Two-point Equidistant with Tissot's Indicatrices of Distortion

The two-point equidistant projection or doubly equidistant projection is a

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...

first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler

azimuthal equidistant projection 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 azimut ...

. In this two-point form, two

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

points are chosen by the mapmaker to configure the projection. Distances from the two loci to any other point on the map are correct: that is, they scale to the distances of the same points on the sphere. The two-point equidistant projection maps a family of confocal

spherical conic In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...

s onto two families of planar ellipses and hyperbolas. The projection has been used for all maps of the Asian continent by the

National Geographic Society The National Geographic Society (NGS), headquartered in Washington, D.C., United States, is one of the largest non-profit scientific and educational organizations in the world. Founded in 1888, its interests include geography, archaeology, and ...

atlases since 1959, though its purpose in that case was to reduce distortion throughout Asia rather than to measure from the two loci. The projection sometimes appears in maps of air routes. The

Chamberlin trimetric projection The Chamberlin trimetric projection is a map projection where three points are fixed on the globe and the points on the sphere are mapped onto a plane by triangulation. It was developed in 1946 by Wellman Chamberlin for the National Geographic Soc ...

is a logical extension of the two-point idea to three points, but the three-point case only yields a sort of minimum error for distances from the three loci, rather than yielding correct distances. Tobler extended this idea to arbitrarily large number of loci by using automated

root-mean-square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...

minimization techniques rather than using closed-form formulae. The projection can be generalized to an

ellipsoid of revolution A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...

by using geodesic distance.Charles Karney, (2011). “Geodesics on an ellipsoid of revolution”. https://arxiv.org/abs/1102.1215

References

* Charles Close (1934). “A doubly equidistant projection of the sphere.” ''The Geographical Journal'' 83(2): 144-145. * Charles Close (1947). ''Geographical By-ways: And Some Other Geographical Essays.'' E. Arnold. *

Waldo R. Tobler Waldo Rudolph Tobler (November 16, 1930 – February 20, 2018) was an American- Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related than distant things" is referred ...

(1966). “Notes on two projections.” ''The Cartographic Journal'' 3(2). 87–89. * François Reignier (1957). ''Les systèmes de projection et leurs applications a la géographie, a la cartographie, a la navigation, a la topométrie, etc...'' Institut géographique national. Map projections Equidistant projections {{Cartography-stub

See also

References