|Reconstructions of Jäger's (left) and Petermann's projections|
On Earth's northern hemisphere, continental areas are clustered around the pole, while south of the Equator wide patches of ocean separate sparse mainlands; moreover, Africa and the Americas narrow down towards the South. This peculiar distribution of lands is the foundation for a group of interrupted star-shaped map projections centered on the north pole with a more or less circular core (often a hemisphere) surrounded by lobes with the less important Antarctica split between their ends. Some star maps invert this pattern, with a Southern polar aspect privileging oceans. Nonpolar aspects are possible but virtually unknown, at least with manual computation.
As a rule, star projections are composite designs; polar azimuthal projections are natural candidates for the core due to their circular parallels. The projection's creator must decide whether the lobes are uniform in size and shape, and how well they preserve the core's features like shape and spacing of parallels.
Star projections should not be mistaken for projections used for representing the celestial sphere; such star maps are at least as old as geographic charts.
|The star projection by Berghaus|
Although preceded by works by Leonardo da Vinci (an octant map, 1514) and Guillaume Le Testu (1556) which could be laid out as a circular arrangement of lobes, the first modern star projection was published in 1865 by G.Jäger. In its polar aspect, the inner hemisphere is an irregular octagon whose vertices are connected to the pole by straight semimeridians with identical scale; all graticule lines are straight lines with linear scale, with parallels broken at the boundary meridians, and meridians at the Equator. Each lobe in the outer hemisphere is a triangle exactly mirroring one of the core sectors.
A more influential design was almost immediately after (1865) proposed by August H.Petermann, a German cartographer and enthusiast of exploration (especially of polar areas), but its inner hemisphere is identical to the polar azimuthal equidistant's. On each lobe, parallels remain circular arcs centered on the pole, spaced the same as in the core but with variable scale; all semimeridians are straight lines. Sources differ on whether the eight lobes are uniform in size.
Neither Jäger's nor Petermann's projections are either conformal or equal-area, but the latter is of course azimuthal in the inner hemisphere.
|Variants of Berghaus's map|
A variation of Petermann's map, Hermann Berghaus's star projection (1879) reduced the number of appendages to five uniform lobes, with boundary meridians at 160°W, 88°W, 16°W, 56°E and 128°E. Of all major land masses, Australia and Antarctica are interrupted. This design became much more popular than Petermann's, appearing in atlases and in the logo of the Association of American Geographers.
Like the original, only the core is azimuthal, and the whole map preserves neither area nor shapes. Further variations with any number of lobes greater than two can easily be done and, although not polyhedral projections by design, assembled into pyramids; with three symmetrical lobes, the map is an equilateral triangle foldable as a regular tetrahedron.
|Maurer's S233 projection, interrupted from 20°W.|
In 1935, Hans Maurer presented a comprehensive catalogue of map projections, organized by hierarchical criteria. His taxonomy included empty categories, i.e., combinations of features not met by actual, existing projections; for illustration, he filled some of those gaps with designs of his own, including the star-shaped S231 and S233.
The projection Maurer named S233 is a regular version of Jäger's map, with six identical lobes. In the polar aspect, all parallels and meridians are straight lines, parallels broken at boundary meridians and uniformly spaced along them; all meridians are broken at and uniformly spaced along the Equator. Each triangular lobe has an exact counterpart mirrored in the inner hemisphere.
|Possible reconstruction of Maurer's S231 projection in north polar form, with lobe divisions starting from 21°W.|
For the much more interesting proposal S231, Lambert's azimuthal equal-area projection was chosen for the inner hemisphere. In each lobe, only the central meridian is straight, and parallels are circular arcs centered on the north pole. Scale along the central meridians is the same in both hemispheres, but mirrored at the Equator; scale is constant along each parallel, and the same on a lobe's and on its counterpart in the core. Therefore the whole map is equal-area.
Projections S231 and S233 were described with six uniform lobes, but can be extended to any number greater than one (S231) or two (S233).
A more recent star projection based on Lambert's equal-area azimuthal was devised by William William-Olsson (1968); however its core is bounded by the 20°N parallel instead of the Equator. The entire map is equal-area, but unlike Maurer's S231, the four lobes are derived from the Bonne/Werner pseudoconic projection: every parallel is a circular arc centered on the north pole with constant scale, and each central meridian is a straight line, with the same scale as on the core at 20°N. Unfortunately, parallel lengths do not coincide at the boundary latitude on the pristine Lambert and Bonne/Werner projections. Matching the lengths at the junction requires increasing the scale of lobe parallels by the secant of half the boundary colatitude (the angular distance from the north pole); for William-Olsson's choice, about 22.077%. Areas are preserved by compressing the central meridians in lobes by the reciprocal amount.
|William-Olsson's equal-area star projection, interrupted from 20°W|
Another hybrid azimuthal/pseudoconic star-shaped proposal is the "tetrahedral" projection introduced in 1942 by John Bartholomew, the fourth of a long lineage of map publishers with the same name and surname, also the author of the oblique Atlantis and the composite "Lotus", "Kite" and "Regional" maps.
The "tetrahedral" map combines a northern polar azimuthal equidistant core limited by the 20°30"N parallel (instead of the Equator adopted by Berghaus) with three identical lobes using the equal-area Werner projection maintaining the same parallel spacing. Again, parallel scales are not the same in the two base projections except at the poles, thus parallels must be lengthened in the lobes by about 26.6%. Since, unlike in William-Olsson's design, parallel spacing is not reciprocally reduced, not even the lobes are equal-area.
|Reconstruction of a north polar tetrahedral projection, interrupted at 150°W, 30°W and 90°E|
Published in both northern and southern polar aspects, this projection apparently owes its name to a fortuitous resemblance of its lobe arrangement to a common lay-out for unfolded tetrahedra; it is unrelated to true polyhedral maps.
In addition to lobes and appendages of compound maps like William-Olsson's and Bartholomew's tetrahedral, the Werner projection has been used alone in star configurations centered on a pole.
|A reconstruction of the interrupted Werner map known as Schjerning's VI projection.|
The sixth and last of Schjerning's projections of 1904 (the first one is a plain conic, the second and third modifications of the azimuthal equidistant, and the fourth and fifth are simple variations of the uninterrupted Werner's) is a south polar star design with three pure Werner lobes of unequal width, each extending to the opposite pole and symmetrical around its straight central meridian. The interruptions at 0°E, 150°E and 70°W emphasize the Atlantic, Indian and Pacific oceans, each in its own lobe. Schjerning added repeated portions, extending lobes in order to complete shorelines around the Atlantic and Pacific.
Relatively straightforward, like its base projection Schjerning's VI preserves areas, and all points are at a correct distance from the central pole. Parallels remain standard lines, and the central meridians suffer from no distortion.
In contrast, J.P.Goode's polar equal-area projection privileges continents and is therefore centered on the north pole; although Werner's is again the sole base projection, the irregular lobes and frequent changes of central meridian make the polar equal-area's construction much more complicated than Schjerning VI's.
After a request for an equal-area design with "true space relations", appropriate for mapping populations of a species of land bird with worldwide distribution apparently originating near the north pole, Goode — better known for his interrupted versions of the sinusoidal, Mollweide, and especially homolosine projections — chose a polar Werner owing to its equidistant parallels. The American lobe, ranging from 170°W to 30°W (plus an extension for Greenland), is split at 100°W south of roughly 10°N. The Eurasian lobe is split at 60°E south of about 30°N. South of about 30°S, New Zealand is further split from Australia at 160°E, and all portions of Antarctica are omitted (Goode mentioned an azimuthal equal-area inset). Parallels remain circular arcs of correct scale; in order to reduce shape distortion, at several places the central longitude of each lobe is shifted — therefore all meridians are broken lines.
Goode published his polar equal-area projection in 1929 with an outline world map, but lacking a detailed description of the lobe interruptions and central longitudes. A possible south-centered version for the oceans was mentioned although not presented.
|Selected Star Projections Compared|
|A fair assessment of star projections requires
normalized versions with identical scaling factor and number of
lobes. On the right, Bartholomew's "Tetrahedral" (green),
William-Olsson's (amber), Petermann/Berghaus (pink) and
Maurer's S231 (grey) with four lobes and matching central
On the diagram above: by construction, the Tetrahedral and Petermann/Berghaus maps share overall dimensions and equidistant parallel spacing. Compare the parallel spacing of the two equal-area projections, identical in the inner core. Lobes in both Peterman/Berghaus and S231 are split at the Equator.
In contrast with other projections in this section, the one referred to here as "Daisy" was not designed to reduce the number of interrupted coastlines. In fact it was not intended to present the Earth at all, and maybe not even to produce flat maps.
|Reconstruction of a Daisy map with connected hemispheres|
In order to store, analyse, process and render NASA's digital imagery of planets and other bodies, the U.S.Geological Survey (USGS) created the long-running Integrated Software for Imagers and Spectrometers (ISIS), still maintained today. A component of the earlier Planetary Image Carthography System (PICS), the "Daisy" software package — credited to Debbie A.Cook — was ported to ISIS in 2000; its purpose is rendering raster data as "flower petal" maps.
Daisy maps comprise individual starlike hemispheres. In each, the central core is a polar azimuthal equal-area projection with a 15° radius, surrounded by twelve identical lobes using the transverse Mercator projection. Unfortunately, because parallels in the latter are not perfectly circular arcs, at each lobe's junction with the core there is either an overlap or two gaps, depending on the implementation (ISIS's documentation for Daisy does not mention the mismatch or how it is handled).
The USGS has publicly released Daisy maps of Mars, Venus and several Jovian moons, intended to be assembled into tennis ball-sized globes; at that scale, the gaps/overlaps are negligible.