Other Interesting Projections
Countless projections were devised in centuries of map-making.
Many designs cannot be readily classified in the main groups (azimuthal, cylindrical, pseudocylindrical, conic or
pseudoconic), even though their design is
similar or derived.
A large number of projections whose graticule lines are
circles or derived conic curves with different radii and centers
are called polyconic (not to be confused with the
particular group of polyconic projections).
This is a broad and artificial category
comprising otherwise unrelated projections.
Projections by Van der Grinten
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| Van der Grinten's first projection
(I) |
Modified van der Grinten's projection
(III) |
An American, Alphons J. van der Grinten published in 1904 and
1905 two projections, the first one devised as early as
1898. Both were designed for the equatorial aspect,
with straight Equator and central meridian; all other parallels
and meridians were circular arcs, with nonconcentric meridians
regularly spaced along the Equator.
Bludau proposed two modifications to the first version; the four
designs soon came to be collectively (and confusingly) called
"van der Grinten" projections:
- the first original projection, bounded by a circle
- Bludau's modification of I, with parallels
crossing meridians at right angles
- Bludau's modification of I, with straight,
horizontal parallels
- the second original projection, bounded by two identical
circles with centers spaced 1.2 radii apart
Van der Grinten's proposals are examples of
conventional designs, derived not from a perspective
process but from an arbitrary geometric construction on the map
plane. They are neither equal-area
nor conformal
(despite a superficial resemblance to projections by
Lagrange,
Eisenlohr and
August), but intended to
"look right", in the sense of conveying the notion of a
round Earth without departing too much from Mercator's familiar
shapes.
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| Van der Grinten's second projection
(IV) |
The best known of all four, van der Grinten's I,
also known simply as the Grinten projection, was widely used,
especially after its choice for reference world maps by the National
Geographic Society from 1922 to 1988. Of the others, only the
III variant saw limited use.
Although the poles can be included in the map, areal distortion
is large at high latitudes, thus most van der Grinten maps are
clipped near parallels 80°N and 80°S.
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| Maurer's "full-globular"
map |
The ancient group of globular projections
includes circular arcs for both meridians and parallels, and maps
ordinarily limited to a single hemisphere.
H. Maurer
presented in 1922 three conventional projections resembling
globular features. The "full-globular" projection
has meridians spaced like in van der Grinten's IV
projection; parallels are equally spaced along the boundary
meridians, and both the central meridian and the Equator have
constant scale. Each boundary meridian spans half the
limiting circle, thus the whole world is set resembling a
double-edged ax.
His two other globular proposals are called "all-globular"
and "apparent-globular".
Beginning in 1943, the notable cartography teacher and
author Erwin Raisz created a series of orthoapsidal
projections mapping the sphere onto
intermediary surfaces. However, instead of "unrolled"
like in cylindrical or conic
maps, each surface is then projected orthographically onto the final
plane.
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| Orthoapsidal ("Armadillo") map on part of a toroidal surface;
tilt angle 20°, central meridian 10°E. Raisz's
original map extended the eastern and western edges, with parallels
spanning about 410° in order to avoid splitting Alaska and
Siberia. |
In the best-known orthoapsidal projection, called
Armadillo (since it vaguely resembles the curling
armored mammal), the
sphere is mapped onto 1/4 of a degenerate torus with radii 1
and 1, which resembles a doughnut with a zero-sized hole.
Parallels and meridians are equidistant circular arcs on the
torus, but nonequidistant elliptical arcs in the final map.
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| Development of the Armadillo projection:
the sphere is mapped to the region resembling half of a
car tire, and that region to the blue projection plane
|
In the conventional form of the Armadillo map, Raisz preferred
10°E as the central meridian; the torus is then tilted 20
degrees and orthographically flattened onto the projection
plane. Southern regions like Patagonia, New Zealand and Antarctica
are hidden from view, and sometimes presented separately.
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| Orthoapsidal map on a half-ellipsoid,
eccentricity 1.75, tilt angle 20°; central
meridian 10°E
|
Raisz also developed a map on one half of an oblate ellipsoid of
rotation; the intermediate process is roughly a three-dimensional
analogue of that
applied by Aitoff to
the
azimuthal equidistant
projection.
Another surface employed by Raisz was one half of a tilted
hyperboloid of rotation of two sheets; in this case, a North
polar map was interrupted in four identical lobes,
resembling Maurer's
S231 projection and,
different from other orthoapsidal designs, showing the whole world.
As drawn by Richard Edes Harrison, this projection was
prominently featured in the cover of Scientific American 233(5);
it is interrupted (at 60°E, 150°E, 120°W
and 30°W) south of, apparently, 10°N.
Harrison, known for his innovative and detailed maps, is quoted as
characterizing it as "the most elegant of all world maps".
Orthoapsidal maps are neither conformal nor equal-area; parallels
and meridians do not necessarily hold properties (like equidistance)
of the intermediary surface.
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| Arden-Close map of Eastern hemisphere, central
meridian 70°E
| Variant Arden-Close map of whole world
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Charles F. Arden-Close designed some map projections by averaging;
his best-known (1943) is a simple arithmetical mean of one hemisphere
of an equatorial
equal-area cylindrical map with its
transverse aspect, the Equator in one map
coinciding with the central meridian in the other. Shaped like a square with
circular corners, the result is neither conformal nor equal-area.
Doubling coordinate values, the method can be easily extended in order
to show the whole world.
 |  |  |  |  | | www.progonos.com/furuti October 7, 2006 |