American polyconic projection

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American polyconic projection of the world American Polyconic projection.jpg
American polyconic projection of the world
American polyconic projection with Tissot's indicatrix of deformation. American polyconic with Tissot's Indicatrices of Distortion.svg
American polyconic projection with Tissot's indicatrix of deformation.

The American polyconic map projection is a map projection used for maps of the United States and regions of the United States beginning early in the 19th century. It belongs to the polyconic projection class, which consists of map projections whose parallels are non-concentric circular arcs except for the equator, which is straight. Often the American polyconic is simply called the polyconic projection.

Map projection Systematic representation of the surface of a sphere or ellipsoid onto a plane

A map projection is a way to "flatten" a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane.

Polyconic projection class

Polyconic can refer either to a class of map projections or to a specific projection known less ambiguously as the American polyconic projection. Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, and the centers of these circles lie along a central axis. This description applies to projections in equatorial aspect.

Contents

The American polyconic projection was probably invented by Ferdinand Rudolph Hassler around 1825. It was commonly used by many map-making agencies of the United States from the time of its proposal until the middle of the 20th century. [1] It is not used much these days, having been replaced by conformal projections in the State Plane Coordinate System.

Ferdinand Rudolph Hassler Swiss-American mathematician and surveyor

Ferdinand Rudolph Hassler was a surveyor who worked mostly in the United States and also in Switzerland. He headed the United States Coast Survey and the Bureau of Weights and Measures.

The State Plane Coordinate System is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First, it uses a simple Cartesian coordinate system to specify locations rather than a more complex spherical coordinate system. By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. Second, the system is highly accurate within each zone. Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping.

Description

The American polyconic projection can be thought of as "rolling" a cone tangent to the Earth at all parallels of latitude. This generalizes the concept of a conic projection, which uses a single cone to project the globe onto. By using this continuously varying cone, each parallel becomes a circular arc having true scale, contrasting with a conic projection, which can only have one or two parallels at true scale. The scale is also true on the central meridian of the projection.

The projection is defined by:

where λ is the longitude of the point to be projected; φ is the latitude of the point to be projected; λ0 is the longitude of the central meridian, and φ0 is the latitude chosen to be the origin at λ0. To avoid division by zero, the formulas above are extended so that if φ = 0 then x = λ  λ0 and y = φ0.

See also

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Rhumb line arc crossing all meridians of longitude at the same angle

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Orthographic projection in cartography map projection of cartography

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The equirectangular projection is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. The projection maps meridians to vertical straight lines of constant spacing, and circles of latitude to horizontal straight lines of constant spacing. The projection is neither equal area nor conformal. Because of the distortions introduced by this projection, it has little use in navigation or cadastral mapping and finds its main use in thematic mapping. In particular, the plate carrée has become a standard for global raster datasets, such as Celestia and NASA World Wind, because of the particularly simple relationship between the position of an image pixel on the map and its corresponding geographic location on Earth.

Sinusoidal projection pseudocylindrical equal-area map projection

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Bonne projection map projection

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Bottomley projection

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Lambert conformal conic projection map projection

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Albers projection map projection

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Hammer projection map projection

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Cylindrical equal-area projection

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Eckert IV projection

The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. In each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert IV is the Eckert III projection.

Eckert II projection

The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to Eckert II is the Eckert I projection.

Armadillo projection

The armadillo projection is a map projection used for world maps. It is neither conformal nor equal-area but instead affords a view evoking a perspective projection while showing most of the globe instead of the half or less that a perspective would. The projection was presented in 1943 by Erwin Raisz (1893–1968) as part of a series of "orthoapsidal" projections, which are perspectives of the globe projected onto various surfaces. This one in the series has the globe projected onto half a torus. Raisz singled it out and named it the "armadillo" projection.

Rectangular polyconic projection

The rectangular polyconic projection is a map projection was first mentioned in 1853 by the U.S. Coast Survey, where it was developed and used for portions of the U.S. exceeding about one square degree. It belongs to the polyconic projection class, which consists of map projections whose parallels are non-concentric circular arcs except for the equator, which is straight. Sometimes the rectangular polyconic is called the War Office projection due to its use by the British War Office for topographic maps. It is not used much these days, with practically all military grid systems having moved onto conformal projection systems, typically modeled on the transverse Mercator projection.

The Eckert-Greifendorff projection is an equal-area map projection described by Max Eckert-Greifendorff in 1935. Unlike his previous six projections, It is not pseudocylindrical.

References

  1. Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 117-122, ISBN   0-226-76747-7.

Eric Wolfgang Weisstein is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He currently works for Wolfram Research, Inc.

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.