HomeSite MapApplicationsMap Projections - ContentsResourcesMap Projections
Equatorial azimuthal equal-area map
The world according to the equatorial aspect of Lambert's azimuthal equal-area projection. Unprojected original satellite data by NASA

Summary

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 and projections 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, Gauss transverse v. Mercator and Petermann v. Berghaus).

Sample graticule Common names Main Features
Azimuthal orthographic azimuthal orthographic, orthographic Azimuthal, "realistic" view of Earth as seen from space infinitely away. Shows at most a single hemisphere
Azimuthal stereographic azimuthal stereographic, stereographic Azimuthal, conformal, preserves all circles; shows at most one 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, preserves distances along any line touching the center of the map
Azimuthal equal-area Lambert's azimuthal equal-area Non-perspective. Unique azimuthal equal-area projection
Ginzburg's azimuthals Ginzburg's azimuthal I and II Non-perspective, neither equal-area nor conformal
Equal-area cylindrical Lambert's equal-area cylindrical; variations by Behrmann, Trystan Edwards, Gall's (orthographic), Peters, Dyer, Tobler/Chen Only possible cylindrical equal-area projection, including scaled variants like Gall's ("Peters") and Hobo-Dyer which differ only in standard parallels.
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 cylindricalBraun'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 the 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; neither equal-area nor conformal.
Trapezoidal Trapezoidal, Donis Pseudocylindrical, meridians are straight lines, sometimes symmetrically broken at the Equator
Mollweide Mollweide, elliptical, Babinet, homolographic, homalographic Pseudocylindrical, equal-area,  meridians are ellipses; full map bounded by 2 : 1 ellipse; sometimes interrupted; variations include Atlantis and Bromley's
Sinusoidal Sanson-Flamsteed, sinusoidal, Mercator equal-area Pseudocylindrical, equal-area, meridians are sinusoids, parallels are equally spaced and standard lines; 2 : 1
Foucault Stereographic 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's P4
Loximuthal Loximuthal Pseudocylindrical, all straight lines passing through intersection of central meridian and a reference parallel are loxodromes with correct scale and azimuth. Usually asymmetrical around the Equator
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 Homolosine 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's stereographic conic Braun's stereographic conic Perspective conic with center of projection at a pole and 30° standard latitude
Albers's equal-area conic 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 the Equator or two parallels have correct length. Neither equal-area 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 Stretchhing of modified equatorial azimuthal equidistant map; boundary is 2 : 1 ellipse; neither equal-area nor conformal
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 Rescaled oblique Hammer projection. Equal-area.
Eckert-Greifendorff Eckert-Greifendorff Similar to Hammer projection, with different rescaling factor and therefore almost straight parallels. Equal-area
Schjerning II Schjerning II (original map includes an arbitrary unspecified 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
Schjerning III Schjerning III Map comprises two circles joined at a point. Distance from the center of the map to every point is correct, but not azimuth. Final map 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. Neither equal-area nor conformal
Starbius-Werner I Stabius-Werner I Pseudoconic, equal-area, parallels are equally spaced circular arcs centered on a pole.
Starbius-Werner II 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.
Starbius-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 conformal 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/Guyou Adams's hemispheres on squares Hemispheres in two squares. Conformal except at square corners. Other aspects by Guyou and Peirce
Adams 1929 Adams's world on a square (1929) Poles at opposite vertices; Equator along a diagonal. Conformal except at four vertices
Adams 1936 Adams's world on a square (1936) Poles at midpoints of opposite edges. Conformal except at poles and four vertices
Xarax Xarax's world in half a hexagon Three-lobed rearrangement of Lee's map on a tetrahedron. Conformal except at midpoints of three longest edges
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 some of Spilhaus's oceanic maps
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, large 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; straight central meridian and Equator. Neither equal-area nor conformal
Van der Grinten III Van der Grinten III Boundary is a circle, meridians are circular arcs; straight horizontal parallels intersect straight central meridian at the same points as in Van der Grinten I. Not conformal or equal-area
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. Neither conformal nor equal-area
Maurer 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 on both outer meridians and Equator.
Bacon Bacon's globular Single hemisphere bounded by a circle. Circular meridians; straight horizontal parallels equidistant along hemisphere boundaries. Neither conformal nor equal-area.
Apian 1 Apian's first globular Single hemisphere bounded by 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 Single hemisphere bounded by a circle. Elliptical meridians; straight horizontal parallels equidistant along central meridian. Neither conformal nor equal-area.
Fournier 1 Fournier's first globular Single hemisphere bounded by a circle. Elliptical meridians; circular parallels. Neither conformal nor equal-area.
Fournier 2 Fournier's second globular Single hemisphere bounded by a circle. Elliptical meridians; straight parallels. Neither conformal nor equal-area.
Nicolosi Globular, "Nicolosi" globular Single hemisphere bounded by a circle. Circular parallels and meridians. Neither conformal nor equal-area. Also attributed to La Hire and al-Biruni.
Ortelius Ortelius's oval Simple extension of Apian's first globular hemisphere. Neither pseudocylindrical, equal-area nor conformal.
Leonardo's Octants Leonardo da Vinci's octant map Octant map, bound by circular arcs; graticule uncertain, probably neither conformal nor equal-area
Jager 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 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
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's S231 Maurer's S231 (equal-area star) Parallels are concentric arcs of circle; central hemisphere is a Lambert azimuthal map. Lobe 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" Core is a partial azimuthal equidistant hemisphere. Lobes are modified Werner maps 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.
Conoalactic Conoalactic Very similar to Berghaus, but the central hemisphere is based on equidistant conic; not to be confused with Cahill's "butterfly" map
Armadillo Armadillo (orthoapsidal on torus) Intermediate projection surface is a torus with radii 1 and 1; final map is projected orthographically; neither equal-area nor conformal
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 Tobler's projection for local maps Fast rendering for small regions. Neither equal-area nor conformal. Parameterized by reference parallel.
Gringorten Gringorten's projection Equal-area on a square.
Lee's projection on a tetrahedron Lee's map on a regular tetrahedron Conformal except at tetrahedral vertices
COBE QSC COBE Quadrilaterized Spherical Cube Cubic, approximately equal-area. Intended for cosmological charts of the universe, not earthly geography
QSC Quadrilaterized Spherical Cube Cubic, equal-area. Modification of COBE QSC
Fuller's cuboctahedral Fuller's Dymaxion™ projection on cuboctahedron Scale preserved along face edges. Neither equal-area nor conformal. Several possible face arrangements
Fuller's Dymaxion 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 on polyhedra Gnomonic projection on polyhedra; adopted by several authors, notably Irving Fisher on the icosahedron and Cahill on octahedra Identical to ordinary gnomonic, with advantages and shortcomings of interruptions, plus arbitrary face arrangements
Fisher's equal-area on polyhedra Fisher's equal-area projection on the icosahedron. Equal-area. Generalized by Snyder for other regular polyhedra
Gnomonic on octahedron Cahill's Butterfly Developed on the regular (usually truncated) octahedron. Basic projection is gnomonic; variants are equal-area or conformal. Further refined by Gene Keyes
Waterman Steve Waterman's projection system Based on a truncated octahedron defined by the centers of packed spheres. Graticule comprises broken straight lines. 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.

HomeSite MapApplicationsMap Projections - ContentsResources  www.progonos.com/furuti    April 20, 2015
Copyright © 1996, 1997, 2008 Carlos A. Furuti (except maps by Halstead, courtesy of the author)