Download 76. The Mercator Projection.

76. The Mercator Projection.
Draw a conformal map (of the globe or sphere) on a rectangular grid.
The Mercator map, which is equally important in both geography and nautical science,
was the creation of Gerhard Kremer (1512-1594), known also as Mercator.
We begin by representing the equator as a horizontal segment AB, whose length
represents the length of the global equator (2=). Divide AB into 360 equal parts and erect
perpendicular (vertical) lines at the dividing points, thereby obtaining the map meridians.
The latitude parallel on the map that corresponds to the global latitude of I is a line parallel
to AB whose distance o from the map equator (AB) is called the exaggerated latitude. The
essence of the problem thus consists of representing o as a function of I.
M
exaggerated latitude
ϕ
1
A
B
In order to solve this problem, we will compare the Mercator map with the Hipparchus or
stereographic map (No. 75) which is also conformal. In the latter, we assume that the north
pole is the center of the projection, the image (map) plane $ is the plane through the
equator (and thus fixed pointwise by the projection), and that the sphere has radius 1.
On the Mercator map partition the vertical line segment of length o from the equator AB
to the Mercator latitude into n equal parts, called 1, 2, . . . , n " 1, n corresponding to geographic
latitudes I 1 , I 2 , . . . , I n"1 , I n I. Consider two parallel map meridians l U1 and l U2
corresponding to geographic meridians l 1 and l 2 , whose difference measured in radians is
/ l 2 " l 1 we will make very small. By this means we obtain a map partition of very small
congruent / by on rectangles.
1
l 1 ' l 2'
k+1
k
A
equator
B
ε
We now do the same on the stereographic map, i.e, we draw the
1. concentric map latitudes corresponding to I 1 , I 2 , . . . , I n"1 , I n I and let their radii
be r 1 , r 2 , . . . , r n r respectively,
2. map meridians l UU1 and l UU2 corresponding to the longitudes l 1 and l 2 . We can think
of l UUi as the radii from the center of the concentric latitude circles in 1.
l 1''
r1
l 2''
r2
ε
equator
O
1
In this manner we obtain a partition of the sector between l UU1 and l UU2 out to r into very
small regions, each of which is approximately rectangular if n is sufficiently large. (/ is the
same on both maps since the unit circle is fixed by the stereographic projection.)
Now consider the region on the stereographic map between that latitudinal circles of
radii r k and r k1 . Its base had length r k / because the central angle remains constant. Since
the Mercator projection (actually its inverse) is to be conformal and the stereographic
projection is conformal, the composite map (via the sphere) from the Mercator map to the
stereographic map is conformal, and it follows that the altitude of this small region on the
stereographic map is approximately r k on or r k1 " r k X r k on or
2
r k1 X r k 1 o
n .
I
r 0 1 and r n r tan 45 ( 2 , the latter from No. 75. Then by repeated substitution
n
2
r X r n"1 Ÿ1 on X r n"2 Ÿ1 on X. . . , we arrive at r X Ÿ1 on where the approximation
o n
o
improves as n v .. By No. 12 (or L’Hopital’s rule), lim
Ÿ1 n e , and we conclude that
nv.
I
r e o . Then o ln r ln tan 45 ( 2 , giving the exaggerated latitude as a function of
geographic latitude.
Note 1. For drawing purposes, the image of a point on the earth with longitude 5 and
latitude I maps to a point on Mercator’s map a distance of 5 from the zero (prime)
I
meridian, and a distance of ln tan =4 2 from the map equatior (line AB above).
Here
€ 5 and I are radian measure of angles, and
€ the radius of the earth is taken as the unit length.
Note 2. Mercator left no writings to indicate how he determined the spacing between the
lines of latitude. He died before the invention of Calculus, so it is unlikely he used
calculus methods.
Note 3. See also Trigonometric Delights by Eli Maor, Princeton U. Press, 2002, Chapter
13 for more on mapmaking.
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