercator map, the straight line
from *A*
(*h*=.711, *L*=0) to *B*
(*h*=.649, *L*=1.694) is
.
The rhumb line on the sphere is therefore

The metric on the sphere is , so the length of the rhumb line is

with

Since *G* is the inverse of a function with derivative
,
the
derivative of *G* is
,
so the first term inside the radical is
,
while the second term
is
.
The radical thus
simplifies to

The length integral is now

The substitution

giving the length as . Converting to degrees gives 78.27 degrees, or 4696 nautical miles.

The moral of the story is that the great circle track from the Farallones to Tokyo is 234 nautical miles shorter than the straight line between them on the Mercator map, as we have computed them.

A great circle track plotted on a Mercator chart. From Dutton's "Navigation and Nautical Astronomy", 7th Ed. Reproduced by permission from the U.S. Naval Institute.

Dutton's rhumb line calculation is different from mine, and takes
into account the earth's eccentricity *e* = .082483399. The
tables used in Mercator sailing, and the java applets available on
the web, take this factor into account: instead of
they use

which is not as simple to invert.

- 1. How to get from here to there?
- 2. The rhumb line
- 3. The great circle track
- 4. Great circle sailing
- 5. Calculation of rhumb line distance

© copyright 2000, American Mathematical Society.