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Figure 10.8 On a line with three given points, A, B, and C, where A is the point of
tangency of the pole with the tangent plane, construct C
′
, the harmonic conjugate
of C, as in this figure. Source: Arlinghaus, S. L. 2007. Geometery/Geography—
Visual Unity. Solstice: An Electronic Journal of Geography and Mathematics.
www-personal.umich.edu/%7Ecopyrght/image/solstice/win07/hyperbolicgeometry.
html
The unfolded animation sequence in
Figures 10.7
through
10.9
offers a sug-
gested direction for the visualization of concepts.
Figure 10.10
offers some
direction for proof.
The Harmonic Map Projection Theorem shows that, in the projective plane,
points on a gnomonic projection are harmonic conjugates of points on an
orthographic projection, with respect to the tangent pole and the corre-
sponding point in a stereographic projection (
Figure 10.10
)
. Whatever we
know to be true about points in a gnomonic projection is therefore known
about the conjugate points in the orthographic projection. The same holds
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