Two-point equidistant projection

The two-point equidistant projection or doubly equidistant projection is a <a href="/facts/Map_projection/XYxklzIu">map projection</a> first described by Hans Maurer in 1919 and Charles Close in 1921. It is a generalization of the much simpler <a href="/facts/Azimuthal_equidistant_projection/gL00mn7A">azimuthal equidistant projection</a>. In this two-point form, two <a href="/facts/Locus_(mathematics)/t4swdGJq">locus</a> 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 <a href="/facts/Spherical_conic/fCqzrPmU">spherical conics</a> onto two families of planar ellipses and hyperbolas.
The projection has been used for all maps of the Asian continent by the <a href="/facts/National_Geographic_Society/mlx2uHa4">National Geographic Society</a> 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 <a href="/facts/Chamberlin_trimetric_projection/G2BxU6Uw">Chamberlin trimetric projection</a> 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. <a href="/facts/Waldo_Tobler/uxwYgfSA">Tobler</a> extended this idea to arbitrarily large number of loci by using automated <a href="/facts/Root_mean_square/Cfmg7dws">root-mean-square</a> minimization techniques rather than using <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form</a> formulae.
The projection can be generalized to an <a href="/facts/Ellipsoid_of_revolution/GJyyLxQ7">ellipsoid of revolution</a> by using geodesic distance.

Two-point equidistant projection open-in-new

Two-point equidistant projection