Map Projections

## Conic Projections

### Introduction

 Map wrapped on a cone

Conic projections, in the normal polar aspect, have as distinctive features:

• meridians are straight equidistant lines, converging at a point which may or not be a pole. Compared with the sphere, angular distance between meridians is always reduced by a fixed factor, the cone constant
• parallels are arcs of circle, concentric in the point of convergence of meridians. As a consequence, parallels cross all meridians at right angles. Distortion is constant along each parallel

For illustration purposes, the resulting shape can be wrapped on a cone set atop the mapped sphere, although very few conic projections are based on true geometric perspective (in other words, the cone is always the result, but seldom directly participates in its construction). Typically the cone intersects the sphere at one or two parallels, chosen as standard lines.

Due to simple construction and inherent distortion pattern, conic projections have been widely employed in regional or national maps of temperate zones (while azimuthal and cylindrical maps were favored for polar and tropical zones, respectively), especially for areas bounded by two not too-distant meridians, like Russia or the conterminous United States. On the other hand, conic projections are seldom appropriate for world maps.

Relatively few projections are called "conic"; nevertheless, many others are ruled by conic principles, since the cone is a limiting case of both the circle (a cone with no height, and cone constant 1) and the cylinder (a cone with vertex at infinity, with standard parallels symmetrical north and south of the Equator). There is only one type of equal-area conic projection, and only one is conformal.

Conic constraints are relaxed by pseudoconic (with curved meridians) and polyconic (with nonconcentric parallels) projections. Conic and coniclike are among the oldest projections, being the base for Ptolemy's maps (ca. 100).

### Equidistant Conic Projections

 Equidistant conic map, standard parallels 60°N and Equator, central meridian 0°. A full map is presented for illustration only, since this projection is seldom used for worldwide maps. Equidistant conic map, standard parallels 30°N and 60°S Euler map, standard parallel 45° should be clipped at limiting parallels 90°N and Equator. Detail of equidistant conic map with standard parallels as chosen by Mendeleyev (90°N and 55°N); central meridian 100°E. Only a tiny missing wedge prevents it from being a full azimuthal map.

The equidistant (also called simple) conic projection has constant parallel spacement, thus scale is the same along all meridians. Commonly one or two parallels are chosen to have the same scale, suffering from no distortion.

Neither equal-area nor conformal (but an acceptable compromise for most temperate countries), this projection is defined arbitrarily instead of by a perspective process. It is the general case of both azimuthal equidistant and equidistant cylindrical projections.

Historically, Ptolemy's first known map resembled an equidistant conic with meridians broken at the Equator. The equidistant concept was adopted and "improved" by many authors, the most famous being Joseph N. de l'Isle (also Delisle), one of a family of cartographers and map publishers. De l'Isle introduced variations to the equidistant model; the projection today bearing his name (ca. 1740) is not a true conic, despite a close resemblance. B. Mead proposed in 1717 a variation with parallels comprising 1°-long straight segments.

Several cartographers kept the general arrangement but studied criteria for standard parallel placement in order to minimize distortion. These include a series by P. Murdock (1758) and Everett (1903), and Euler's projection (1777). Russian usage of the equidistant conic led to other derivations, including by V.V. Kavrayskiy (ca. 1930, 62°N and 47°N) and by D. Mendeleyev (1907, 90°N and 55°N) of chemistry fame.

### Equal-area Conic Projections by Lambert and Albers

 Map in Albers's conic projection, rendered with standard parallels 60°N and 30°N; reference parallel 45°N, central meridian 0° Map in Lambert's equal-area conic projection, standard parallels 90°N and the Equator Detail of map of Europe using Lambert's equal-area conic with proposed cone constant 7/8; central meridian 30°E

The German Heinrich C. Albers published his equal-area conic projection in 1805. As usual, there is little distortion along the central parallel and none on the standard ones.  The standard parallels may lie on different hemispheres, but if equidistant from the Equator, the projection degenerates into an equal-area cylindrical.

This projection was commonly applied to official American maps after usage of the polyconic projection declined.

In a particular case of Albers's conic projection, either 90°N or 90°S is chosen as a standard parallel, and therefore meridians converge at a pole.  Published by Lambert in 1772, this projection preserves areas, thus parallels are farther apart near the vertex, getting closer together towards the non-standard pole.  When 0° is chosen as the other standard parallel, the result is a cone constant of 1/2 and a semicircular map. Lambert himself chose a constant of 7/8 for his map of Europe: the resulting standard parallel, roughly 48°35'N, lies between Paris and Munich.

This projection was employed much less frequently than Albers's.  In fact, it is probably the least known of Lambert's projections.

### Lambert's Conformal Conic Projection

 Conformal conic map with standard parallels 50°N and 10°S, clipped at 50°S.

The same paper (1772) with Lambert's equal-area conic projection included his conformal conic design: Lambert explicitly investigated a conic approach as intermediary between the then known conformal projections, azimuthal stereographic and Mercator's. These are in fact special cases of the conformal conic, obtained respectively when one pole is the single standard parallel and when the standard parallels are symmetrically spaced above and below the Equator.

This projection remained essentially ignored until World War I, when it was employed by the French military. Since then, it has become one of the most widely used projections for large-scale mapping, second only to Mercator's.

Like in all conformal projections, scale distortion is greatly exaggerated in the borders of a worldwide map, although less than in Mercator's. Meridians converge at the pole nearest the standard parallels; the opposite pole lies at infinity and can not be shown. Scale distortion is constant along each parallel. Meridian scale is less than true between the standard parallels, and greater "outside" them.

### Perspective Conic Projections

#### Braun Stereographic Conic Projection

 Braun stereographic conic map
Actually the only conic projection presented here which is defined by a simple geometric construction, the stereographic projection created by C. Braun (1867) encloses the globe in a cone aligned with the north-south axis, 1.5 times as tall as the globe and tangent at the 30°N parallel.  The projection center is the South pole and the resulting map fits a perfect semicircle.

### Polyconic Projections

 Three partial equidistant conic maps, each based on a different standard parallel, therefore wrapped on a different tangent cone (shown on the right with a quarter removed plus tangency parallels). When the number of cones increases to infinity, each strip infinitesimally narrow, the result is a continuous polyconic projection.
Cartographers apply the name polyconic to:
• a specific map projection associated with F. R. Hassler, also called American polyconic
• a few projections derived from Hassler's, all geometrically inspired by stacked, overlapping cones; they include the rectangular polyconic
• a more general group of projections with nonconcentric circular parallels in the normal aspect

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.

#### (American) Polyconic Projection

 Polyconic map, central meridian 100°W, emphasizing its classic use: mapping the United States.

In ordinary conic projections, only one or two parallels, where the conic and spherical surfaces coincide, have correct scale. However, the map may be divided in strips of similar latitude, each fitted to a different cone. Cone constant varies from one at poles to infinity at the Equator, so the strips are not continuous, 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 central meridian has constant correct scale, the result is the classic or common polyconic projection, also called American polyconic.

Most authors credit the Swiss Ferdinand R. Hassler with designing the classic polyconic (ca. 1820) while leading the government agency called, for most of its history, the U.S. Coast and Geodetic Survey. 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 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.

#### Rectangular (War Office) Polyconic Projection

 Rectangular or War Office polyconic map, central meridian 100°W, Equator as standard parallel. Note how meridians are broken at poles.

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, but changes 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 (common case) or two parallels symmetrical about the Equator have true length.

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

 www.progonos.com/furuti    August 28, 2012