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Conic Projections

Introduction

cones and spheres
Polar maps wrapped on cones

Conic map projections, in the normal and almost universal polar aspect, have as distinctive features:

For illustration purposes, the resulting shape can be wrapped on a cone set atop the mapped sphere, although all important conic projections are not based on true geometric perspective — in other words, the conic surface is always the projection's result, but seldom directly participates in its construction.

Typically, either one or two parallels are chosen to be standard lines; in perspective projections, they define where the cone actually intersects the sphere (the tangent and secant cases, respectively). In non-perspective designs, there is no such assurance, although the names tangent and secant are traditionally retained.

Cones: two parallel paths from azimuthal to cylindrical
Comparing conic projections
Conic projections are general cases of azimuthal and cylindrical projections. All maps above occupy the same area, because the three projections used are equal-area and were applied at identical scaling factors. The appearance of a particular conic map is affected by the cone constant, which in turn is determined by the standard parallels (highlighted in green above); whether the map converges at an apex or comprises a truncated cone (frustum) depends on a pole being or not a standard parallel.
Comparing conic projections
Because Albers's projection is not defined by a perspective process, when visualized as actual cone models those maps do not exactly fit an underlying sphere. For instance, in the version with standard parallels at 90°N and 30°S they can not both coincide in place with the sphere's counterparts, even though their corresponding lengths are identical.

Owing to a simple construction and inherent distortion pattern, conic projections have been widely employed in national or large-scale regional maps of temperate zones — while azimuthal and cylindrical maps are favored for polar and tropical zones, respectively — especially for areas bounded by two moderately close meridians, like Russia or the conterminous United States. As a consequence, the ellipsoidal case has been developed for noteworthy conic maps. On the other hand, conic projections are seldom appropriate for uninterrupted world maps, with one hemisphere necessarily suffering from much higher distortion than the other.

Relatively few projections are called "conic"; nevertheless, many others are ruled by conic principles, since both the disc and the cylinder are limiting cases of the cone: the first a flattened cone with a standard parallel at a pole and constant 1, the other a cone with apex at infinity, constant 0 and standard parallels symmetrical north and south of the Equator. There is only one type of equal-area conic projection, and only one is conformal.

The conic constraints are relaxed by pseudoconic (with curved meridians) and polyconic (with nonconcentric parallels) projections; except for a non-strict definition of polyconics, circular parallels are retained. Conic and coniclike are among the oldest projections, being the base for Ptolemy's maps (ca. 100 AD).

Perspective Conic Projections

Perspective conic projections in cross-section
Cross-sections of perspective conic projections tangent at 30°N (a latitude arbitrarily chosen, except for Braun's stereographic). Light rays in red are shown for parallels spaced 30° apart. The tangency latitude determines the extension of the blue mapping surface and also the thin blue contour which delimits the mappable range.

Compared with the plane and cylinder, the cone offers more freedom regarding the source and direction of light rays. However, perspective conic projections have never been significant. They are rarely, if ever, mentioned in cartography textbooks — not even, like the central cylindrical, as a negative example. Despite their geometric simplicity, they offer few interesting properties, even when compared with the very common equidistant conic, which is simpler still.

Simple Perspectives

From top to bottom, orthographic, stereographic and centrographic perspective conic maps, all with a single standard parallel 30°N. The orthographic and centrographic maps cannot present areas south of 60°S; the stereographic version can show the whole world but was arbitrarily clipped at 60°S, and the centrographic, at 35°S. Compare parallel spacing and the mapped range in the maps and in the cross-sections above. Orthographic conic map
Stereographic conic map
Centrographic conic map

Conventionally, the geometry of perspective conic projections is defined by a line connecting the center of the sphere to the tangency point (or the angular midpoint for the secant case), therefore normal to the mapping surface. In the orthographic conic projection, all rays are parallel to that line; in the stereographic, they shoot from the antipodal point; and in the centrographic, from the center of the sphere.

The orthographic conic generalizes both the azimuthal orthographic and Lambert's cylindrical equal-area projections but except in that latter's particular case it is not itself equal-area and its hemisphere opposite the tangent point is restricted to 90° of latitude beyond that point. The stereographic conic bridges the gap between the azimuthal stereographic and Carl Braun's stereographic cylindrical, and like the latter, can show the whole world. The centrographic, a general case of both the azimuthal gnomonic and the already mentioned central cylindrical projection, is restricted to the same mappable range as the orthographic, but in practice is clipped to little more than one hemisphere.

Other interpretations are possible: for instance, an orthographic projection could have rays normal to the polar axis, thus making the whole world mappable. Or Braun's stereographic, centered on a pole instead of on an antipode.

Historically, a centrographic geometric model was the base for P.Murdoch's three conic projections of 1758. A secant case was described by C.Colles in 1794. None of them received much attention.

Braun's Stereographic Conic Projection

Braun's stereographic conic map
Braun's stereographic conic map
The stereographic conic projection published by Braun (1867, together with his stereographic cylindrical) is a perpective design tangent at 30°N, defined by light rays shooting not from the antipodal point at 60°S, but from the South pole.

HomeSite MapHybrid Pseudocylindrical ProjectionsMap Projections - ContentsNon-perspective Conic Projections  www.progonos.com/furuti    November 18, 2013
Copyright © 1996, 1997, 2003, 2013 Carlos A. Furuti