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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
June 2007: http://www.youtube.com/watch?v=JX3VmDgiFnY
projection "light" at the top of the sphere in the video sends points
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).
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:
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).
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).
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.
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).
1a. Projection is from the north pole.
1b. Projection is from the center of the globe.
1c. Projection is from a point at infinity.
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
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
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].
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
respect to A and B. Follow the animations in
Figures 3a and 3b to understand the
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
3a. C and C' are harmonic conjugates
3b. C and C' are harmonic conjugates
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.
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.
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].
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].
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
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
looking at an infinity of projection centers spread along a bounded
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
and that also embrace the concept of infinity.
- 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/
- Arnold, D. N. and Rogers,
J. 2007. "Möbius Transformations Revealed."
- Coxeter, H. S. M. 1961. The Real Projective Plane,
Cambridge University Press.
B. 1982. The
Fractal Geometry of Nature.
W. H. Freeman.
W. A. Die Zauberflöte.
- 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 &
was a Pirelli INTERNETional
Award Semi-Finalist, 2001 (top 80 out of over 1000 entries worldwide)
article in Solstice was a Pirelli
INTERNETional Award Semi-Finalist, 2003 (Spatial Synthesis Sampler).
is listed in the Directory of Open
Journals maintained by the University of Lund where it is
as a "searchable" journal.
is listed on the journals section of the website of the American
- Solstice is listed
is listed in the EBSCO
is listed on the website of the Numerical Cartography Lab of The Ohio
to all Solstice contributors.
An Electronic Journal of Geography and Mathematics,
XVIII, Number 2
of Mathematical Geography (IMaGe).
rights reserved worldwide, by IMaGe and by the authors.
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