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Interrupted and Compound Conic Maps

Interrupted centrographic conic map
Views of a double cone inscribed in a sphere
Flat map of a centrographic conic projection interrupted at the Equator with conic constant 0.707, and several views of the assembled double cone inscribed in a semitransparent sphere.

Introduction

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.

Lambert's Equal-area Conic

Interrupted transverse equal-area conic map
Lambert's equal-area map in the transverse aspect, with two hemispheres and conic constant 0.707 as suggested by O.S.Adams. Original standard parallels highlighted.

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.

Lambert's Conformal Conic

More Star Maps

Interrupted conic conformal map
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.

Interrupted conic conformal map
With a constant 2/3 (like in Funke's star charts of 1777), this interrupted transverse conic conformal map resembles Cahill's butterfly maps, as suggested by John Savard (Savard selected a standard parallel equivalent to 40°, making the map slightly less symmetric; in a variation he also rearranged portions of the map in order to reduce interruptions in Antarctica).
Double conic conformal maps
Rendered with identical equatorial diameters, a sphere and three double cones using the conformal conic projection with constants 2/3, 3/4 and 5/6.

Doubled Double Cones

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.

Interrupted conic conformal map
Double conic map with doubled hemispheres as suggested by Clements and built by the U.S. Coast and Geodetic Survey as a large geared drawing board for shipping courses.

The Conoalactic Projection

Conoalactic map
Conoalactic map
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 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 DO To be continued...

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