Global Positioning System Reference
In-Depth Information
Technical Appendix
In this appendix I break my rule about presenting nonmathematical elucidations of
navigational science in this topic. I made this rule, you may recall, because the math
involved in our subject can get a little hairy and, frankly, quite tedious. However, the
Mercator projection is historically important, and explanations of how the gyrocom-
pass works are often not accurate.
The Mercator Projection
By showing you the mathematical derivation of the Mercator projection, I am
conveying the essence of cartography: projecting points on a sphere onto points on
a plane. Each of the projections we saw in chapter 4 has a di√erent formula, some
of which are much more complicated than the Mercator. Here you have the full
derivation.
Consider figure A1. The Mercator projection is a cylindrical projection, so that
the earth is mapped onto a cylinder that can then be unrolled to form a flat
rectangular map. In figure A1(a) we have wrapped the earth (assumed to be
spherical) with a large piece of paper, which forms a cylinder touching the globe at
the equator. The essence of the Mercator projection is easy to describe. First, each
point on the equator of the globe is already mapped onto a point on the cylinder,
defining the equator on our map. Now let us project point
A
on the globe onto
point
A
% on the cylinder. The large arrow in figure A1(a) shows the projection of
A
to
A
%. Imagine a plane that passes through three points on the globe: the North and
South Poles and the point
A
. This plane will cut the cylinder along a straight line.
The line will be vertical and, when the cylinder is unfolded, will be a meridian of
the map. In this way, points on the globe are easily projected onto the correct
longitude on the cylinder.
What about latitude? The plane cuts the globe as shown in figure A1(a). The
length of the cut from the equator to the point
A
is
s
. On the cylinder, we measure
from the equator northward (in this case, because
A
is north of the equator) a
distance
s
and mark the point
A
% on the cylinder. In other words, the distance on
