SOLSTICE: An Electronic Journal of Geography and Mathematics. (Major articles are refereed; full electronic archives available)  Sandra Lach Arlinghaus

Benoit Mandelbrot has shown us the power of computer-generated images to make mathematical concepts come alive that had otherwise been tucked away in mathematics classrooms [Mandelbrot, 1982].   What is accurate in terms of definition/proof presentation may not penetrate the minds of those less gifted in mastering the mathematical notation and its underlying system of logic.  Beautiful images, conceived by those who understand both the abstract concepts and their representation using current technological capability, can go a long way toward making heretofore inaccessible concepts become accessible.

A recent example of such transformation in accessibilty appeared in YouTube this past summer [Arnold and Rogers, 2007].  Imagine being able to see Möbius transformations come alive as an animated sequence showing their relations to each other in the plane and then, unifying those representations by lifting them into the third dimension on a sphere.  Take a look at this award-winning work, now.  Use the link above to view it at low resolution on YouTube.
Original Source Douglas N. Arnold and Jonathan Rogness, June 2007:  http://www.youtube.com/watch?v=JX3VmDgiFnY

The projection "light" at the top of the sphere in the video sends points on the sphere to points in the plane via stereographic projection, one of a class of "true" perspective projections employed by map makers to send part of the surface of a spherical globe to a plane.  Three members of this projection class are shown below (Figure 1).
In all cases, the plane into which points are projected is tangent to the globe at S, the south pole of the globe.  The projections are:

• Stereographic Projection, in which the center of projection is at the north pole of the globe.  A point P on the globe is sent to a point P(stereo) in the plane (Figure 1a).
• Gnomonic Projection, in which the center of projection is at the center of the globe.  A point P on the globe is sent to a point P(gnomon) in the plane (Figure 1b).
• Orthographic Projection, in which the center of projection is at a point at infinity.  A point P on the globe is sent to a point P(ortho) in the plane (Figure 1c).
Each of the first three images shows an animation suggesting how a point is sent from the globe to the plane.  When a large number of points, outlining regions of various sorts, is sent from the globe to the plane, a geographic outline map is created. Figure 1a.  Projection is from the north pole. Figure 1b.  Projection is from the center of the globe. Figure 1c.  Projection is from a point at infinity.

Clearly, there is an infinite number of choices available, up and down the white vertical line, for centers of projection.  Figure 2 shows one relationship among the three projections of Figure 1.  Perspective projections are only one style of projection; there are an infinity of other styles possible and within most of those, another infinity of ways to create maps.  Readers wishing to discover more about the characteristics of individual projections, and classes of projections, are referred to the vast literature on the subject of cartography, the science and art of making maps [Snyder, 1993; Yang, Snyder, Tobler, 2000]. Figure 2.  Relationship among different perspective projection types.

As Arnold and Rogness suggest, there may be more to learn by looking at the relationship between plane and spherical expressions.  To consider this idea, turn to the underlying mathematics...that of projective geometry which explores the theory of perspectivity and related concepts [Coxeter, 1961].  Perspective projection through P, in the sequence below, shows how one might find, given three collinear points A, B, and C, a fourth point C' that is independent of arbitrary choices made during the process of construction.  The points C and C' are harmonic conjugates with respect to A and B.  Follow the animations in Figures 3a and 3b to understand the process. Figure 3a.  C and C' are harmonic conjugates Figure 3b.  C and C' are harmonic conjugates
In 1986, an essay appeared couched in the language of projective geometry in which a theorem, linking harmonic conjugacy to perspective map projection, was proved to show the following (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].
The animation in Figure 4 illustrates both of these points and it does so more clearly than did the original text which used notation and static grayscale images only. Figure 5.  Animation displaying content of Harmonic Map Projection Theorem (1986).  Link to a Quicktime movie (works better in some browsers than others--try Firefox if Internet Explorer is a problem;  Music [Mozart, The Magic Flute].

Naturally, it also follows that all the theorems of projective geometry that apply to harmonic conjugates also apply to all projections that satisfy the conditions of this theorem.  Thus, for example, the entire set of perspective projections may be derived in the projective plane, alone, from the subset of projections with centers of projection contained within the sphere of projection.  The unbounded problem of looking at an infinity of projection centers spread along an unbounded ray is thus converted to one of looking at an infinity of projection centers spread along a bounded line segment.

Further, the language of duality of projective geometry then applies to the geometry of all perspective map projections.
At the broader level of identifying future research topics, this geo/metric/graphic unity of harmonic conjugacy and perspective projection suggests possible advantages in employing this highly symmetric geometry, that does not distinguish the ordinary from the infinite, to understand and to analyze other geographical problems that exhibit symmetry in underlying relations and that also embrace the concept of infinity.

References

• Arlinghaus, S. 1986.  "The Well-tempered Map Projection."  Essays on Mathematical Geography.  Monograph #3, Institute of Mathematical Geography, pp. 1-27.  http://www.imagenet.org/  "Books"
• Arnold, D. N. and Rogers, J.  2007.  "Möbius Transformations Revealed."
• Coxeter, H. S. M.  1961.  The Real Projective Plane, Cambridge University Press.
• Mandelbrot, B.  1982.  The Fractal Geometry of Nature.  W. H. Freeman.
• Mozart, W. A.  Die Zauberflöte.  http://www.amazon.co.uk/exec/obidos/clipserve/B000001GXI002029/026-8133798-8086815
• Snyder, J. P.  1993.  Flattening the Earth:  Two Thousand Years of Map Projections.  University of Chicago Press.
• Yang, Q. H.; Snyder, J. P.; Tobler, W. R.  2000.  Map Projection Transformation:  Principles and Applications, Taylor & Francis.

Notes

• Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2001 (top 80 out of over 1000 entries worldwide)
• One article in Solstice was a Pirelli INTERNETional Award Semi-Finalist, 2003 (Spatial Synthesis Sampler).
• Solstice is listed in the Directory of Open Access Journals maintained by the University of Lund where it is maintained as a "searchable" journal.
• Solstice is listed on the journals section of the website of the American Mathematical Society, http://www.ams.org/
• Solstice is listed in Geoscience e-Journals
• Solstice is listed in the EBSCO database.
• IMaGe is listed on the website of the Numerical Cartography Lab of The Ohio State University:  http://ncl.sbs.ohio-state.edu/4_homes.html
Congratulations to all Solstice contributors. Solstice:  An Electronic Journal of Geography and Mathematics,  Volume XVIII, Number 2 Institute of Mathematical Geography (IMaGe). All rights reserved worldwide, by IMaGe and by the authors. Please contact an appropriate party concerning citation of this article: sarhaus@umich.edu http://www.imagenet.org
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