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Polyconic Projections

Introduction

From the equidistant conic to the American polyconic
Equidistant conic map in 3 sectors Equidistant conic map in 12 sectors
Equidistant conic map in 21 sectors Equidistant conic map in 30 sectors
It helps to visualize the polyconic concept with a series of cones or frustums mapped as stacked conic maps, each with standard parallels best matching a range of latitudes. As the number of frustums grows to infinity, each sector slimming to a wisp, the result approaches a continuous polyconic map where, since many gaps must be filled at the periphery, it is obvious how scale distortion grows. However, if equidistant conic sectors are used, there is no distance distortion along any parallels, and none at all along the central meridian.

Cartographers apply the name polyconic to:

Quite heterogeneous, the latter group includes many designs with little or no relationship with cones other than the name. They include works by McCaw, Ginzburg and Salmanova. Some authors extend the definition to include projections like Aitoff's and Hammer's, with noncircular parallels.

(American) Polyconic Projection

Polyconic map
The ordinary polyconic projection is usually applied to local or regional, rather than world maps.

In ordinary conic projections, only one or two parallels — where in true perspective projections the conic and spherical surfaces coincide — are standard lines with correct scale. However, the map may be split along parallels in several strips, each fitted to a different cone; this is similar to the polycylindrical concept. The cone constant is made to change from one at the poles to zero at the Equator, so the strips do not touch, except along the central meridian. When infinitely many cones are used, each optimally tangent to a thin strip containing a single parallel, the gaps disappear; if the base projection is the equidistant conic, the central meridian has constant correct scale and the now continuous result is the classic or common polyconic projection, also known as the American polyconic.

On the left: superimposing the compound equidistant conic map on the polyconic illustrates their close relationship.
Below: moving the central straight meridian of the polyconic is similar to rearranging the tangency point of the equidistant conic sectors. This kind of generalization is more of a novelty, perhaps more useful in the transverse aspect.
Equidistant conic map in 30 sectors superimposed on polyconic map
Equidistant conic map in 30 sectors Generalized polyconic map

The Swiss Ferdinand R. Hassler is credited with designing the American polyconic projection (ca. 1820) while leading the government agency called, for most of its history, the U.S. Coast and Geodetic Survey — today the National Geodetic Survey, part of the National Oceanic and Atmospheric Administration. Applied in ellipsoidal form to most official large-scale maps until about 1920, it was adopted by several other countries and official agencies.

The classic polyconic projection has circular parallels (except the Equator), all with constant and correct scale, but not concentrical. The curvature of each parallel is the same of its counterpart on a cone tangent at its latitude. That same scale applies to the straight central meridian; all other meridians are curved. Neither equivalent nor conformal, this projection is better suited for local or regional maps. Compared with the equal-area conic and conformal conic projections, which together with the transverse Mercator largely superseded it in American agencies, its chief disadvantage is the much wider range of scale variation.

Polyconic map
Rectangular or War Office polyconic map, with the Equator as standard parallel.

Rectangular (War Office) Polyconic Projection

Also developed (1853) at the U.S. Coast and Geodetic Survey, the best-known modification of Hassler's projection was widely employed for large-scale mapping by the British War Office, thus its common name. It is also called rectangular polyconic due to graticule angles, not overall map shape.

In the rectangular polyconic projection, parallels are circular arcs, again equally spaced along the straight central meridian. However, their scale is not constant, changing in order to make each meridian cross every parallel at right angles. This is not a sufficient condition for conformality, neither is the result equivalent. Besides, only the Equator (the common case) or two parallels symmetrical about the Equator have true length.

Polyconic map, North America Rectangular polyconic map, North America
Ordinary (left) and rectangular (right) polyconic maps of North America with Tissot indicatrices.

Usage of the rectangular polyconic projection is similar to the classic polyconic's; in fact, for small regions they are barely distinguishable.

Modified Polyconic Projection

A modification of the classic polyconic projection by Charles Lallemand was used in the International Map of the World series, a set of separate sheets in the 1 : 1000000 scale (the project is also known as the Map of the Million or the Millionth Map of the World) with standardized notation and nomenclature. Initially based on the British Ordnance Survey ca. 1913, it was later maintained by the United Nations. Considerable difficulties had to be overcome as each country was initially supposed to contribute its own maps; in practice many nations had neither resources nor expertise to conduct their own geographic surveys. Around 1980 the project seemingly became stagnant with less than half of the planned maps actually published. However, the indexing system for sheets is still used today in other projects.

In Lallemand's officialy adopted projection, each sheet comprises a single quadrangle bounded by two parallels separated by 4°, both standard lines. All parallels are nonconcentric circular arcs, with curvature identical to the classic polyconic's. However, meridians are straight lines, but not converging at a point. Near the Equator, the boundary meridians are spaced by 6°, and the two 2° apart from the center are standard lines. The width of quadrangles changes to 12° north of 60°N and south of 60°S, and again to 24° after the 76°latitudes.

Due to the choice of standard parallels and straight meridians, any pair of neighboring sheets may be joined with no gaps; however, gaps cannot be avoided if sheets are joined in more than one direction at once. Like the ordinary and rectangular polyconics, the projection is neither conformal nor equal-area.



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Copyright © 1996, 1997, 2003 Carlos A. Furuti