Graphics Programs Reference
In-Depth Information
4.14 Map Projections
According to [wikipedia 05], the ancients generally believed that the Earth is flat, but by
the time of Pliny the Elder (the first century a.d.) its spherical shape had already been
generally acknowledged. Many scientists and cartographers strongly
believed in a round Earth, which led Columbus to risk his life, in
1492, trying to reach Japan by going west. Today, most of us believe
that the Earth is a sphere (more accurately, a spheroid, since it is
slightly flattened at the poles), but there is still a persistent minority
that believes otherwise (see [flatearthsociety 05] for an interesting
example). Regardless of anyone's beliefs or convictions, our aim in
this section is to describe the chief methods for projecting a sphere
on a flat plane.
The equation of a sphere of radius R centered on the origin is x 2 + y 2 + z 2 = R 2 .
This is a special case of the ellipsoid
x 2
a 2
+ y 2
b 2
+ z 2
c 2
x 2 + y 2
a 2
+ z 2
c 2
= 1 and the spheroid
=1 .
The sphere may also be described in spherical coordinates as (compare with Equa-
tion (4.3))
x = R cos θ sin φ,
y = R sin θ sin φ,
z = R cos φ,
where θ is the longitude (or azimuthal coordinate), which varies from 0 to 2 π ,and φ is
the colatitude (or polar coordinate, the latitude measured from the north pole), which
varies from 0 to π .
Exercise 4.15: Look up (in a dictionary or on the Internet) the definitions of latitude,
longitude, antipode, and graticule.
First, let's convince ourselves that projecting a sphere on a plane is a practical,
important problem. After all, we have globes of the Earth, so perhaps we don't need
maps as well. A globe is a true representation of the Earth's surface because it maintains
the true scale of areas and distances and shows the correct shapes of regions and the
correct angles between lines. However, its use is limited. Only one half of a globe can be
viewed at a time. Normally, the size or scale of a globe is too small to show the details
of a small region, such as a town, and large globes are expensive and di cult to handle.
Maps, on the other hand, are much more versatile. A flat map is portable because it can
be rolled or folded. It is easy to print maps in large quantities and store them digitally
in a computer where they can be edited, processed, displayed, and printed.
There is vast literature on map projections, map making, and cartographic tech-
nique. Distilling it to just four items yields, in the opinion of this author, [Pearson 90]
(very mathematical), [Snyder 87], [Snyder 93], and [Furuti 97].
The main problem with mapping a globe is the fact that a sphere is an undevel-
opable surface. Any attempt to open, unfold, or unroll a sphere to lie flat results in
stretching and deforming it in some way. (This is also mentioned in Section 4.6.) Thus,
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