Integrating Past, Present, and Future Approaches - Spatial Mathematics: Theory and Practice through Mapping - page 220

statement that is true in the projective plane and an equally true statement
will be the result. There are other pairs of dual terms, as well: “Concurrent”
and “collinear,” for example. The Principle of Duality is a “meta” concept
about the language in which the mathematics is written. Projective geometry
is a symmetric geometry that does not distinguish the ordinary from the
infinite. What might be the implications of this broad geometry for the geo/
metry/graphy of mapping?
10.3.2 Perspective projections
The material below shows that the entire set of perspective map projections
may be derived in the projective plane. It is done using only the subset of
projections with centers of projection contained within the sphere of projec-
tion. Because the transformation takes place in non-Euclidean, rather than
Euclidean geometry, the unbounded problem of looking at an infinite number
of projection centers spread along an unbounded ray is converted to one of
looking at an infinity of projection centers spread along a bounded line seg-
ment. An unbounded, intractable problem becomes bounded, and perhaps
manageable. The language of duality then applies to the geometry of all per-
spective map projections and offers the potential to analyze geographical
problems that exhibit symmetry in underlying relations and simultaneously
embrace the concept of infinity. Reduction of the complex to the simple, or
the unbounded to the bounded, is a powerful ally.
In the last chapter, we displayed three azimuthal projections, in which the
parallels were projected. They were gnomonic, stereographic, and ortho-
graphic projections. All three are simultaneously visualized in Figure 10.7 .
Clearly, there are an infinite number of choices available, up and down the
white vertical ray, for centers of projection. Figure 10.7 shows the relation-
ships among the three centers of projection:
• The three projected points are collinear.
• The closer the center of projection is to the point of tangency, the
farther the projected image of a point P is from the point of tangency.
It is not hard to imagine the pattern continuing in a natural way with varying
choices for the center of projection on the white ray yielding expected posi-
tions on the line of projection for projections of P . A term for this style of
projection, rooted in geometric language, is “perspective” projection.
10.3.3 Harmonic conjugates
One concept from projective geometry, that permits collapsing the unbounded
infinity of perspective projection centers into a bounded infinity of them,
involves the construction of “harmonic conjugates.” The construction involves
perspective projection through a point P not on a given line through points
A , B , and C . It shows how to determine, uniquely, a fourth point C ′ that is