Conic projections are conceptually simple to build in the normal aspect, but are seldom employed to represent more than one hemisphere. However, if interrupted at or near the Equator, the overall distortion of full-world conic maps can become acceptable.
Conic projections have also been combined with other designs in hybrid maps; most commonly, pseudoconic projections since they also have circular parallels.
There is a long tradition of conic hemispheres being applied to celestial charts, maybe even predating the widespread usage of conic projections for geographical purposes. Wilhelm Schickard used the simple equidistant conic as early in 1623 (revised in 1687, with a cone constant of 2/3), followed by Johann-Jacob Zimmermann in 1692 (1706 as cited by de Lalande) and Christoph Cellarius in 1705. Augustus De Morgan presented (1836) the outline of a star map as a double cone which, if assembled, could be inscribed in a sphere, with identical polar and equatorial diameters. The projection was presumably the centrographic conic with 45° standard parallels, but only the ecliptic and a partial graticule were drawn.
In 1945, Oscar S.Adams suggested parameters for minimizing the angular distortion in world maps using the equal-area conic projection. However, he preferred an interrupted version in two hemispheres in the transverse aspect, which with a suitable choice of central meridians leaves the Americas and most of Eurasia/Africa uncut. He also proposed (polar) standard parallels at 24°28′11″N and S in order to minimize maximum angular distortion across each hemisphere: all inhabited continents are crossed by an arc of zero distortion. Alternatively, a standard latitude of 30°N and S yields a cone constant of 0.75 and easier computation.
|Instead of a celestial chart, here an ordinary Earth map helps assessing Lambert's conformal conic projection when applied to a double cone, with constant 5/6 as suggested by Lambert.|
The title of Johann Lambert's seminal paper of 1772 already stated his concern of properly representating both the Earth and the celestial sphere. He mentioned Zimmermann's star cones with a 5/6 constant and proposed his conformal conic projection as a better option, suggesting constants of 3/4, 4/5 and 5/6. In fact, one of the first recorded applications of Lambert's conic orthomorphic projection was a double star cone published by Christlieb Benedict Funke (or Funks) in 1777, with a cone constant of 2/3.
For constellation guides, fidelity of shape is more important than preserving areas or distances, and the conformal conic is a very good alternative to the azimuthal stereographic, another popular choice.
The U.S.Shipping Board and the U.S.Coast and Geodetic Survey collaborated in a large application of Lambert's conformal conic projection, as mentioned by Deetz and Adams (1921). A.B.Clements of the Board suggested a map with circular doubled hemispheres, i.e., conic maps with constant 1/2 but repeated, thus comprising complete circles. Each hemisphere remains conformal, except at the poles.
The Survey designed and built a map with tangent hemispheres 54 inches in diameter and connected by gears, which could be turned in synchronism at will. Therefore, shipping routes drawn on the map could cross the Equator at any point and still be a single, continuous line.
Created by Anton Steinhauser, the conoalactic projection (ca. 1883) closely follows the principles of Petermann's and Berghaus's starlike 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 conic portion. The boundary meridians are 90°W, 0°, 90°E and 180°. Parallels in lobes remain circular arcs centered on the north pole.
The north 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.To be continued...