# List of map projections

Last updated

This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable. Because there is no limit to the number of possible map projections, [1] there can be no comprehensive list.

## Table of projections

ProjectionImageTypePropertiesCreatorYearNotes
Equirectangular
= equidistant cylindrical
= rectangular
= la carte parallélogrammatique
CylindricalEquidistant Marinus of Tyre c.120Simplest geometry; distances along meridians are conserved.

Plate carrée: special case having the equator as the standard parallel.

Cassini
= Cassini–Soldner
CylindricalEquidistant César-François Cassini de Thury 1745Transverse of equidistant projection; distances along central meridian are conserved.
Distances perpendicular to central meridian are preserved.
Mercator
= Wright
CylindricalConformal Gerardus Mercator 1569Lines of constant bearing (rhumb lines) are straight, aiding navigation. Areas inflate with latitude, becoming so extreme that the map cannot show the poles.
Web Mercator CylindricalCompromise Google 2005Variant of Mercator that ignores Earth's ellipticity for fast calculation, and clips latitudes to ~85.05° for square presentation. De facto standard for Web mapping applications.
Gauss–Krüger
= Gauss conformal
= (ellipsoidal) transverse Mercator
CylindricalConformal Carl Friedrich Gauss 1822This transverse, ellipsoidal form of the Mercator is finite, unlike the equatorial Mercator. Forms the basis of the Universal Transverse Mercator coordinate system.
Roussilhe oblique stereographic Henri Roussilhe1922
Hotine oblique Mercator CylindricalConformalM. Rosenmund, J. Laborde, Martin Hotine1903
Gall stereographic
CylindricalCompromise James Gall 1855Intended to resemble the Mercator while also displaying the poles. Standard parallels at 45°N/S.
Miller
= Miller cylindrical
CylindricalCompromise Osborn Maitland Miller 1942Intended to resemble the Mercator while also displaying the poles.
Lambert cylindrical equal-area CylindricalEqual-area Johann Heinrich Lambert 1772Standard parallel at the equator. Aspect ratio of π (3.14). Base projection of the cylindrical equal-area family.
Behrmann CylindricalEqual-area Walter Behrmann 1910Horizontally compressed version of the Lambert equal-area. Has standard parallels at 30°N/S and an aspect ratio of 2.36.
Hobo–Dyer CylindricalEqual-area Mick Dyer 2002Horizontally compressed version of the Lambert equal-area. Very similar are Trystan Edwards and Smyth equal surface (= Craster rectangular) projections with standard parallels at around 37°N/S. Aspect ratio of ~2.0.
Gall–Peters
= Gall orthographic
= Peters
CylindricalEqual-area James Gall 1855Horizontally compressed version of the Lambert equal-area. Standard parallels at 45°N/S. Aspect ratio of ~1.6. Similar is Balthasart projection with standard parallels at 50°N/S.
Central cylindrical CylindricalPerspective(unknown)c.1850Practically unused in cartography because of severe polar distortion, but popular in panoramic photography, especially for architectural scenes.
Sinusoidal
= Sanson–Flamsteed
= Mercator equal-area
PseudocylindricalEqual-area, equidistant(Several; first is unknown)c.1600Meridians are sinusoids; parallels are equally spaced. Aspect ratio of 2:1. Distances along parallels are conserved.
Mollweide
= elliptical
= Babinet
= homolographic
PseudocylindricalEqual-area Karl Brandan Mollweide 1805Meridians are ellipses.
Eckert II PseudocylindricalEqual-area Max Eckert-Greifendorff 1906
Eckert IV PseudocylindricalEqual-area Max Eckert-Greifendorff 1906Parallels are unequal in spacing and scale; outer meridians are semicircles; other meridians are semiellipses.
Eckert VI PseudocylindricalEqual-area Max Eckert-Greifendorff 1906Parallels are unequal in spacing and scale; meridians are half-period sinusoids.
Ortelius oval PseudocylindricalCompromise Battista Agnese 1540

Meridians are circular. [2]

Goode homolosine PseudocylindricalEqual-area John Paul Goode 1923Hybrid of Sinusoidal and Mollweide projections.
Usually used in interrupted form.
Kavrayskiy VII PseudocylindricalCompromise Vladimir V. Kavrayskiy 1939Evenly spaced parallels. Equivalent to Wagner VI horizontally compressed by a factor of ${\displaystyle {\sqrt {3}}/{2}}$.
Robinson PseudocylindricalCompromise Arthur H. Robinson 1963Computed by interpolation of tabulated values. Used by Rand McNally since inception and used by NGS in 1988–1998.
Equal Earth PseudocylindricalEqual-areaBojan Šavrič, Tom Patterson, Bernhard Jenny2018Inspired by the Robinson projection, but retains the relative size of areas.
Natural Earth PseudocylindricalCompromise Tom Patterson 2011Computed by interpolation of tabulated values.
Tobler hyperelliptical PseudocylindricalEqual-area Waldo R. Tobler 1973A family of map projections that includes as special cases Mollweide projection, Collignon projection, and the various cylindrical equal-area projections.
Wagner VI PseudocylindricalCompromise K. H. Wagner 1932Equivalent to Kavrayskiy VII vertically compressed by a factor of ${\displaystyle {\sqrt {3}}/{2}}$.
Collignon PseudocylindricalEqual-area Édouard Collignon c.1865Depending on configuration, the projection also may map the sphere to a single diamond or a pair of squares.
HEALPix PseudocylindricalEqual-area Krzysztof M. Górski 1997Hybrid of Collignon + Lambert cylindrical equal-area.
Boggs eumorphic PseudocylindricalEqual-areaSamuel Whittemore Boggs1929The equal-area projection that results from average of sinusoidal and Mollweide y-coordinates and thereby constraining the x coordinate.
Craster parabolic
=Putniņš P4
PseudocylindricalEqual-areaJohn Craster1929Meridians are parabolas. Standard parallels at 36°46′N/S; parallels are unequal in spacing and scale; 2:1 aspect.
McBryde–Thomas flat-pole quartic
= McBryde–Thomas #4
PseudocylindricalEqual-areaFelix W. McBryde, Paul Thomas1949Standard parallels at 33°45′N/S; parallels are unequal in spacing and scale; meridians are fourth-order curves. Distortion-free only where the standard parallels intersect the central meridian.
Quartic authalic PseudocylindricalEqual-areaKarl Siemon

1937

1944

Parallels are unequal in spacing and scale. No distortion along the equator. Meridians are fourth-order curves.
The Times PseudocylindricalCompromiseJohn Muir1965Standard parallels 45°N/S. Parallels based on Gall stereographic, but with curved meridians. Developed for Bartholomew Ltd., The Times Atlas.
Loximuthal PseudocylindricalCompromiseKarl Siemon 1935

1966

From the designated centre, lines of constant bearing (rhumb lines/loxodromes) are straight and have the correct length. Generally asymmetric about the equator.
Aitoff PseudoazimuthalCompromise David A. Aitoff 1889Stretching of modified equatorial azimuthal equidistant map. Boundary is 2:1 ellipse. Largely superseded by Hammer.
Hammer
= Hammer–Aitoff
variations: Briesemeister; Nordic
PseudoazimuthalEqual-area Ernst Hammer 1892Modified from azimuthal equal-area equatorial map. Boundary is 2:1 ellipse. Variants are oblique versions, centred on 45°N.
Strebe 1995 PseudoazimuthalEqual-areaDaniel "daan" Strebe1994Formulated by using other equal-area map projections as transformations.
Winkel tripel PseudoazimuthalCompromise Oswald Winkel 1921Arithmetic mean of the equirectangular projection and the Aitoff projection. Standard world projection for the NGS since 1998.
Van der Grinten OtherCompromise Alphons J. van der Grinten 1904Boundary is a circle. All parallels and meridians are circular arcs. Usually clipped near 80°N/S. Standard world projection of the NGS in 1922–1988.
Equidistant conic
= simple conic
ConicEquidistantBased on Ptolemy's 1st Projectionc.100Distances along meridians are conserved, as is distance along one or two standard parallels. [3]
Lambert conformal conic ConicConformal Johann Heinrich Lambert 1772Used in aviation charts.
Albers conic ConicEqual-area Heinrich C. Albers 1805Two standard parallels with low distortion between them.
Werner PseudoconicalEqual-area, equidistant Johannes Stabius c.1500Parallels are equally spaced concentric circular arcs. Distances from the North Pole are correct as are the curved distances along parallels and distances along central meridian.
Bonne Pseudoconical, cordiformEqual-area Bernardus Sylvanus 1511Parallels are equally spaced concentric circular arcs and standard lines. Appearance depends on reference parallel. General case of both Werner and sinusoidal.
Bottomley PseudoconicalEqual-area Henry Bottomley 2003Alternative to the Bonne projection with simpler overall shape

Parallels are elliptical arcs
Appearance depends on reference parallel.

American polyconic PseudoconicalCompromise Ferdinand Rudolph Hassler c.1820Distances along the parallels are preserved as are distances along the central meridian.
Rectangular polyconic PseudoconicalCompromise U.S. Coast Survey c.1853Latitude along which scale is correct can be chosen. Parallels meet meridians at right angles.
Latitudinally equal-differential polyconic PseudoconicalCompromiseChina State Bureau of Surveying and Mapping1963Polyconic: parallels are non-concentric arcs of circles.
Nicolosi globular Pseudoconical [4] Compromise Abū Rayḥān al-Bīrūnī; reinvented by Giovanni Battista Nicolosi, 1660. [1] :14c.1000
Azimuthal equidistant
=Postel
=zenithal equidistant
AzimuthalEquidistant Abū Rayḥān al-Bīrūnī c.1000Distances from center are conserved.

Used as the emblem of the United Nations, extending to 60° S.

Gnomonic AzimuthalGnomonic Thales (possibly)c.580 BCAll great circles map to straight lines. Extreme distortion far from the center. Shows less than one hemisphere.
Lambert azimuthal equal-area AzimuthalEqual-area Johann Heinrich Lambert 1772The straight-line distance between the central point on the map to any other point is the same as the straight-line 3D distance through the globe between the two points.
Stereographic AzimuthalConformal Hipparchos*c.200 BCMap is infinite in extent with outer hemisphere inflating severely, so it is often used as two hemispheres. Maps all small circles to circles, which is useful for planetary mapping to preserve the shapes of craters.
Orthographic AzimuthalPerspective Hipparchos*c.200 BCView from an infinite distance.
Vertical perspective AzimuthalPerspectiveMatthias Seutter*1740View from a finite distance. Can only display less than a hemisphere.
Two-point equidistant AzimuthalEquidistantHans Maurer1919Two "control points" can be almost arbitrarily chosen. The two straight-line distances from any point on the map to the two control points are correct.
Peirce quincuncial OtherConformal Charles Sanders Peirce 1879Tessellates. Can be tiled continuously on a plane, with edge-crossings matching except for four singular points per tile.
Guyou hemisphere-in-a-square projection OtherConformal Émile Guyou 1887Tessellates.
Lee conformal world on a tetrahedron PolyhedralConformal L. P. Lee 1965Projects the globe onto a regular tetrahedron. Tessellates.
AuthaGraph projection Link to file PolyhedralCompromise Hajime Narukawa 1999Approximately equal-area. Tessellates.
Octant projection PolyhedralCompromise Leonardo da Vinci 1514Projects the globe onto eight octants (Reuleaux triangles) with no meridians and no parallels.
Cahill's butterfly map PolyhedralCompromise Bernard Joseph Stanislaus Cahill 1909Projects the globe onto an octahedron with symmetrical components and contiguous landmasses that may be displayed in various arrangements.
Cahill–Keyes projection PolyhedralCompromise Gene Keyes 1975Projects the globe onto a truncated octahedron with symmetrical components and contiguous land masses that may be displayed in various arrangements.
Waterman butterfly projection PolyhedralCompromise Steve Waterman 1996Projects the globe onto a truncated octahedron with symmetrical components and contiguous land masses that may be displayed in various arrangements.
Quadrilateralized spherical cube PolyhedralEqual-areaF. Kenneth Chan, E. M. O'Neill1973
Dymaxion map PolyhedralCompromise Buckminster Fuller 1943Also known as a Fuller Projection.
Myriahedral projections PolyhedralCompromise Jarke J. van Wijk 2008Projects the globe onto a myriahedron: a polyhedron with a very large number of faces. [5] [6]
Craig retroazimuthal
= Mecca
RetroazimuthalCompromiseJames Ireland Craig1909
Hammer retroazimuthal, front hemisphere Retroazimuthal Ernst Hammer 1910
Hammer retroazimuthal, back hemisphere Retroazimuthal Ernst Hammer 1910
Littrow RetroazimuthalConformal Joseph Johann Littrow 1833on equatorial aspect it shows a hemisphere except for poles.
GS50 OtherConformal John P. Snyder 1982Designed specifically to minimize distortion when used to display all 50 U.S. states.
Wagner VII
= Hammer-Wagner
PseudoazimuthalEqual-areaK. H. Wagner1941
Atlantis
= Transverse Mollweide
PseudocylindricalEqual-areaJohn Bartholomew1948Oblique version of Mollweide
Bertin
= Bertin-Rivière
= Bertin 1953
OtherCompromiseJacques Bertin1953Projection in which the compromise is no longer homogeneous but instead is modified for a larger deformation of the oceans, to achieve lesser deformation of the continents. Commonly used for French geopolitical maps. [7]

*The first known popularizer/user and not necessarily the creator.

## Key

### Type of projection

Cylindrical
In standard presentation, these map regularly-spaced meridians to equally spaced vertical lines, and parallels to horizontal lines.
Pseudocylindrical
In standard presentation, these map the central meridian and parallels as straight lines. Other meridians are curves (or possibly straight from pole to equator), regularly spaced along parallels.
Conic
In standard presentation, conic (or conical) projections map meridians as straight lines, and parallels as arcs of circles.
Pseudoconical
In standard presentation, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as circular arcs.
Azimuthal
In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, concentric circles. They are radially symmetrical. In any presentation (or aspect), they preserve directions from the center point. This means great circles through the central point are represented by straight lines on the map.
Pseudoazimuthal
In standard presentation, pseudoazimuthal projections map the equator and central meridian to perpendicular, intersecting straight lines. They map parallels to complex curves bowing away from the equator, and meridians to complex curves bowing in toward the central meridian. Listed here after pseudocylindrical as generally similar to them in shape and purpose.
Other
Typically calculated from formula, and not based on a particular projection
Polyhedral maps
Polyhedral maps can be folded up into a polyhedral approximation to the sphere, using particular projection to map each face with low distortion.

### Properties

Conformal
Preserves angles locally, implying that local shapes are not distorted and that local scale is constant in all directions from any chosen point.
Equal-area
Area measure is conserved everywhere.
Compromise
Neither conformal nor equal-area, but a balance intended to reduce overall distortion.
Equidistant
All distances from one (or two) points are correct. Other equidistant properties are mentioned in the notes.
Gnomonic
All great circles are straight lines.
Retroazimuthal
Direction to a fixed location B (by the shortest route) corresponds to the direction on the map from A to B.

## Notes

1. Snyder, John P. (1993). Flattening the earth: two thousand years of map projections. University of Chicago Press. p. 1. ISBN   0-226-76746-9.
2. Donald Fenna (2006). Cartographic Science: A Compendium of Map Projections, with Derivations. CRC Press. p. 249. ISBN   978-0-8493-8169-0.
3. Furuti, Carlos A. "Conic Projections: Equidistant Conic Projections". Archived from the original on December 20, 2013. Retrieved February 11, 2020.CS1 maint: unfit url (link)
4. Jarke J. van Wijk. "Unfolding the Earth: Myriahedral Projections".
5. Carlos A. Furuti. "Interrupted Maps: Myriahedral Maps".
6. Rivière, Philippe (October 1, 2017). "Bertin Projection (1953)". visionscarto. Retrieved January 27, 2020.

## Related Research Articles

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

In cartography, 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. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

The Robinson projection is a map projection of a world map which shows the entire world at once. It was specifically created in an attempt to find a good compromise to the problem of readily showing the whole globe as a flat image.

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.

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.

A gnomonic map projection displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by casting surface points of the sphere onto a tangent plane, each landing where a ray from the center of the sphere passes through the point on the surface and then on to the plane. No distortion occurs at the tangent point, but distortion increases rapidly away from it. Less than half of the sphere can be projected onto a finite map. Consequently, a rectilinear photographic lens, which is based on the gnomonic principle, cannot image more than 180 degrees.

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.

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.

The sinusoidal projection is a pseudocylindrical equal-area map projection, sometimes called the Sanson–Flamsteed or the Mercator equal-area projection. Jean Cossin of Dieppe was one of the first mapmakers to use the sinusoidal, appearing in a world map of 1570.

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.

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.

In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth is preserved in the image of the projection, i.e. the projection is a conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle.

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

The equidistant conic projection is a conic map projection known since Classical times, Ptolemy's first projection being derived from it.

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.

The Eckert VI projection is an equal-area pseudocylindrical map projection. The length of polar line is half that of the equator, and lines of longitude are sinusoids. 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 VI is the Eckert V 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.

In cartography, the loximuthal projection is a map projection introduced by Karl Siemon in 1935, and independently in 1966 by Waldo R. Tobler, who named it. It is characterized by the fact that loxodromes from one chosen central point are shown straight lines, correct in azimuth from the center, and are "true to scale" in the sense that distances measured along such lines are proportional to lengths of the corresponding rhumb lines on the surface of the earth. It is neither an equal-area projection nor conformal.

In map projections, an interruption is any place where the globe has been split. All map projections are interrupted at at least one point. Typical world maps are interrupted along an entire meridian. In that typical case, the interruption forms an east/west boundary, even though the globe has no boundaries.