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Oblique Projections

An oblique aspect of the the Hammer projection

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)

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.

Map in Atlantis projection

The Atlantis map, an oblique version of Mollweide's projection, was named after an ocean, not a myth

Simplified “Oceanic” maps
Oblique August projection

August version

Hammer projection in oblique Oceanic aspect

Hammer version

Simplified reconstructions of "Oceanic" maps by Spilhaus using Hammer's and August's projections (the original maps have very complicated borders shaped by shorelines, not a round frame). Contrast the same region and aspect in equal-area and conformal projections (the 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.

Schjerning IV projection

The Schjerning IV projection

The pseudoconic equal-area Schjerning IV projection, published in 1904, is an oblique aspect of the well-known Werner projection, with a 0° central meridian but centered on London instead of a pole.

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.

An oblique aspect of the the Hammer projection

Oblique Hammer map centered on Eurasia

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.
Briesemeister projection

Map in Briesemeister's projection, a simple modification of Hammer's projection clearly presenting all land masses except Antarctica, with a double pole

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.

Oblique equidistant cylindrical projection
Oblique azimuthal orthographic projection
Above, an equidistant cylindrical map with Campinas, Brazil infinitely stretched on the bottom edge; its antipode lies at the top edge. Using the superimposed reddish graticule, both the angular (horizontal) and geodesic (vertical) distance from either Campinas or its antipode to any other point on the globe can be found directly. On the left, a similar map with identical graticules, but using the azimuthal orthographic projection.
While the azimuthal equidistant projection preserves distances radially from the center, the equidistant cylindrical projection preserves distances vertically. In particular, as suggested by Botley in 1951, 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.
HomeSite MapOther ProjectionsMap Projections - ContentsInterrupted Maps — June 16, 2018
Copyright © 1996, 1997, 2009 Carlos A. Furuti