Lunar and Planetary Science XXXIII (2002)
1794.pdf
MAPS WITH CONSTANT SCALE NATURAL BOUNDARIES AND THE ASTEROID EROS.
Chuck Clark, Architect,1 (1100 Alta Avenue, Atlanta, GA 30307, [email protected])
Summary: All scientists are familiar with the fundamental problem of representing curved or complex
volumes in two dimensions, i.e., on a map.
Maps of Eros and other irregularities presented
here were hand-plotted by an architect inspired by
anamorphic sculpture,1 a topographic paper by James
Clerk Maxwell and its extension by Marston Morse,2, 3
and the respect Morse shared with Robert Frost, the
poet and philosopher,--for accuracy in uses, such as
mathematics or poetry--of the imagination.4
They present collectively several advantages—
greatly improved conformality for example--over conventional projections for the representation of various
irregular surfaces or parameters.5, 6 A map’s perimeter,
which is constructed prior to delineating a map’s field,
is laid out as surveyors’ meets-and-bounds, with occasional adjustment of azimuth.. These maps have constant scale natural boundaries (CSNB).
CSNB projections offer a new perspective, literally,
on objects, processes and structures of irregular solid
surfaces.
Properties: 1.Maps can be folded up to 3-D replicas.
Antipodal geometry is preserved; that is, the maps are
conformal for antipodal areas unlike, for example, a
cylindrical projection of Eros.7 For spheres, globes are
the only true representation of the body, but for extremely irregular solids like Eros, CSNB maps can in
principle be transformed to solids. Flat sheets can be
economically printed with ortho-normal photos, then
folded into photoreal models. (See Fig. 2.)
FIGURE 2: Eros model with folded Eros maps
2. Maps can be “zipped up,” without loosing foldability. This advantage, allowing large-region unification,
follows from the first one. (See Fig. 3.) Highly zipped
maps, when folded up, have curious characteristics—
weak and strong properties—which may mimic actual
conditions.10
FIGURE 1: Eros map, centered on Psyche,
map bounded by topographic ridges
The maps presented here have been drawn graphically by hand and measure.7 However, they can be
mathematically reproduced in c-code8 and can be digitally bridged—gridded for data insertion—with waterlining programs.9
For planetary scientists, CSNB projections may be
useful in representing geomorphic regions—as expressed by topography, debris, faulting, volcanism and
other parameters. (See Fig. 1.) Most Eros investigations use stereographic views, which are unexcelled for
small-region picturation but not for proportionate assessment of large-region events, issues or ideas, especially as one suspects global components at play.
FIGURE 3: Eros map, zipped up
Lunar and Planetary Science XXXIII (2002)
1794.pdf
MAPS WITH CONSTANT SCALE NATURAL BOUNDARIES AND EROS, Chuck Clark
3. Maps can represent solids with holes or deep fissures. For example it is difficult to construct a global
weathermap because of up- and down-welling, i.e.,
holes, between stratified altitudes. CSNB can handle
such problems. An Eros-related application of CSNB
might be representation of debris fields from Psycheimpact in combination with topography. Another possible application, in physical chemistry, might be conformal representation of electron-density topography.
4. Maps can be used to compare several different parameters. On Mars, they might be useful for comparing
on a flat map erosion regions at various altitudes. In
geophysics, they might be useful for showing seismic
tomography.10 (See Fig. 4.)
FIGURE 4: Earth tectonic activity,
map bounded by spreading ridges
5. Maps permit geometric harmonic analysis. This is
useful for natural boundaries that derive from force or
motion. A medial axis, described by waterlining from
the map’s boundary, locates centroids and directs vectors.9 (See Fig. 5.)
References: [1] Clark C. S. (1979) Anamorphic Dining
Room, High Musem of Art, Atl, GA. [2] Maxwell J. C.
(1870) On Hills and Dales, Phil. Mag., 40, 269. [3] Morse,
H. C. M. (3 PM Oct 8, 1950 delayed, finding a room with a
piano) Some Reflections on Evaluations in Mathematics and
the Arts, Kenyon Col., published (1959) with Author’s Note,
“which might interest Robert Frost,” Bull. Atom. Sci. , ; ibid
(1965) Pits, Peaks and Passes, a lecture on critical point
theory, video, MAA; [4] Frost R. L. (8 PM Oct 8, 1950)
REMARKS, at “The Poet and Reality,” a conference in
honor of Robert Frost, Kenyon Col., Gambier, OH. [5]
Krantz S. C. (Sep-Oct 1999) Conformal Mappings, Amer.
Sci., 84, 436. [6] Spilhaus A. F. and Snyder, J. P. (Oct 1991)
World Maps With Natural Boundaries, GSIJ-ACSM, 18, 4.
[7] NEARmissionNASA (2000) A Roadmap for Eros, image
of the day, Oct. 2, Sci. Mag. [8] Msezane A. Z., Handy, C.
and Smith T. (Dec 1998) comments on my lecture: True
Natural Shape Projection of Objects (CSNB), Cen. For
Theoretical Studies of Physical Systems, CAU, Atl. GA. [9]
Christensen, A. H. J. (Jun 1999) The Revival of a Victorian
Art: Waterlining with a Computer, Brit. Cart. Soc. 36, 1.
[10] Lowman P. D. et al. (1999) A Digital Tectonic Activity
Map of the Earth, J. of Geosci. Ed., 47, 5, 489.
Acknowledgements: to jim hagan, architect from Texas, for
provoking my recent interest in mapping; to Mark Robinson,
Northwestern U., Peter Thomas, Cornell U., Louise Prockter,
Andrew Cheng and Robert Farquhar, JHUAPL--NEAR mission team—for their encouragement, and a model of 433Eros; to Paul Lowman, Goddard Space Flight Cen. and
Dave McAdoo, NOAA, for encouragement, review and
comments; to Steve Krantz, Wash. U. in St.L., Albert Christensen, Silver Springs, MD, and Athelstan Spilhaus, Middleburg VA, for interest and comments; to David Finkelstein,
Ga Tech, for steering me to Morse when all I had was Maxwell, and for the suggestion of CSNB mapping of electrondensity topography; to Robert Frost, “Maine man from New
Hampshire,” for 1. guidance in the science and arts of observation, 2. provoking the Anamorphic Dining Room,1 and 3.
sparking, in 1954 in need of “a heron’s map to Brazil,” my
initial interest in mapping; and to Marston Morse, a cofounder of the Institute for Advanced Studies, for teaching
me, age two, to count my age without forgetting “zero.”
Caveats: In spite of a surveyor’s accuracy at the edges of
CSNB maps, Christensen says “I find no numbers here,” and
Finkelstein says “I see no math.” Morse’s reflections, on
“the eternal battle between geometry and numbers,”3 apparently remain pertinent today.
FIGURE 5: Earth tectonic activity with
edge harmonics and an event in India