Lambert cylindrical equal-area projection

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Comparison of the Lambert cylindrical equal-area projection and some cylindrical equal-area map projections with Tissot indicatrix, standard parallels and aspect ratio Tissot indicatrix world map cyl equal-area proj comparison.svg
Comparison of the Lambert cylindrical equal-area projection and some cylindrical equal-area map projections with Tissot indicatrix, standard parallels and aspect ratio
Lambert cylindrical equal-area projection of the world Lambert cylindrical equal-area projection SW.jpg
Lambert cylindrical equal-area projection of the world
Lambert cylindrical equal-area projection of the world, central meridian at 160degW to focus the map on the oceans. Oceans base map.svg
Lambert cylindrical equal-area projection of the world, central meridian at 160°W to focus the map on the oceans.
Lambert cylindrical equal-area projection with Tissot's indicatrix of deformation Tissot indicatrix world map Lambert cyl equal-area proj.svg
Lambert cylindrical equal-area projection with Tissot's indicatrix of deformation
How the Earth is projected onto a cylinder Cilinderprojectie-constructie.jpg
How the Earth is projected onto a cylinder

In cartography, the Lambert cylindrical equal-area projection, or Lambert cylindrical projection, is a cylindrical equal-area projection. This projection is undistorted along the equator, which is its standard parallel, but distortion increases rapidly towards the poles. Like any cylindrical projection, it stretches parallels increasingly away from the equator. The poles accrue infinite distortion, becoming lines instead of points.

Contents

History

The projection was invented by the Swiss mathematician Johann Heinrich Lambert and described in his 1772 treatise, Beiträge zum Gebrauche der Mathematik und deren Anwendung, part III, section 6: Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten, translated as, Notes and Comments on the Composition of Terrestrial and Celestial Maps. [1]

Lambert's projection is the basis for the cylindrical equal-area projection family. Lambert chose the equator as the parallel of no distortion. [2] By multiplying the projection's height by some factor and dividing the width by the same factor, the regions of no distortion can be moved to any desired pair of parallels north and south of the equator. These variations, particularly the Gall–Peters projection, are more commonly encountered in maps than Lambert’s original projection due to their lower distortion overall. [1]

Formulae

where φ is the latitude, λ is the longitude and λ0 is the central meridian. [1]

See also

Related Research Articles

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Orthographic projection in cartography map projection of cartography

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Mollweide projection map projection

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Equirectangular projection map projection that maps meridians and parallels to vertical and horizontal straight lines, respectively, producing a rectangular grid

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

The Bottomley map projection is an equal area map projection defined as:

Tissots indicatrix

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

Lambert conformal conic projection map projection

A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten.

Albers projection map projection

The Albers equal-area conic projection, or Albers projection, is a conic, equal area map projection that uses two standard parallels. Although scale and shape are not preserved, distortion is minimal between the standard parallels.

Lambert azimuthal equal-area projection map projection

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection.

Van der Grinten projection map projection

The van der Grinten projection is a compromise map projection, which means that it is neither equal-area nor conformal. Unlike perspective projections, the van der Grinten projection is an arbitrary geometric construction on the plane. Van der Grinten projects the entire Earth into a circle. It largely preserves the familiar shapes of the Mercator projection while modestly reducing Mercator's distortion. Polar regions are subject to extreme distortion.

Hammer projection map projection

The Hammer projection is an equal-area map projection described by Ernst Hammer in 1892. Using the same 2:1 elliptical outer shape as the Mollweide projection, Hammer intended to reduce distortion in the regions of the outer meridians, where it is extreme in the Mollweide.

Cylindrical equal-area projection

In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.

Eckert IV projection equal-area pseudocylindrical map projection devised by Max Eckert-Greifendorff

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. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III.

Eckert II projection equal-area pseudocylindrical map projection devised by Max Eckert-Greifendorff

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.

Central cylindrical projection

The central cylindrical projection is a perspective cylindrical map projection. It corresponds to projecting the Earth's surface onto a cylinder tangent to the equator as if from a light source at Earth's center. The cylinder is then cut along one of the projected meridians and unrolled into a flat map.

Gall stereographic projection Cylindrical map projection

The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection.

References

  1. 1 2 3 Snyder, John Parr (1987). Map Projections: a Working Manual. U.S. Government Printing Office. pp. 76–85.
  2. Ward, Matthew O.; Grinstein, Georges; Keim, Daniel (2015). Interactive Data Visualization: Foundations, Techniques, and Applications, Second Edition. CRC Press. pp. 226–227. ISBN   978-1-4822-5738-0.