Map Projections: Summary

Map Projections

Summary

The world according to Lambert's azimuthal equal-area map, in the equatorial aspect

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 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. Show true directions from single tangent point; in polar aspects all parallels are circular, and meridians straight lines uniformly spaced; unclipped map is circular. Projection can usually be defined by tangent point at a plane and location of light source (projection center)
Cylindrical	Use a cylinder as intermediate projection surface; in equatorial aspect all parallels are straight horizontal, all meridians are straight vertical and uniformly spaced; unclipped map is rectangular
Conic	Use a conic intermediate projection surface; in polar aspect all parallels are concentric arcs of circle, all meridians are straight lines perpendicular to every parallel; unclipped maps are circular sectors
Pseudocylindrical	In equatorial aspect all parallels are straight horizontals; meridians are arbitrary curves, equally spaced along every parallel
Pseudoconic	In polar or equatorial aspects all parallels are circular arcs, while meridians are arbitrary curves
Arbitrary	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".

Projections Classified by Property

Category	Properties
Equal-area	Any region in the map has area linearly proportional to the corresponding region on the sphere; also called equivalent or authalic. Generally more useful for geographical 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.
Aphylactic	Some authors use this name for those projections which are neither conformal nor equivalent.

Projections in a Nutshell

Sample graticule	Common names	Main Features
	azimuthal orthographic	Azimuthal, "realistic" view of Earth as seen from space
	azimuthal stereographic	Azimuthal, conformal, circle-preserving; shows at most a hemisphere
	gnomonic, central	Azimuthal, all great circles map to straight lines; extreme distortion far from the center; shows less than one hemisphere
	azimuthal equidistant, zenithal equidistant	Azimuthal, distances from center preserved, navigational purposes
	Lambert's azimuthal equal-area	Only possible azimuthal equal-area
	globular projections by Bacon, Apian, Fournier, Ortelius, Nicolosi	General class of arbitrary projections bounded by a hemisphere in a circle; graticule comprising simple curves
	Lambert's equal-area cylindrical; variations by Behrmann, Trystan Edwards, Gall (isographic), Peters, Dyer, Tobler/Chen	Only possible cylindrical equal-area projection, including scaled variants like Gall's and Hobo-Dyer
	Gall's stereographic cylindrical	Neither conformal nor equal-area
	Braun's stereographical cylindrical	Neither conformal nor equal-area
	central cylindrical, centrographic cylindrical	Neither conformal nor equal-area; not to be confused with Mercator's. Transverse aspect is the Wetch projection
	equirrectangular, equidistant cylindrical, plain chart, plane chart; special case is simple cylindrical or plate carrée; Cassini (transverse aspect)	Cylindrical, very fast and easy to compute; in the most common case maps into a rectangle with aspect ratio 2 : 1 (twice wide as tall)
	Mercator, cylindrical conformal; transverse ellipsoidal form called Gauss conformal or Gauss-Krüger	Only possible conformal cylindrical projection; foundation of the UTM grid
	Miller	Cylindrical, arbitrary compromise to Mercator
	Mollweide, elliptical, Babinet, homolographic, homalographic	Pseudocylindrical, equal-area, meridians are ellipses; full map bounded by 2 : 1 ellipse; sometimes interrupted; variations include Atlantis
	Sanson-Flamsteed, sinusoidal, Mercator equal-area	Pseudocylindrical, equal-area, meridians are sinusoids, parallels are equally spaced and standard lines; 2 : 1
	Collignon	Pseudocylindrical, equal-area, meridians are straight lines. Two main variants, with triangular frame or symmetrical diamond with meridians broken at Equator
	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
	Flat polar quartic	Pseudocylindrical, equal-area, poles are 1/3 as long as the Equator
	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	Pseudocylindrical, equal-area, 2 : 1, poles are half as long as the Equator, meridians are straight lines broken at Equator.
	Eckert III	Pseudocylindrical, 2 : 1, meridians are elliptical arcs (boundary is circular). Parallels are equally spaced.
	Eckert IV	Pseudocylindrical, equal-area, 2 : 1, meridians are elliptical arcs, circular at boundary.
	Eckert V	Pseudocylindrical, 2 : 1, meridians are sinusoids, parallels are equally spaced. Particular case of Winkel's first projection
	Eckert VI	Pseudocylindrical, equal-area, 2 : 1, poles are half as long as the Equator, meridians are sinusoids.
	Robinson, orthophanic	Pseudocylindrical, compromise. Neither conformal nor equal-area
	Winkel I	Pseudocylindrical (generalizes Eckert V), averages Sanson-Flamsteed and equidistant cylindrical, meridians are sinusoids
	Winkel II	Pseudocylindrical, averages equidistant cylindrical and a modified elliptical projection
	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	Pseudocylindrical, equal-area, meridians are sinusoids
	Quartic authalic	Pseudocylindrical, equal-area, meridians are 4th order polynomials
	Equidistant conic	Conic, constant meridian scale; limiting cases are azimuthal equidistant and cylindrical equidistant projections. Many variations, mostly in choice of standard parallels (Murdock, Euler). Others include de l'Isle's coniclike projection.
	Braun stereographic conic	Conic, semicircular shape
	Albers equal-area conic	Conic, equal-area; limiting cases are Lambert's equal-area conic and cylindrical projections.
	Lambert's equal-area conic	Conic, equal-area; limiting case of Albers's conic, with a pole as standard parallel
	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, 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
	Aitoff	Stretching of modified equatorial azimuthal equidistant map; boundary is 2 : 1 ellipse
	Wagner IX, Aitoff-Wagner	Modified Aitoff projection; 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
	Briesemeister	Equal-area, simple oblique stretching of Hammer projection
	Eckert-Greifendorff	Rescaled modification of Hammer projection. Equal-area
	Winkel Tripel	Arithmetic mean of Aitoff and equidistant cylindrical projections
	Stabius-Werner I	Pseudoconic, equal-area, parallels are equally spaced circular arcs centered on a pole
	Werner, Stabius-Werner II, cordiform	Pseudoconic, equal-area, parallels are equally spaced circular arcs and standard lines, centered on a pole
	Stabius-Werner III	Pseudoconic, equal-area, parallels are equally spaced circular arcs centered on a pole
	Bonne	Pseudoconic, equal-area, parallels are equally spaced circular arcs and standard lines. General case of both Werner and sinusoidal
	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	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's world on a square	Poles at opposite vertices; Equator at a diagonal. Conformal except at four vertices
	Adams's world on a square	Poles at midpoints of opposite edges. Conformal except at poles and four vertices
	Xarax's world in half a hexagon	Three-lobed world map. Conformal except at North pole and meridian breaks at each lobe
	Eisenlohr	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 Spilhaus's oceanic map
	"Lagrange"	Map is bounded by a circle; meridians and parallels are circular arcs, except central meridian and Equator. Conformal except at the poles
	Van der Grinten, Van der Grinten I	Boundary is a circle, meridians and parallels are circular arcs, except central meridian and Equator
	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	Bounded by two intersecting circles, meridians are arcs of circle equally spaced along Equator, parallels are arcs of circle
	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.
	Jäger Star	Graticule comprising only straight lines. Eight unequal lobes, each symmetrical in core and outer hemisphere. Parallels linearly spaced in each lobe.
	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
	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	Graticule comprises straight lines, with constant spacing. Neither conformal nor equal-area. Symmetrical case of Jäger's projection.
	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.
	"Tetrahedral"	Central core is part of azimuthal equidistant hemisphere. Outer lobes are modified Werner with expanded parallel scaling. Neither conformal, equal-area nor polyhedral.
	Leonardo da Vinci's octant map	Octant map, bound by circular arcs. Neither conformal nor equal-area.
	Armadillo (orthoapsidal on torus)	Intermediate projection surface is a torus with radii 1 and 1; final map is projected orthographically
	Orthoapsidal on ellipsoid	Intermediate projection surface is an ellipsoid; final map is projected orthographically
	Arden-Close	Arithmetical mean of equal-area cylindrical map and its transverse aspect; neither equal-area nor conformal.
	Trapezoidal	Pseudocylindrical, meridians are straight lines, sometimes symmetrically broken at the Equator
	Lee's map on a regular tetrahedron	Conformal except at tetrahedral vertices
	COBE Quadrilaterized Spherical Cube	Cubic, approximately equal-area.
	Quadrilaterized Spherical Cube	Cubic, equal-area.
	Steve Waterman's projection system	Based on a truncated octahedron defined by the centers of packed spheres. 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.