Oblique Projections
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Hammer map with coordinates rotated 45°
counterclockwise.
Every piece on Earth is represented (Asia
was not cut off, but cut up to the middle
and stretched out)
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Most maps published in atlases are oriented with the North Pole
at top, South Pole at bottom and the Atlantic Ocean somewhere
in the middle. Supposing a perfectly spherical Earth, such
setting is just a convention - one could first rotate the globe
in any way and afterwards project the rotated coordinates as
usual. A Middle-Age world map drawn by Western ambassadors to
China, surely due to political reasons, put the Northern
Hemisphere at the bottom and China nearer the center than
usual.
If neither Equator nor the central meridian are aligned
with and centered on the map axes, the result is commonly
called an oblique projection
(or, more properly, an oblique map).
Although general properties of the original projection (like area and shape
equivalence) still hold, those depending on the
graticule orientation
are generally not preserved.
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| The Atlantis map was named after an ocean, not a myth |
Oceanic maps by Spilhaus using
Hammer (left) and August (right) projections. Contrast the same
region and aspect in equal-area and conformal projections (scale in
the Hammer map was enlarged 75% so map sizes would be similar) |
A common reason for tilting a projection is moving a large,
important area to the places of lesser distortion. The
Atlantis map (Bartholomew, 1948)
presents the Atlantic Ocean in a long, continuous strip aligned with
the map's major dimension. Also
clearly showing the Arctic "ocean" as a rather small extension of
the larger Atlantic, it is an oblique
Mollweide projection
centered at 30°W, 45°N.
Two other maps emphasizing sea regions
were announced by Athelstan Spilhaus in 1942, one using Hammer's and
the other
August's conformal projection, both
with 15°E, 70°S as the map center: very few oceanic sites
(notably the Caribbean sea)
are interrupted, and relative sizes of oceans are clearly
expressed. With modern computers, finding the appropriate rotation
parameters for such a "good" distribution of features is fairly easy;
one can only imagine the laborious process employed by the original author.
Later, Spilhaus also published an ocean-themed world map using a conformal
projection in a square and then extended his ideas using
interrupted maps.
Of course, sometimes a fundamental requirement (like keeping
coordinate lines straight or parallel, or preserving correct
directions along the meridians in azimuthal projections) prevents
adoption of oblique maps.
A fact often overlooked is that points at the borders of any
world map are represented at least twice, since in the original
sphere the "edges" are joined (an unrelated phenomenon occurs in
cylindrical and other flat-polar projections like Eckert's
and mine, which stretch the poles into line segments).
We are used to that obvious
scale
distortion (points in a neighborhood get widely separated in the map), so
we hardly notice it in conventional maps, save maybe at the extreme
tips of Siberia and Alaska.
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| Hammer map centered on Eurasia, with a double pole
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Oblique maps often make this kind of distortion obvious: notice the
New Zealand islands in the Atlantis map, and the east and west coasts
of North America in Spilhaus's maps. Rotating a projection in order to
put a temperate region at the center can easily create a two-pole map.
Although one could probably use a polyconic projection
for this purpose, the map above nicely shows Eurasia with a
shape nearer to reality than usual, at expense of North America
and Antarctica and, as shown by the projected graticule,
dramatic angular deformation at the southern Indian Ocean.
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Briesemeister map, a simple modification of Hammer's
projection clearly presenting
all land masses except Antarctica, with a double pole
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A similar, very simple modification of Hammer's projection
was published by William Briesemeister in 1948: the map is first
projected obliquely with 10°E 45°N as the central
point; then it is linearly stretched to an aspect ratio of
7 : 4 (compare with 2 : 1 for both
Mollweide and conventional Hammer). As a result, parallels
in the vicinity of the North Pole are almost circular.
Still another oblique version of Hammer's method is the
Nordic projection (Bartholomew 1950), centered
at 0°W, 45°N: it closely resembles Briesemeister's map,
without the rescaling.
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An equidistant cylindrical map with Campinas, Brazil infinitely stretched
on the bottom edge; its antipode lies at the top edge
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While the azimuthal
equidistant projection preserves distances radially
from the center, the
equidistant
cylindrical projection preserves
distances vertically. In particular, distances of any point to
the projected "poles" can be directly read as the y-coordinate
from the top or bottom of the map. The bearing from either
projected "pole" is also immediately available as the
x-coordinate from the center, since meridian spacing is uniform
in the original map. Thus, this map has the same purpose
(actually it is also appropriate for the antipodal point) as the
familiar azimuthal
map, but shows the bearing without a protractor.
 |  |  |  |  | | www.progonos.com/furuti March 22, 2008 |