Global Positioning System Reference
In-Depth Information
the earth's circumference at the equator. Now scale the map down—say, by a
factor of 6,336,000—and we have a Mercator projection map of a sensible size (in
this case, 1 inch represents 100 miles). Note from figure A1(b) one disturbing
feature of the Mercator projection: the North Pole is mapped onto a line (the top
edge of the map), not onto a single point. The South Pole is similarly mapped onto
the bottom edge of the map. This reflects the distortion of Mercator maps at high
latitudes.
So much for the geometry: now for an algebraic analysis of Mercator projec-
tions that provides us with the mapping formulas and also shows that this projec-
tion preserves angles. We want to project a location on the globe, say at latitude f
and longitude l, onto a position (
x,y
) on the map. Mapping longitude is easy, as we
have seen: mathematically, we say that
x
= l, meaning that the horizontal coordi-
nate on the map is just the angle of longitude. Mapping latitude is trickier. Figure
A1(a) shows a small square on the globe that is mapped onto the cylinder. To
preserve angles, the square must be mapped to a square so that the local shape of
things is maintained. Say the center of the square is at latitude f on the globe; you
can see from figure A1(b) that if the globe is of radius
R
, then the parallel at
latitude f is a circle of radius
R
cos f. On the map, parallels are horizontal lines of
equal length (and meridians are vertical lines of equal length) because the map is
rectangular. Thus, parallels at latitude f on the globe must be stretched by a factor
of 1/cos f when they appear on the map.
Let us say that the small square of figure A1(a) has sides that are along merid-
ians and parallels. If the square extends over a small angular range of longitudes,
say
d
f, then the length of each side of the square is
R d
f. But the parallels at
latitude f are stretched, and so, on the map, the square is of width
R d
f/cos f.
(This stretching must occur if the square is to remain a square when mapped.) I
will denote this small width
dy
. Thus,
dy/d
f
=
R/cos
f. This equation is a
di√erential equation for the vertical coordinate (
y
) of a point on the map. It can be
integrated (solved) to yield
y
=
R
ln[tan (
1
⁄
2
f+
p
⁄
4
)]. Thus, the formula for a
Mercator projection is as follows:
2
(
x,y
) = (l, ln[tan(
1
⁄
2
f+
p
⁄
4
)]).
(A1)
Note that the analysis presented here has made use of logarithms and of calculus.
Neither of these mathematical ideas was known when Edward Wright provided
the first mathematical analysis of the Mercator projection in 1599. The following
century brought more elegant analyses, by James Gregory and Isaac Barrow.
In figure A2 you can see what a rhumb line looks like on a linear scale (no
projection) and on a Mercator projection map. Here, I assume that a course
is followed that cuts meridians at 45\. The bearing must change on the linear
2. The formula for an oblique Mercator projection is more complicated. See, for example,
Eric W. Weisstein, ''Mercator Projection,'' at the website MathWorld—A Wolfram Web Resource,
