Mapmaking - The Science of Navigation - page 104

Local Projections
In the figure we see that projecting from one plane to another does not lead to
distortions. Any shape that is inscribed in the top plane is faithfully mapped
onto the bottom plane, via a projection from any chosen point above the
top plane. All angles and shapes are preserved; the only map feature that
changes as a result of this projection is the scale.
Contrast this plane-to-plane projection with a sphere-to-plane projection.
In part (b) of the illustration, we have a cross section of such a projection: the
sphere appears as a circular arc and the plane as a straight line. Now you can
see that distances are distorted. (Distance AB equals distance BC , but DE does
not equal EF .) However, if the projection is restricted to small regions of the
sphere nearest the plane (regions with a linear extent that is small compared
to the sphere radius) then the sphere surface appears nearly flat, and so
distortions are small. The projection of part (c) resembles that of part (a).
The message to take away from all this is that low-distortion map areas are
those in the region where sphere and plane meet—or near the closest point, if
they do not meet, as in part (c) of the illustration. The same applies to a cone:
in this case, as we have seen, the tangential contact region forms a circle, not
a point, and so for conical projections, regions near the contact circle suffer
little distortion.
A subclass of projections called secant projections increases the area of
globe that suffers from little distortion by changing the projection geometry.
If, in figure 4.5a, the sphere does not sit on the plane, but instead sinks into it a
little, then the contact point becomes a more extended contact circle. Careful
choice of geometry (how much the sphere sinks into the plane) can result in
small distortions over quite a wide area of the map. Similarly for conic and
cylindrical projections: the spheres of figure 4.5b and 4.5c can be allowed to
penetrate the cone and cylinder, thus increasing the contact points and the
area of the globe that can be projected onto a map with little distortion.
We have seen that there are three different types of distortion: distance,
area, and shape or direction. All three of these are small in regions of the
globe that are near the point(s) of contact for the projection. For some projec-
tions, as we have seen, one or other of the distortions may be reduced or be
absent farther away from the points of contact.