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Summary

Equatorial azimuthal equal-area map
The world according to Lambert's azimuthal equal-area map, in the equatorial aspect. Unprojected data by NASA

Several approaches attempt to classify projections. Most are orthogonal, thus any single projection may belong simultaneously to different categories. In others, like biology-inspired Maurer's, a branching taxonomy is applied.

Please note that, for ease of definition and visualization, some categories are informally described below in terms of parallels and meridians, thus some properties may seem to depend on the particular aspect used for the map. For instance, the coordinate lines in cylindrical maps cross at right angles in equatorial, but not in polar or oblique maps, although all other properties still hold; after all, the coordinate grid is only a set of conventional lines.

Projections Classified by Geometry

Category Properties
Azimuthal   Also called zenithal. Shows true directions (azimuths) from a single point; in polar aspects all parallels are circular, and meridians are straight lines uniformly spaced and concurring at a point; an unclipped world map is a disc.
Cylindrical Defined by analogy to a cylinder as an intermediate projection surface; in the equatorial aspect all parallels and meridians are straight lines; meridians are orthogonal to parallels and uniformly spaced; an unclipped world map is rectangular
Conic By analogy to a conic intermediate projection surface; in the polar aspect all parallels are concentric arcs of circle, while meridians are straight lines perpendicular to every parallel, uniformly spaced by less than on Earth; unclipped maps are circular or annular sectors
Pseudocylindrical In the equatorial aspect all parallels are straight parallel lines; meridians are arbitrary curves, equally spaced along every parallel
Pseudoconic In the polar aspect all parallels are concentric circular arcs, while meridians are arbitrary curves
Arbitrary or compromise Parallels and meridians are arbitrary curves; usually no purely geometric construction is defined. Some authors call "arbitrary", "conventional" or "compromise" any projection not derived from geometric devices, but custom-fit to a purpose

In a sense, the cone includes as extreme cases both the cylinder (a cone with vertex at the infinite) and the plane (a cone with zero height). Therefore, the conic group generalizes the azimuthal and cylindrical and, broadly, pseudocylindrical and pseudoconic projections. Also, some consider a polyconic group to include projections where parallels are derived from circles, including modified azimuthals like Hammer's and Aitoff's. Actually, many so-called "azimuthal", "conic" or "cylindrical" projections are not built on a pure projective process using solids, but are so classified due to geometrical properties of the mapped coordinate grid.

Also, a projective, geometric or perspective projection can be described in exact analogy to a geometric set-up of light rays connecting the original surface to the map surface. Some authors call other projections "mathematical".

Polar azimuthal equal-area map Equal-area sinusoidal map
Eisenlohr's conformal map
In azimuthal projections, the angles between straight lines radiating from the center of projection (which may or may not coincide with the center of the map) are the same for the corresponding lines on Earth. On the azimuthal equidistant map on the left, distances along those lines are also directly proportional to those on Earth.
On the equal-area sinusoidal map (top right), for any two identical boundaries (like the blue squares), their counterparts on Earth will enclose the same area, although they will not necessarily have the same shape. That pseudocylindrical projection preserves angles only along the two axes (green); it is equidistant (red) only along those axes and all lines perpendicular to the minor axis.
Unlike almost every other conformal projections, Eisenlohr's (bottom right) preserves small angles at every point: the green lines are perpendicular on both Earth and map. Areal and distance distortion are large, but less than in typical conformal maps.

Projections Classified by Property

Category Properties
Equal-area   Any region in the map has area directly proportional to the corresponding region on the sphere; also called equivalent or authalic. Generally more useful for statistical comparisons and didactic purposes.
Equidistant On the map there are two sets of points A and B, such that, along a selected set of lines (not necessarily straight), distances from any point in A to another in B are proportional to the distances between corresponding points on the sphere, again along those corresponding lines.  In other words, scale is constant on those lines, which are called standard.  Most projections have such sets but few are actually called "equidistant".
Conformal In any* small region of the map, two concurrent lines have the same angle as corresponding lines on the sphere, thus shapes are locally preserved. Also called orthomorphic or autogonal. Most important for navigational purposes and large-scale mapping, especially in the ellipsoidal case.
*In almost every conformal projection, at least one point (frequently a pole is chosen) either can not be represented or fails conformality.
Aphylactic Some authors use this name for those projections which are neither conformal nor equivalent.

Projections in a Nutshell

Projections enumerated below are described in further detail. This is just a small sample of all existing designs, not necessarily the most important or most commonly used; as such, the selection is necessarily subjective. Equally arbitrary is whether changing the aspect or another minor detail is enough to justify a separate entry (cf. e.g., Cassini's v. equidistant cylindrical and Gauss transverse v. Mercator).

Sample graticule Common names Main Features
orthographic
azimuthal orthographic, orthographic Azimuthal, "realistic" view of Earth as seen from space infinitely away. Shows at most a single hemisphere
stereographic
azimuthal stereographic, stereographic Azimuthal, conformal, preserves all circles; shows at most a hemisphere
gnomonic
gnomonic, central, centrographic, gnomic Azimuthal, all great circles map to straight lines; extreme distortion far from the center; shows less than one hemisphere
general vertical perspective
general vertical perspective Azimuthal, general case of orthographic, stereographic, gnomonic azimuthal projections. Most realistic view from space directly towards center of Earth. Parameterized by distance of projection center; special cases by La Hire, Parent, Lowry, Fischer, Gretschel, James, Clarke ("Twilight") and others
azimuthal equidistant
azimuthal equidistant, zenithal equidistant Non-perspective azimuthal, distances from center are preserved, navigational purposes
azimuthal equal-area
Lambert's azimuthal equal-area Non-perspective. Unique azimuthal equal-area projection
azimuthals by Ginzburg
Ginzburg's azimuthal I and II Non-perspective, neither equal-area nor conformal
cylindrical equal-area
Lambert's equal-area cylindrical; variations by Behrmann, Trystan Edwards, Gall (orthographic), Peters, Dyer, Tobler/Chen Only possible cylindrical equal-area projection, including scaled variants like Gall's and Hobo-Dyer
Gall's stereographic cylindrical
Gall's stereographic cylindrical Neither conformal nor equal-area. Variations include Braun's stereographic cylindrical and the BSAM cylindrical.
Braun's stereographic cylindrical
Braun's stereographic cylindrical Neither conformal nor equal-area, special case of Gall's stereographic
Central cylindrical
central cylindrical, centrographic cylindrical Neither conformal nor equal-area; not to be confused with Mercator's. Transverse aspect is the Wetch projection
equidistant cylindrical
equirrectangular, equidistant cylindrical, plain chart, plane chart; special cases are simple cylindrical (plate carrée), Gall's isographic and Cassini Cylindrical, very fast and easy to compute, neither conformal nor equal-area; in the most common case maps into a rectangle with aspect ratio 2 : 1 (twice wide as tall)
Cassini
Cassini Transverse aspect of the plate carrée
Gall isographic
Gall's isographic Special case of the equidistant cylindrical, standard parallels 45°N and 45°S
Mercator
Mercator, cylindrical conformal; transverse ellipsoidal form called Gauss conformal or Gauss-Krüger Only possible conformal cylindrical projection; transverse aspect is the foundation of the UTM grid
Miller
Miller Cylindrical, arbitrary compromise to Mercator
Mollweide
Mollweide, elliptical, Babinet, homolographic, homalographic Pseudocylindrical, equal-area,  meridians are ellipses; full map bounded by 2 : 1 ellipse; sometimes interrupted; variations include Atlantis
Sinusoidal
Sanson-Flamsteed, sinusoidal, Mercator equal-area Pseudocylindrical, equal-area, meridians are sinusoids, parallels are equally spaced and standard lines; 2 : 1
Foucault's stereographic equivalent
Foucault's stereographic equivalent Pseudocylindrical, equal-area, parallels are spaced like in the equatorial aspect of the azimuthal stereographic
Collignon
Collignon Pseudocylindrical, equal-area, meridians are straight lines. Two main variants, with triangular frame or symmetrical diamond with meridians broken at Equator
Craster Parabolic
Craster parabolic Pseudocylindrical, equal-area, meridians are parabolas. Same as Putniņš P4
Loximuthal
Loximuthal Pseudocylindrical, all straight lines passing through intersection of central meridian and a reference parallel are loxodromes with correct scale and azimuth
quartic authalic
Quartic authalic Pseudocylindrical, equal-area, meridians are 4th order polynomials; limiting case of Hammer and Eckert-Greifendorff
Flat-polar quartic
Flat polar quartic Pseudocylindrical, equal-area, poles are 1/3 as long as the Equator
Nell Nell's pseudocylindrical Pseudocylindrical, equal-area, polelines
Nell-Hammer Nell-Hammer Pseudocylindrical, equal-area, polelines
Eckert I Eckert I Pseudocylindrical, 2 : 1, poles are half as long as the Equator, meridians are straight lines broken at Equator. Parallels equally spaced.
Eckert II Eckert II Pseudocylindrical, equal-area, 2 : 1, poles are half as long as the Equator, meridians are straight lines broken at Equator.
Eckert III
Eckert III Pseudocylindrical, 2 : 1, meridians are elliptical arcs (boundary is circular). Parallels are equally spaced.
Eckert IV
Eckert IV Pseudocylindrical, equal-area, 2 : 1, meridians are elliptical arcs, circular at boundary.
Eckert V
Eckert V Pseudocylindrical, 2 : 1, meridians are sinusoids, parallels are equally spaced. Particular case of Winkel's first projection
Eckert VI Eckert VI Pseudocylindrical, equal-area, 2 : 1, poles are half as long as the Equator, meridians are sinusoids.
Rosen Rosén's pseudocylindrical Pseudocylindrical, equal-area, based on sinusoidal: poles are mapped to parallels arcsin(0.8) N and S of base projection
Robinson
Robinson, orthophanic Pseudocylindrical, compromise. Neither conformal nor equal-area
Kavrayskiy V
Kavrayskiy V Pseudocylindrical, equal-area
Kavrayskiy VII
Kavrayskiy VII Pseudocylindrical, compromise, elliptical meridians
Goode
Goode homolosine Pseudocylindrical, equal-area, hybrid joining Mollweide at poles, Sanson-Flamsteed at the equatorial band, almost always interrupted
Boggs
Boggs eumorphic Pseudocylindrical, equal-area, arithmetic average of Mollweide and Sanson-Flamsteed projections. Usually interrupted
Sinu-Mollweide
Sinu-Mollweide Pseudocylindrical, equal-area, hybrid fusion of Mollweide and (in lower portion) Sanson-Flamsteed projections. Usually oblique and interrupted
Winkel I
Winkel I Pseudocylindrical (generalizes Eckert V), averages Sanson-Flamsteed and equidistant cylindrical, meridians are sinusoids
Winkel II
Winkel II Pseudocylindrical, averages equidistant cylindrical and a modified elliptical projection
HEALPix
HEALPix Pseudocylindrical, equal-area, hybrid of Lambert's equal-area cylindrical and interrupted Collignon's projection; designed for raster processing of astronomical and cosmological data in the FITS grid.
Pseudo-Eckert
Pseudo-Eckert Pseudocylindrical, equal-area, meridians are partial sinusoids
Perspective Conic
Perspective (orthographic, stereographic or centrographic) conic Conic, true perspective. Used by Murdoch and Colles
Equidistant Conic
Equidistant conic Conic, constant meridian scale; limiting cases are azimuthal equidistant and cylindrical equidistant projections. General case of Schjerning's I projection. Many variations, mostly in choice of standard parallels (Murdock, Euler); others include de l'Isle's coniclike projection.
Braun stereographic conic
Braun's stereographic conic Perspective conic with center of projection at a pole and 30° standard latitude
Albers
Albers equal-area conic Conic, equal-area; limiting cases are Lambert's equal-area conic and cylindrical projections.
Lambert's equal-area conic
Lambert's equal-area conic, isospherical stenoteric Conic, equal-area; limiting case of Albers's conic, with a pole as standard parallel
Conformal conic
Lambert's conformal conic, orthomorphic conic Conic, conformal; limiting cases are azimuthal stereographic and Mercator projections
polyconic
Polyconic, American Polyconic Polyconic, parallels are nonconcentric arcs of circle with correct scale. Neither conformal nor equal-area.
rectangular polyconic
Rectangular Polyconic, War Office Polyconic, parallels are nonconcentric circular arcs crossing all meridians at right angles; either Equator or two parallels have correct length. Neither equivalent nor conformal
Wiechel
Wiechel Pseudoazimuthal; modified azimuthal equal-area projection, no longer azimuthal. Only interesting in polar aspect, where meridians are circular arcs with standard scale. Usually clipped to a single hemisphere
Aitoff
Aitoff Stretching of modified equatorial azimuthal equidistant map; boundary is 2 : 1 ellipse
Hammer
Hammer, Hammer-Aitoff, Aitoff-Hammer Modified from azimuthal equal-area equatorial map; equal-area, boundary is 2 : 1 ellipse; variations include Briesemeister, oceanic and Nordic
Briesemeister
Briesemeister Equal-area, simple oblique stretching of Hammer projection
Eckert-Greifendorff
Eckert-Greifendorff Rescaled modification of Hammer projection. Equal-area
Schjerning2
Schjerning II (original map includes an arbitrary enlargement) Azimuthal equidistant inner hemisphere. The outer hemisphere completes a 2:1 ellipse. Distances correct from center of map, azimuth correct only in inner hemisphere. Not conformal or equal-area
Schjerning2
Schjerning III Map comprises two circles joined at a point. Distance from every point to the center of the map is correct, but not azimuth. Final map is centered on London. Neither conformal nor equal-area
Wagner IX
Wagner IX, Aitoff-Wagner Modified Aitoff projection; neither equal-area nor conformal
Winkel Tripel
Winkel Tripel Arithmetic mean of Aitoff and equidistant cylindrical projections
StabiusWerner I
Stabius-Werner I Pseudoconic, equal-area, parallels are equally spaced circular arcs centered on a pole.
Werner
Werner, Stabius-Werner II, cordiform Pseudoconic, equal-area, parallels are equally spaced circular arcs and standard lines, centered on a pole. Schjerning IV is an oblique aspect; Schjerning V has shortened parallels, Schjerning VI is interrupted.
Werner III
Stabius-Werner III Pseudoconic, equal-area, parallels are equally spaced circular arcs centered on a pole
Bonne
"Bonne" Pseudoconic, equal-area, parallels are equally spaced circular arcs and standard lines. Appearance depends on reference parallel. General case of both Werner and sinusoidal
Lagrange
"Lagrange" Meridians and parallels are circular arcs, except the central meridian and a base parallel which are straight. Conformal except at the poles. The basic case, developed by Lambert, is circular
Gilbert/DeLucia/Snyder
DeLucia/Snyder's orthographic projection of Gilbert's double world sphere Graticule comprises elliptical arcs. Neither conformal nor equal-area.
Peirce Quincuncial
Peirce Quincuncial World map in a square, central hemisphere in an inner square. Conformal except at edge midpoints. Other aspects by Guyou and Adams
Guyou
Guyou World map in 2:1 rectangle. Conformal except at hemisphere corners. Other aspects by Peirce and Adams
Adams hemispheres
Adams's hemispheres on squares Hemispheres in two squares. Conformal except at square corners. Other aspects by Guyou and Peirce
Adams maps
Adams's world on a square Poles at opposite vertices; Equator at a diagonal. Conformal except at four vertices
Adams maps
Adams's world on a square Poles at midpoints of opposite edges. Conformal except at poles and four vertices
Xarax's projection
Xarax's world in half a hexagon Three-lobed world map derived from Lee's map on a tetrahedron. Conformal except at North pole and meridian breaks at each lobe
Eisenlohr
Eisenlohr Fully conformal, no singular points. Scale constant along boundary. Optimal range of scale distortion for a conformal design
August
August, August epicycloidal Conformal everywhere, with no singular points. Map bounded by a epicycloid. Base for Spilhaus's oceanic map
Van der Grinten I
Van der Grinten, Van der Grinten I Boundary is a circle, meridians and parallels are circular arcs, except central meridian and Equator. Not conformal, great area distortion far from Equator.
Van der Grinten II
Van der Grinten II Boundary is a circle, meridians and parallels are circular arcs intersecting at right angles, except central meridian and Equator
Van der Grinten III
Van der Grinten III Boundary is a circle, meridians (except central) are circular arcs; parallels are horizontal lines intersecting central meridian at same points as in Van der Grinten I
Van der Grinten IV
Van der Grinten IV Bounded by two intersecting circles, meridians are arcs of circle equally spaced along Equator, parallels are arcs of circle
Maurer's full globular
Maurer's full-globular Meridians along lines of Van der Grinten's IV, outer meridians bounded by half limiting circles.  Parallels are arcs of circle, equally spaced both on outer meridians and Equator.
Bacon Bacon's globular Bounded by a single hemisphere in a circle. Circular meridians; straight horizontal parallels equidistant along hemisphere boundaries. Neither conformal nor equal-area.
Apian 1 Apian's first globular Bounded by a single hemisphere in a circle. Circular meridians; straight horizontal parallels equidistant along central meridian. Neither conformal nor equal-area. Extended by Ortelius and Agnese.
Apian 2 Apian's second globular Bounded by a single hemisphere in a circle. Elliptical meridians; straight horizontal parallels equidistant along central meridian. Neither conformal nor equal-area.
Fournier 1 Fournier's first globular Bounded by a single hemisphere in a circle. Elliptical meridians; circular parallels. Neither conformal nor equal-area.
Fournier 2 Fournier's second globular Bounded by a single hemisphere in a circle. Elliptical meridians; straight parallels. Neither conformal nor equal-area.
Nicolosi Globular, "Nicolosi" globular Bounded by a single hemisphere in a circle. Circular parallels and meridians. Neither conformal nor equal-area. Also attributed to La Hire and al-Biruni.
Jäger
Jäger star Graticule comprising only straight lines. Eight unequal lobes, each symmetrical in core and outer hemisphere. Parallels linearly spaced in each lobe. Neither conformal nor equal-area
Petermann star
Petermann star Parallels are concentric, equally spaced arcs of circle, meridians are straight lines (most broken at the Equator). Neither conformal nor equal-area. Sometimes described with unequal lobes
Berghaus
Berghaus star Five-lobed version of Petermann's projection
Conoalactic
Conoalactic Very similar to Berghaus, but northern hemisphere is based on equidistant conic; not to be confused with Cahill's "butterfly" map
Maurer's S233
Maurer's S233 Graticule comprises straight lines, with constant spacing. Neither conformal nor equal-area. Symmetrical case of Jäger's projection
Maurer
Maurer's S231 (equal-area star) Parallels are concentric arcs of circle; northern hemisphere is a Lambert azimuthal map. Southern meridians are curved. Equal-area
William-Olsson
William-Olsson Combines Lambert azimuthal projection in partial inner hemisphere with lobes using rescaled Werner projection. Equal-area
Tetrahedral
Bartholomew's "Tetrahedral" Central core is part of azimuthal equidistant hemisphere. Outer lobes are modified Werner with expanded parallel scaling. Neither conformal, equal-area nor polyhedral
Daisy
"Flower-petal", daisy Transverse Mercator lobes; central core uses an azimuthal equal-area projection. Part of the USGS's ISIS package.
da Vinci
Leonardo da Vinci's octant map Octant map, bound by circular arcs. Neither conformal nor equal-area
Armadillo Armadillo (orthoapsidal on torus) Intermediate projection surface is a torus with radii 1 and 1; final map is projected orthographically
Half-Ellipsoidal Orthoapsidal on ellipsoid Intermediate projection surface is an ellipsoid; poles may be points or lines, and meridians may optionally have constant scale; the final map is projected orthographically
Arden-Close Arden-Close Arithmetical mean of equal-area cylindrical map and its transverse aspect; neither equal-area nor conformal.
Tobler's local projection Tobler's projection for local maps For small regions only. Neither equal-area nor conformal. Parameterized by reference parallel.
Gringorten projection Gringorten's projection Equal-area on a square.
Trapezoidal Trapezoidal, Donis Pseudocylindrical, meridians are straight lines, sometimes symmetrically broken at the Equator
Lee's tetrahedral conformal
Lee's map on a regular tetrahedron Conformal except at tetrahedral vertices
COBE Quadrilaterized Spherical Cube COBE Quadrilaterized Spherical Cube Cubic, approximately equal-area. Intended for cosmological charts of the universe, not earthly geography
Quadrilaterized Spherical Cube Quadrilaterized Spherical Cube Cubic, equal-area. Modification of COBE QSC
Fuller's projection on cuboctahedron Fuller's Dymaxion™ projection on cuboctahedron Scale preserved along face edges. Neither equal-area nor conformal. Several possible face arrangements.
Fuller's projection on icosahedron Fuller's Dymaxion™ Air-Ocean World Map on an icosahedron Scale preserved along face edges. Neither equal-area nor conformal. Several possible face arrangements.
Gnomonic projection on polyhedra Gnomonic projection on polyhedra; adopted by several authors, notably Irving Fisher on the icosahedron Identical to ordinary gnomonic, with advantages and shortcomings of interruptions.
Gnomonic projection on polyhedra Fisher's equal-area projection on the icosahedron. Equal-area. Generalized by Snyder for other regular polyhedra
Waterman Steve Waterman's projection system Based on a truncated octahedron defined by the centers of packed spheres. Neither conformal nor equal-area.
Halstead Kent Halstead's equidistant projection Equidistant along all meridians and parallels, which are broken and interrupted to reduce shearing. Neither conformal nor equal-area.
Halstead Kent Halstead's Composite World projection Interrupted, based on Lambert's azimuthal projection. Mostly equal-area, except at lobe boundaries.

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Copyright © 1996, 1997, 2008 Carlos A. Furuti (except maps by Halstead, courtesy of the author)