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Non-perspective Conic Projections

Introduction

Schjerning I projection, a particular case of equidistant conic map
Schjerning's first projection, or the north polar equidistant conic with cone constant 1/2. A rare case of conic map designed for the whole world.

Historically, only three fundamental types of conic projection have gained substantial adoption. They are defined by:

As a customary compromise in cartography, once a main feature is designated, projection parameters — in this case, one or two standard parallels — may be determined in order to reduce overall deformation. Mainly this means:

Optimal selection of parallels may be obtained by analytical or numerical methods like least squares; sometimes criteria can be distilled into rules of thumb — e.g., set standard parallels at 1/6 and 5/6 between the minimum and maximum latitudes of interest (Deetz and Adams).

The Equidistant Conic and Variations

Equidistant conic map
Equidistant conic map, standard parallels 30°N and 45°S.

The conic projection with the easiest construction method is the simple or equidistant conic, with uniformly spaced parallels. Neither equal-area nor conformal, except along the standard parallels, but an acceptable compromise for most temperate countries, it is the general case of both azimuthal equidistant and equidistant cylindrical projections.

Its origins can be traced back at least to Claudius Ptolemy's first partial world map, which is purely conic only north of the Equator. It has been applied since at least the 17th century to Earth and star maps, occasionally with modifications, some of which are perhaps due to the difficulty of drafting arcs of large diameter, which frequently occur with small cone constants. Usage persists today in local and regional maps.

Equidistant conic map
Equidistant conic map, central meridian 90°E. Standard parallels 90°N and 55°N, after Mendeleev. Cone constant 0.939.
Equidistant conic map
Equidistant conic map, central meridian 95°E. Given limiting parallels 70°N and 40°N, Euler's constraints set the cone constant to 0.810.
Equidistant conic map
Equidistant conic map, standard parallels 62°N and 47°N after Kavraiskiy. Central meridian 95°E, cone constant 0.812.

Some equidistant conic maps credit the projection to "Delisle" (or "de L'isle", or other variants). Ironically, although Joseph Nicholas de L'Isle did employ the (true) equidistant conic in many maps published by his family business, the projection he actually created (ca. 1745) is not a strictly conic modification, because its meridians connect points equally spaced along two limiting straight lines instead of parallels; therefore they do not in general converge at a point.

Another modification, by Mead (1717), was based on trapezoidal cells or quadrangles, each 1°-wide. In each cell the limiting parallels and the central meridians are standard lines. The map, no longer conic, would resemble part of a spider web. Both Mead's and the proper "De L'isle" projection are completely obsolete today.

The first projection suggested by Wilhelm Schjerning (first published in 1882, then again in 1904 with his elliptic and cordiform/oblique proposals) is a north polar equidistant conic with constant 1/2.

The distinguished mathematician Leonhard Euler presented in 1777 criteria for choosing the cone constant given two limiting parallels: in the result, sometimes called Euler's projection, given a longitudinal range, the distance error along the extreme latitudes is the same but opposite magnitude as the error along the central parallel.

Russia, like the former Soviet Union, encompasses a vast east-west range with intermediate to high latitudes, an ideal subject for conic maps. Accordingly, several Russian and Soviet cartographers have explored criteria for optimizing the placement of standard parallels; notable examples include Vitkovskiy (1907), Mendeleev (1907), Mikhaylov (1911-2), Krasovskiy (1922-5) and Kavrayskiy (1934). Dmitri I. Mendeleev, more famous after his contributions to chemistry, preferred standard latitudes 90°N and 55°N. Vladimir V. Kavrayskiy, author of various other projections, favored 62°N and 47°N.

Equal-area Conic Projections by Lambert and Albers

Lambert's equal-area conic map
Albers's equal-area conic map
Top and above, equal-area conic maps with identical cone constant 0.707; top: Lambert's projection, standard parallels 90°N and 24°28'11"N; above: Albers projection, standard parallel 45°N.
Map of Europe in Lambert's equal-area conic map
On the right, approximate (the essential standard latitudes are correct, but the central meridian is approximate because Lambert's coordinate grid used a reference meridian other than Greenwich) reconstruction of Lambert's map of Europe using his equal-area conic projection with a 7/8 constant.

Below, Lambert's equal-area conic projection applied to the whole world, standard parallels 90°N and 18°25'S, central meridian 10°E as proposed by O.S.Adams.

Lambert's conic map as proposed by Adams

Among several other topics, Johann Lambert's extensive monograph of 1772 considered what happens when meridians are represented as straight lines separated by a constant angle other than real — as in azimuthal projections — but not parallel as in cylindrical projections: the outcome can incidentally be wrapped over a cone. Lambert, born in Alsace (then part of Switzerland), applied calculus to solve the case of preserving the fidelity of areal relationships with either 90°N or 90°S as a standard parallel — i.e., one pole is a point and the cone ends at an apex; the resulting proposal, his conic equal-area, or Lambert's isospherical stenoteric (the name used by Adrien Germain and O.S.Adams, probably because it preserves the sphere's area while narrowing meridian spacing) projection, was later generalized by Albers.

Lambert's paper was illustrated with a small map of Europe, for which he conjectured a range of latitudes of 30°N to 70°N would be enough, and the second standard parallel, with no shape distortion, could be placed about 50°N. For easier calculation, a cone constant of 7/8 was chosen, determining the actual standard latitude to be 48°35'25"N, somewhere between Paris and Munich. Another simple choice is 0°, which yields a semicircular world map.

For a whole-world equal-area conic map, Oscar S.Adams (1945) suggested 90°N and 18°25'S as standard parallels, in order to minimize shape distortion between the north pole and the 50°S parallel, considered regions of human importance. However, he preferred a transverse interrupted version in two hemispheres.

A German, Heinrich C.Albers published his conic equal-area projection in 1805 as a general case of Lambert's conic, with one or two standard parallels not necessarily on a pole. If the standard parallels are equidistant from the Equator or coincide with it, the design degenerates into Lambert's equal-area cylindrical projection, or one of its rescaled variants.

Not much is known about Albers, and for a long time his design received little attention from the cartographic literature; only relatively recently it became a common choice for equal-area (i.e., usually small-scale and statistical) maps of U.S. government agencies and elsewhere. An influential proponent was Adams, who developed criteria for minimizing distortion and equations for the ellipsoidal case (ca.1927).

Areal v. Angular Distortion
Map of Australia in Albers's equal-area conic map Map of Australia in Albers's equal-area conic map
Map of Australia in Lambert's conformal conic map Map of Australia in Lambert's conformal conic map
Tissot's indicatrices enable to compare the distortion in maps of Australia drawn with Albers's equal-area (top) and Lambert's conformal (bottom) conic projections. Points along green standard parallels — 15°S and 35°S (left) and 25°S (right) — suffer from no distortion, areal or angular. Notice in the equal-area conic how the direction of shape distortion is orthogonal between the standard parallels and elsewhere; notice how areal distortion changes from compression to expansion between the parallels and elsewhere in the conformal conic.

Lambert's Conformal Conic Projection

Europe map using Lambert's conformal conic projection
Approximate reconstruction of Lambert's conformal conic projection applied to Europe. Standard parallels are exact, but Lambert's prime meridian was not based on Greenwich.

On the same monograph of 1772, Lambert created the bridge between the polar azimuthal stereographic and the equatorial Mercator's projections, again using calculus, this time to ensure angles are locally preserved. Developed for both spherical and ellipsoidal cases, the conic conformal projection, also known as the conical orthomorphic, is today the most important and commonly employed of all of Lambert's original proposals, comparable to the transverse Mercator. He illustrated it with a small map of Europe, with the same choice of limiting and standard parallels as of the equal-area version; in the conformal case, the cone constant is 3/4 instead of 7/8. The stereographic and Mercator's are the conformal conic's limiting cases, respectively when a pole is the single standard latitude and when both standard parallels are symmetrically spaced around the Equator (with a suitable rescaling if not coinciding).

Surprisingly, this projection scarcely saw practical usage and its true authorship remained virtually unrecognized until World War I, when it was adopted in French battle maps (not to be confused with the French military Système Lambert of 1918, a grid standard based on the nonconformal, non-equivalent minimum error projection by Tissot) and by the American USGS. Apparently it was independently developed by Charles L.Harding in the spherical case and applied to star atlases from 1808 to 1822. The great mathematician J.Carl F.Gauss — himself an important contributor to the field of conformal mapping and author of the fundamental theorem which implies that no flat map can represent a sphere without distortion — acknowledged Harding's work, but not Lambert. John Herschel reinvented Lambert's spherical equations and proposed a world map with cone constant 1/3 (1859); he also offered criteria for choosing standard parallels in order to reduce global scale variation. George Boole, the forerunner of modern digital computer logic, extended Herschel's equations to the ellipsoid (ca. 1860). Harding, Gauss, Herschel have been variously credited as creators of the projection, even in specialized treatises like Thomas Craig's (1882).

World map using Lambert's conformal conic projection as proposed by Herschel
Lambert's conformal conic projection applied to a world map, with cone constant 1/3 as suggested by Herschel. Conformal everywhere, except at the North Pole (where the sum of meridian spacings comprises 120° instead of 360°) and the South Pole, which cannot be shown. Large scale distortion is obvious near the visible pole and in the (arbitrarily clipped) hemisphere opposite it.

As usual in conformal projections, Lambert's conic is better used in large-scale topographic mapping; uninterrupted world maps present too large a range of scales. It can be constructed with either one or two standard parallels; at almost every point the scale, due to conformality, is uniform at every direction, less than true between the standard parallels, greater elsewhere; only the standard parallels are free of any distortion. Conformality fails at both poles: around one, the sum of all meridian angles is less than 360°, while the other one lies at infinity.

More recently, the conformal conic has become a standard of many official mapping agencies; in the USGS, it superseded the American polyconic. It is also the base for the bipolar oblique conic, a compound of two circular sectors using oblique projections focused on the Americas; in each section, the original parallels lie roughly aligned with the continental "crescents": concave towards the Northeast in North America, to the Southwest in the South. A narrow compromise strip across Central America and the Caribbean where the components meet is nonconformal, although the two standard parallels are skillfully chosen to connect exactly. Published in 1941 and the base for several other works, the bipolar oblique conic map was developed by Osborn Miller and William Briesemeister; although released in several sheets, the relatively small scale meant only spherical equations were needed.



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