Map Projections

Star Projections

Introduction

Since most continental area is concentrated around the North Pole, some star-like interrupted projections were designed having part or all of the northern hemisphere as a more or less circular shape and splitting the remaining surface in lobes placed around it instead of in a row.  Of course, for ordinary maps only the north polar aspect is interesting and Antarctica is always divided. A few projections emphasize oceanic areas by inverting this pattern.

Most star maps are actually composites, employing azimuthal or near-azimuthal projections for the central "core", which is bounded by an arbitrary parallel. The lobes are projected differently but frequently retain the core's parallel spacing.

Early Maps: Projections by Jäger and Petermann

Star-shaped maps probably date from works by Leonardo da Vinci (1514) and Guillaume Le Testu (1556).

	Possible reconstructions of Jäger's (left) and Petermann's projections

One of the earliest modern designs was the compromise projection published by G.Jäger (1865). In its polar aspect, all meridians in the core hemisphere are straight lines evenly spaced, but with different scale; the Equator and eight arbitrary meridians define eight triangular sectors. Parallels in this hemisphere are straight lines broken at sector boundaries, perpendicular to and equally spaced along each sector's central meridian. The outer hemisphere comprises eight triangles mirroring the core sectors, therefore with simple straight lines for parallels.

Since the meridians bounding sectors are not evenly spaced, the core octagonal hemisphere is irregular, and lobes differ in width and length; later star projections usually have symmetric lobes.

Petermann almost immediately (1865) reproduced Jäger's proposal but greatly modified the map with distinct projection methods for the core and outer hemisphere:

• the core hemisphere uses an ordinary azimuthal equidistant projection
• the outer hemisphere is divided in eight identical lobes (although I have found conflicting descriptions)
• in each lobe, the central meridian is a straight line; all other meridians are straight lines broken at the Equator
• in each lobe, parallels are concentric arcs of circle spaced exactly like in the core hemisphere

Almost unused, Jäger's and Petermann's projections are neither conformal nor equal-area.

Classic Berghaus map

Berghaus's Star Projection

Although other arrangements are possible, Berghaus recommended interrupting the southern hemisphere at 56°E, 128°E, 160°W, 88°W and 16°W.  As a result, of all major land masses only Australia and Antarctica are divided.

Berghaus map variants

Out of curiosity, I used Berghaus's approach with other numbers of lobes, although less than five is prone to greater distortion and much more causes excessive land cutting, thus defeating the projection's purpose. 

Several star maps can be cut, folded and assembled into pyramids; the 3-point Berghaus map is an equilateral triangle foldable into a regular tetrahedron.

Conoalactic Projection

Conoalactic map, interrupted in South hemisphere at 90°W, 0°, 90°E

Introduced by Steinhauser in 1883, the conoalactic projection is actually very similar to Petermann's and Berghaus's in construction. The northern hemisphere is based on a simple equidistant conic projection, while the southern hemisphere is split in four lobes having equidistant parallels and straight meridians broken at the Equator.  It is neither equal-area nor conformal; scale is constant along the standard parallels 0°N and 4°17'53"N and in the four unbroken meridians. Other meridians preserve distances only in the northern hemisphere.

Notwithstanding its unusual shape, this projection can also be classified as star-like.

Maurer's Star Projections

Possible reconstruction of Maurer's S231 projection in North polar form, with lobe divisions starting from 21°W.

Hans Maurer developed a monumental effort to catalogue and organize every possible projection according to hierarchical categories and rigidly defined criteria (1935). He also designed many new projections, sometimes for illustrating gaps in his classification system.

The two projections Maurer called S231 and S233 are star-shaped. The first extends the northern hemisphere of a Lambert azimuthal map and is also equal-area, even in the interrupted hemisphere.  The meridians in each lobe are not straight but delicately curved lines.  Parallels are concentric arcs of circle symmetrically spaced around the Equator.  Meridian spacing along parallels is also symmetric above and below the Equator.

Maurer's S233 projection, interrupted from 20°W.

Although described with six lobes, both star projections can be easily generalized to any number, at least two (S231) or three (S233).

"Tetrahedral" Projection

Reconstruction of a North polar tetrahedral projection, interrupted starting at 30°W

Although it has no relation to true polyhedral maps, this projection is called "tetrahedral", maybe due to a vague resemblance to the face arrangement of a tetrahedron's fold-out. Neither equal-area nor conformal, it was used in both North and South polar aspects.

www.progonos.com/furuti    February 20, 2008