Global Positioning System Reference
In-Depth Information
AZIMUTHAL PROJECTIONS
The idea underlying mathematical projections from a point on a sphere to
a point on a plane is illustrated by the arrow in figure 4.5a. The geometry is
much simpler to grasp than the algebra.
Let us imagine that we shoot an arrow out from the center of the globe
shown in figure 4.5a. This arrow will intercept the surface of the sphere at a
particular point on its southern hemisphere and then intercept the plane.
We say that the point on the sphere has been projected onto the plane. It is
obvious from the illustration that every single point on the southern half of
the sphere can be projected in this way onto some point of the plane. In an
azimuthal projection (which is the kind of projection that Al-Biruni con-
cerned himself with), the projection is a systematic way of sending every
point on the globe (that we want to map) onto a point on the flat plane, or
sheet of paper.
There is nothing unique about the map center point chosen for the
azimuthal projection shown in figure 4.5a. I happened to choose the South
Pole, but I could have chosen any other position on the globe as the center
point. Thus, for example, if we wanted to map the Arctic regions, we might
place the flat plane on top of the sphere, so that the center point—the point
of contact between sphere and plane—is the North Pole. For such a map,
distortions will be small near the North Pole and will increase further
south. In the case of projections from either the South or the North Pole
center points, the equator is infinitely far away, so this type of projection is
useful only for latitudes near the pole.
Projections from a center on the globe onto a plane are called gnomonic ,
and gnomonic maps have certain advantages. You can see that, for the
gnomonic projection of figure 4.5a, Antarctica will be well represented.
Near the point of contact, the projection is faithful; distortions increase as
distance away from the point of contact increases. Gnomonic projections
are good for depicting areas like Antarctica that spread significantly both
north-south and east-west.
In addition, gnomonic projections turn great circles into straight lines.
It is well known that the shortest distance between two points on a sphere
is the line formed by the great circle that connects them. 10 Thus, on maps
formed from gnomonic projections, the shortest distance between the
10. Imagine two points, A and B , marked on the surface of a sphere. These points,
plus the center of the sphere, define a plane. Imagine this plane cutting the sphere. The
 
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