Equidistant conic projection

Last updated
The world on an equidistant conic projection. 15deg graticule, standard parallels of 20degN and 60degN. Equidistant conic projection SW.JPG
The world on an equidistant conic projection. 15° graticule, standard parallels of 20°N and 60°N.
The equidistant conic projection with Tissot's indicatrix of deformation. Standard parallels of 15degN and 45degN. Equidistant Conic with Tissot's Indicatrices of Distortion.svg
The equidistant conic projection with Tissot's indicatrix of deformation. Standard parallels of 15°N and 45°N.

The equidistant conic projection is a conic map projection commonly used for maps of small countries as well as for larger regions such as the continental United States that are elongated east-to-west. [1]

Contents

Also known as the simple conic projection, a rudimentary version was described during the 2nd century CE by the Greek astronomer and geographer Ptolemy in his work Geography . [2]

The projection has the useful property that distances along the meridians are proportionately correct, and distances are also correct along two standard parallels that the mapmaker has chosen. The two standard parallels are also free of distortion.

For maps of regions elongated east-to-west (such as the continental United States) the standard parallels are chosen to be about a sixth of the way inside the northern and southern limits of interest. This way distortion is minimized throughout the region of interest.

Transformation

Coordinates from a spherical datum can be transformed to an equidistant conic projection with rectangular coordinates by using the following formulas, [3] where λ is the longitude, λ0 the reference longitude, φ the latitude, φ0 the reference latitude, and φ1 and φ2 the standard parallels:

where

Constants n, G, and ρ0 need only be determined once for the entire map. If one standard parallel is used (i.e. φ1 = φ2), the formula for n above is indeterminate, but then

[4]

The reference point (λ0, φ0) with longitude λ0 and latitude φ0, transforms to the x,y origin at (0,0) in the rectangular coordinate system. [4]

The Y axis maps the central meridian λ0, with y increasing northwards, which is orthogonal to the X axis mapping the central parallel φ0, with x increasing eastwards. [4]

Other versions of these transformation formulae include parameters to offset the map coordinates so that all x,y values are positive, as well as a scaling parameter relating the radius of the sphere (earth) to the units used on the map. [1]

The formulae used for ellipsoidal datums are more involved. [5]

See also

Related Research Articles

In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.

Orthographic projection in cartography map projection of cartography

The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

Craig retroazimuthal projection

The Craig retroazimuthal map projection was created by James Ireland Craig in 1909. It is a modified cylindrical projection. As a retroazimuthal projection, it preserves directions from everywhere to one location of interest that is configured during construction of the projection. The projection is sometimes known as the Mecca projection because Craig, who had worked in Egypt as a cartographer, created it to help Muslims find their qibla. In such maps, Mecca is the configurable location of interest.

Mollweide projection map projection

The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for global maps of the world or night sky. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.

Azimuthal equidistant projection azimuthal map projection

The azimuthal equidistant projection is an azimuthal map projection. It has the useful properties that all points on the map are at proportionally correct distances from the center point, and that all points on the map are at the correct azimuth (direction) from the center point. A useful application for this type of projection is a polar projection which shows all meridians as straight, with distances from the pole represented correctly. The flag of the United Nations contains an example of a polar azimuthal equidistant projection.

Scale (map) Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

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.

Bonne projection map projection

The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvano and Honter's usages were approximate, however, and it is not clear they intended to be the same projection.

Bottomley projection

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

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.

Aitoff projection map projection

The Aitoff projection is a modified azimuthal map projection proposed by David A. Aitoff in 1889. Based on the equatorial form of the azimuthal equidistant projection, Aitoff first halves longitudes, then projects according to the azimuthal equidistant, and then stretches the result horizontally into a 2:1 ellipse to compensate for having halved the longitudes. Expressed simply:

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.

Hammer retroazimuthal projection

The Hammer retroazimuthal projection is a modified azimuthal proposed by Ernst Hermann Heinrich Hammer in 1910. As a retroazimuthal projection, azimuths (directions) are correct from any point to the designated center point. Additionally, all distances from the center of the map are proportional to what they are on the globe. In whole-world presentation, the back and front hemispheres overlap, making the projection a non-injective function. The back hemisphere can be rotated 180° to avoid overlap, but in this case, any azimuths measured from the back hemisphere must be corrected.

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.

American polyconic projection class of map projections

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.

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. 1 2 "Simple Equidistant Conic Map Projection". Simulator Manual. PowerWorld Corporation. Archived from the original on 22 May 2020. Retrieved 21 May 2020.
  2. Snyder 1993, p. 111.
  3. Weisstein, Eric. "Conic Equidistant Projection". Wolfram MathWorld. Wolfram Research. Retrieved 20 May 2020.
  4. 1 2 3 Snyder 1993, p. 113.
  5. Snyder 1993, p. 114–115.

Sources