Map Projections

Star Projections

Reconstructions of Jäger's (left) and Petermann's projections

Introduction

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 Northern 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

Early Projections by Jäger and Petermann

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

Berghaus's Projection

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.

Maurer's Star Maps

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 Northern 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).

William-Olsson's Projection

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 Northern 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 Northern 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

The "Tetrahedral" Projection

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 starting at 30°W

Published in both Northern and Southern polar aspects, this projection apparently owes its name to a fortuit resemblance of its lobe arrangement to a common lay-out for unfolded tetrahedra; it is unrelated to true polyhedral maps.

The Conoalactic Projection

Maps in the conoalactic projection with original lobe divisions (top) and (above) boundary meridians shifted West by 15° to reduce shearing of Africa and South America.

Created by Anton Steinhauser, the conoalactic projection (ca. 1883) closely follows the principles of Petermann's and Berghaus's earlier proposals, but with an equidistant conic projection instead of the azimuthal equidistant for the inner hemisphere. It comprises four identical lobes with straight meridians; the central meridians copy the constant scale of the core. The boundary meridians are 90°W, 0°, 90°E and 180°. Parallels in lobes remain circular arcs centered on the Northern Pole.

The Northern Pole is mapped to a point and the Equator spans 240°; the second standard parallel is at approximately 4°17'52.90"N. This projection is neither equal-area nor conformal.

www.progonos.com/furuti    October 26, 2012