Mapmaking - The Science of Navigation - page 109

est time. Yet, it is not di≈cult to appreciate why the projection dominated
cartography and navigation for centuries: straight loxodromes won out
over almost everything else in the eighteenth and nineteenth centuries.
The mathematics of Mercator projection is more complicated than the
simple geometric point projections discussed above, or the so-called ''nor-
mal'' Lambert projection. In this case, I have made an exception to my
general rule of not presenting you with the mathematics of navigation and
have provided a derivation of the Mercator projection in the technical
appendix. You will see how the straight loxodrome property is built in, and
the derivation will give you an idea of the type of math that is involved in
mapping, whatever the projection may be. The message to take away from
the appendix is that a map projection can be mathematically fine-tuned to
cater to a particular need—to minimize a particular distortion (zero angle
distortion in the case of Mercator). The price to be paid is an increase in
the other types of distortion (area and distance, in the case of Mercator).
The transverse Mercator projection rotates the axis of the cylinder 90\ so
that it is perpendicular to the earth's north-south axis. In this way the polar
regions can be accurately represented. In addition, a well-chosen trans-
verse or oblique Mercator projection allows accurate mapping of countries
such as Chile, which stretches along a meridian with little east-west ex-
tent. 11 Another advantage of transverse Mercator maps is that they can be
joined vertically without worrying about changing scale across the maps.
DYMAXION MAPS
Now we turn to the oddball family of projections promised earlier. We have
seen that projections from a sphere onto a plane (which well approximate
projections from the globe onto a paper map) necessarily introduce distor-
tions. We have also seen that these distortions are reduced in regions
where the map (be it a plane, or a sheet rolled into a cylinder or cone) is
close to the globe and is tangential or almost tangential to it. Well, there
exist in mathematics a group of beautiful 3-D constructions, the regular
polyhedra, which can serve to replace the cylinders and cones. True, we
cannot simply construct a polyhedron from a flat piece of paper by curling
it up, as we did with the cylinder and cone: to make a polyhedron, we have
to cut the paper. But it can be done in such as way that the paper is not
stretched and remains in one piece, as we will see.
11. A transverse Mercator projection (also known as the Gauss-Krüger projection ) sets the
cylinder angle perpendicular to the earth's axis, whereas an oblique projection, mentioned
earlier, sets it at some other angle between 0\ and 90\.