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Figure 10.7 Simultaneous display of gnomonic, stereographic, and orthographic
projections. Source: Arlinghaus, S. L. 2007. Geometery/Geography—Visual Unity.
Solstice: An Electronic Journal of Geography and Mathematics. Volume XVIII, No.
edu/%7Ecopyrght/image/solstice/win07/hyperbolicgeometry.html
independent of arbitrary choices made during the process of construction.
The given point
C
and the constructed conjugate point
C
′
are said to be har-
monic conjugates with respect to
A
and
B
. The determination of the conjugate
point is unique. It is independent of support points generated within the con-
struction. The mechanics of this construction are suggested in
Figure 10.8
and the detail is available elsewhere (Coxeter, 1965).
Figure 10.9
illustrates
how stereographic projection fits with the construction in the tangent plane.
10.3.4 Harmonic map projection theorem
• Centers of projection that are inverses in relation to the poles of a
sphere are harmonic conjugates in the projection plane in relation to
the projected images of the poles of the sphere.
• As a special case of the observation above, it follows that gnomonic
and orthographic projections, with inverse centers of projection in the
sphere, are composed of points that are harmonic conjugates of each
other in the plane (Arlinghaus, 1986).
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