 |  |  |  |  | Map Projections |
| | | | | Map Projections |
| The "van der Grinten" projections, clockwise from top left: van der Grinten's I, van der Grinten's II (by Bludau), van der Grinten's III (by Bludau) and the "apple-shaped" van der Grinten's IV. |
Alois Bludau proposed in 1912 two modifications to the first version; the four designs soon came to be collectively - and confusingly - called "van der Grinten" projections:
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 (in this aspect, they resemble earlier globular projections) without departing too much from Mercator's familiar shapes.
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.
| Maurer's "full-globular" map |
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.
| 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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| Schematic 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.
| 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.
| Arden-Close's projection in hemispherical and whole-world maps | |
| Conventional Eastern hemisphere, central meridian 70°E | Western hemisphere, central meridian 110°W |
| Extended to whole world | Whole world, transverse aspect |
Sometimes, simplifying an existing projection may actually enhance its usefulness, or at least make it easier to use. That was Waldo Tobler's conclusion after looking for a projection suitable for efficiently presenting a small area such as the U.S.'s State of Michigan on a computer screen (1974). The requisites were fast computation, reasonable fidelity of shape and area, easily calculated distortion, simple parameterization, and exact and easily computed inverse equations, in order to quickly correlate screen and world coordinates.
| Changing the reference parallel (30°N above): | ||||
| 90°N | 75°N | 60°N | ||
| 45°N | 30°N | 15°N | 0° | 15°S |
| 30°S | 45°S | 60°S | 75°S | 90°S |
| Although Tobler never intended his projection for local maps to be used in world charts, it's an interesting exercise presenting how its distortion changes with the reference latitude. | ||||
After researching several conformal and equal-area approaches, Prof. Tobler decided for a previous projection by Tissot (1881), neither conformal nor equal-area, also designed for local maps. Tissot's projection is defined by a power series, but Tobler retained only the linear terms, which directly led to inverse equations. The projection is parameterized by a reference latitude, which also helped optimizing distortion. The trivial case, centered at the Equator, is identical to the Plate Carrée.
| | | | | www.progonos.com/furuti August 24, 2009 |