Mapmaking - The Science of Navigation - page 101

point of contact and any other point on the map is a straight line. This
property makes gnomonic maps useful for plotting airplane routes from
an airport, with the airport latitude and longitude as the gnomonic cen-
ter point.
The gnomonic projection has certain disadvantages, as I have indicated.
One of those is that only half the globe can be projected onto the plane (i.e.,
the paper). We can get around this problem by projecting, not from the
center of the earth, but from the opposite position on the globe. In the case
shown in figure 4.5a, the projection would be from the North Pole, and
every point on the globe except the North Pole would be projected onto the
plane. Projections like this one, with the projection point placed on the
globe in a position that is diametrically opposite the center point, are
called stereographic projections. In practice they are useful only for re-
gions near the center point, where the map thus formed is approximately
conformal—that is, it does not distort shapes or directions very much.
For azimuthal projections (onto a plane) the projection point does not
have to be the center of the globe, or a point on the surface of the globe. If
we choose the projection point to be infinitely far away from the globe,
then the resulting map is an orthographic projection, shown in figure 4.6. It
looks like a drawing of the earth, because perspective is accurately de-
picted. Distortions are minimal near the center of an orthographic projec-
tion and increase toward the edge. Such distortions are easy for us to
accommodate mentally because we recognize that the map is an image of
the earth and we can allow for the perspective distortion.
CONICAL PROJECTIONS
Now we turn to projections from a spherical globe onto the curved surface
of a cone. This time we figuratively shoot an arrow out from the South Pole
(from the bottom of the sphere shown in fig. 4.5b). You can see that the
cone is completely filled by this projection: every point on the surface of
the cone corresponds to a point on the sphere. All but one point on the
sphere is mapped onto the cone. The South Pole of the sphere is unique in
this projection because it projects to more than one point: it can be thought
of as projecting to every point on the circle that defines the bottom edge of
the cone. Unfolding the cone, we then have a flat map.
intersection of plane and sphere is a circle—in fact, a great circle, because its center is the
center of the sphere. The section of this great circle connecting A and B is the shortest
distance between the two points that lies on the surface of the sphere.