Map Projections - Spatial Mathematics: Theory and Practice through Mapping - page 196

on the North Pole. It is the map projection (when centered on
Pyongyang) that should initially have been used to measure true
nuclear range distances from a center point of Pyongyang!
• If, on the other hand, parallels are also projected, then the point
from which they are projected will produce maps with differ-
ing spacings of parallels, different coverage of area, and of dif-
ferent appearance. Theoretically, there is an infinite number of
points which might serve as centers for projection. These are often
referred to as “perspective” projections. They are easy to visualize
geometrically and thus seem particularly appropriate to consider
in a book on spatial mathematics! Three common ones are:
− Gnomonic ( Figure 9.2b shows a polar gnomonic): The center
of projection is the center of the sphere. The derived map cov-
ers less than the surface of a hemisphere and is bounded by,
and does not include, the Equator. Great circle routes on these
maps appear as straight lines; useful for planning routes of
various sorts.
− Stereographic ( Figure 9.2c shows a polar stereographic): The
center of projection is the pole opposite the tangent pole. The
derived map covers all points on the spherical surface except
the projection pole. Thus, if the projection pole is the North
Pole, and the projection plane is tangent at the South Pole, all
points on the spherical surface, except the North Pole, project
in a one-to-one transformation to the plane. Do you now see
why, in Chapter 4, it takes the same number of colors to color
a map in the plane and on the spherical surface?
− Orthographic ( Figure 9.2d shows a polar orthographic): The
center of projection is the point at infinity opposite the tangent
pole. The derived map covers all points on a hemispherical
surface and is bounded by, and includes, the Equator.
Map projections that do not fit within these three classes of cylindrical, conic,
or azimuthal are described as Pseudo or Miscellaneous projections. Though
few maps are truly the result of such simple geometric projection (most are
derived from mathematical formulas), the geometric strategy is a useful way to
visualize and understand the transformation process. Many maps are actually
derived from mathematical formulas, which in some way imitate elements of
these processes, but these calculated maps are not true geometric projections.
Before moving on, let us first consider a method to formally capture what we
can all see intuitively in maps of varying underlying projections: Distortion!
9.4 Sampling projection distortion
The Tissot Indicatrix is the classical way to sample projection distortion
(Tissot, 1881). A sequence of circles of constant radius is centered on graticule