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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


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, 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
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 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 the simple cylindrical (plate carrée), Gall's isographic and Cassini
Cylindrical, true scale along all meridians and one or two parallels; neither conformal nor equal-area; in the most common case maps into a rectangle with aspect ratio 2 : 1
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, 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
Cylindrical, arbitrary compromise to Mercator; neither equal-area nor conformal
Trapezoidal, Donis
Pseudocylindrical, meridians are straight lines, sometimes symmetrically broken at the Equator
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
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
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
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's pseudocylindrical
Pseudocylindrical, equal-area, polelines
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
Rosén's pseudocylindrical
Pseudocylindrical, equal-area, based on sinusoidal: poles are mapped to parallels arcsin(0.8) N and S of base projection
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 eumorphic
Pseudocylindrical, equal-area, arithmetic average of Mollweide and Sanson-Flamsteed projections. Usually interrupted
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
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
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, 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
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
Stretchhing of modified equatorial azimuthal equidistant map; boundary is 2 : 1 ellipse; neither equal-area nor conformal
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
Rescaled oblique Hammer projection. Equal-area
Similar to Hammer projection, with doubled 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
Pseudoconic, equal-area, parallels are equally spaced circular arcs and standard lines. Appearance depends on reference parallel. General case of both Werner and sinusoidal
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
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
World map in 2:1 rectangle. Conformal except at hemisphere corners. Other aspects by Peirce and Adams
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's world in half a hexagon
Three-lobed rearrangement of Lee's map on a tetrahedron. Conformal except at midpoints of three longest edges
Fully conformal, no singular points. Scale constant along boundary. Optimal range of scale distortion for a conformal design
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
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
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'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
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
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 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
Parallels are concentric arcs of circle; central hemisphere is a Lambert azimuthal map. Lobe meridians are curved. Equal-area
Combines Lambert azimuthal projection in partial inner hemisphere with lobes using rescaled Werner projection. Equal-area
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
“Flower-petal”, Daisy
Transverse Mercator lobes; central core uses an azimuthal equal-area projection. Part of the USGS's ISIS package
Very similar to Berghaus, but the central hemisphere is based on equidistant conic; not to be confused with Cahill's “butterfly” map
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
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
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'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 Quadrilaterized Spherical Cube
Cubic, approximately equal-area. Intended for cosmological charts of the universe, not earthly geography
Quadrilaterized Spherical Cube
Cubic, equal-area. Modification of COBE QSC
Fuller's cuboctahedral
Fuller's Dymaxion™ projection on a 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
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
Kent Halstead's equidistant projection
Equidistant along all meridians and parallels, which are broken and interrupted to reduce shearing. Neither conformal nor equal-area
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)