Polar maps wrapped on cones |

In their normal and almost universally used polar aspect, the distinctive features of conic map projections are:

- meridians are straight equally-spaced lines, converging at a
point which may or not be a pole. Compared with the sphere, the
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. In consequence, every parallel crosses all meridians at right angles, and the pattern of distortion is the same along each parallel. Given the same constant, different cone projections are distinguished by parallel spacing only

For illustration purposes, any conic map can be wrapped on
a cone, although
all important conic
projections are not based on a simple perspective model
— in other words, the conic surface is always the projection's
*result*, but seldom directly participates in its
geometric *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, pointed or not: two parallel paths from azimuthal to cylindrical |

Conic projections are general cases of azimuthal and cylindrical projections. All maps above occupy the same area, because the three projections used (actually all particular versions of Albers's conic) are equal-area and were applied at identical scaling factors. The general appearance of a 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. |

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 inherently simple construction and 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. The ellipsoidal case has been developed for noteworthy large-scale 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 azimuthal disc and the cylinder are limiting cases of the cone: the first is a flattened cone with a standard parallel at a pole and cone 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* projections correlate corresponding points on globe
and map using a set of straight lines with a common origin, by analogy to
light rays of geometric perspective.
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 and still simpler
equidistant conic.

Conventionally, the geometry of perspective conic projections is
defined by a line touching the mapping surface at the tangency point
(or the angular midpoint for the secant case). In
the *orthographic* conic projection, all rays are parallel to
that line and normal to the surface; 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 map |

Copyright © 1996, 1997, 2003, 2013 Carlos A. Furuti