|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.
|Schematic development of the Armadillo projection: the sphere is mapped to the region resembling half of a car tire, and, after the torus is tilted, that region to the blue projection plane.|
|If the torus is left untilted, the result resembles the third and fourth projections by Eckert|
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.
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.
|Orthoapsidal map on a half-ellipsoid, eccentricity 1.75, tilt angle 20°; central meridian 10°E|
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):|
|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.
|Reconstruction of Gringorten's square map, with a superimposed 10x10 grid in normal (top) and transverse (above left) aspects. Above right, with alternative meridian placement. Below, quadrants rearranged giving each hemisphere an entire square.|
After looking for map projections suitable for global climate analysis,
climatologist Irving I.Gringorten published in 1972 a very
distinctive design, today nearly forgotten.
He required a world map to be square, in order to efficiently use printed space in reports, articles and books; it should be equal-area so, for instance, the density and distribution of weather probes and stations can be correctly estimated at a glance; it should also avoid excessive shape distortion; and finally, should minimize continental interruption.
Details of Gringorten's projection were devised for a polar aspect
of a spherical Earth, with a pole-centered hemisphere on an inner
square, while the other hemisphere is split into four right triangles,
its pole repeated across four corners.
Except for interruptions, both hemispheres are fully
symmetrical, an arrangement similar to Peirce's quincuncial map.
Each hemisphere comprises four triangular quadrants with a vertex on a pole, symmetrical around the central meridian. In each quadrant, parallels are elliptical arcs, straight on the Equator and concave towards the pole. At quadrant boundaries, parallel arcs join with first order continuity. Given those constraints, there is more than one equal-area solution for meridian placement, unfortunately none satisfying another desirable attribute: meridians crossing the Equator at straight angles, thus unbroken across hemispheres.
The mathematics required by Gringorten's projection is rather complex and must be calculated by numerical approximation. Gringorten's paper includes a table of computed coordinates and a north polar map, with southern lobes bounded by meridians 20°W, 70°E, 160°E and 110°W. The map is superimposed with a grid of 100 numbered cells, proposed as an aid for quickly locating points in addition to the ordinary latitude/longitude coordinate system. Antarctica is interrupted, but this inconvenience can be alleviated by an inset, by a second map with hemispheres swapped, or by moving three grid cells from corners to around the pole.
The author suggested other variations, like an oblique aspect and an alternative solution for meridian placement. Another immediate modification is rearranging the quadrant lay-out analogously to Peirce's and Guyou's maps: for instance, setting each hemisphere in a whole square.