Map Projections - Spatial Mathematics: Theory and Practice through Mapping - page 208

South Pole) projects the entire spherical surface, except one point—the North
Pole—to the plane. The reason that the North Pole does not project into the
plane is that the line projecting the North Pole runs parallel to the tangent
plane. In Euclidean geometry, a line parallel to a given plane never intersects
it. Stereographic projection is the best we can do, in terms of getting all the
points on the globe (but one) into the plane. While this feature of the stereo-
graphic projection might seem to make it attractive, the distortion introduced
in this projection might not be desired in a number of applications.
Euclid's Parallel Postulate states that given a line l and a point P not on l , there
exists exactly one line m through P that does not intersect l . It is more com-
mon perhaps to think of this as “parallel lines never meet.” Euclid's Parallel
Postulate is behind almost all of the decisions we make in contemporary map-
ping and is responsible for the infinity of selections available. It has been with
us for thousands of years and our minds are conditioned through intellectual
institutions to think naturally in a world that embraces it as a foundation.
There is, however, a whole class of geometries that deliberately violate this
postulate (Coxeter, 1965; 1998). If an infinite set of points at infinity is added
to the plane, then the line through North Pole (the “parallel”) and a line in
the tangent plane intersect at a point at infinity. The concept of “parallel” has
been discarded. This sort of geometry, based on a simple violation of Euclid's
Parallel Postulate, which denies the presence of parallels, is called “elliptic”
geometry. So, we have a situation with no parallels (elliptic geometry) and
with a unique parallel (Euclidean geometry). We can also have a situation with
multiple parallels. Imagine multiple lines passing through P that bend and
become asymptotic to l (Figure 9.12).
These lines are all “parallels.” This sort of geometry, based on violating
Euclid's Parallel Postulate from above (more than one parallel to a given line)
is called “hyperbolic” geometry. These local geometries have found applica-
tion in physics and some other contexts (Minkowski, 1907/1915). What might
be discovered by such alteration of the foundations of the world of mapping?
Figure 9.12 Two parallels to line l pass through point P. Based on material from
Arlinghaus, S. L. 1986. “The Well-tempered Map Projection.” Essays on Mathematical
Geography. Monograph #3, pp. 1-27. Ann Arbor: Institute of Mathematical Geography.
http://www-personal.umich.edu/%7Ecopyrght/image/monog03/fulltext.pdf