Two-point equidistant projection

{{Short description|Two-point equidistant map projection}} [[File:Two-point equidistant projection SW.jpg|300px|thumb|Two-point equidistant projection of Eurasia. All distances are correct from the two points (45°N, 40°E) and (30°N, 110°E).]] [[File:Two-point Equidistant with Tissot's Indicatrices of Distortion.svg|300px|thumb|Two-point equidistant projection of the entire world with Tissot's indicatrix of deformation. The two points are Rome and Luoyang.]]

The '''two-point equidistant projection''' or '''doubly equidistant projection''' is a map projection first described by Hans Maurer in 1919 and Charles Close in 1921.<ref>Hans Maurer (1919). „Doppelbüschelstrahlige, orthodromische“ statt „doppelazimutale, gnomonische“ Kartenentwürfe. Doppel-mittabstandstreue Kartogramme. (Bemerkungen zu den Aufsätzen von W. Immler und H. Thorade. Ann. d. Hydr. usw 1919, S. 22 und 35.), ''Annalen der Hydrographie und Maritimen Meteorologie'', 47 (3–4), 75–8.</ref><ref>{{Cite journal |last=Close |first=Charles |date=1921 |title=Note on a Doubly-Equidistant Projection |url=https://www.jstor.org/stable/1780793 |journal=The Geographical Journal |volume=57 |issue=6 |pages=446–448 |doi=10.2307/1780793 |jstor=1780793 |bibcode=1921GeogJ..57..446C |issn=0016-7398}}</ref> It is a generalization of the much simpler azimuthal equidistant projection. In this two-point form, two locus 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 conics onto two families of planar ellipses and hyperbolas.<ref>{{Cite journal |last=Cox |first=J. F. |date=1946-10-01 |title=The doubly equidistant projection |url=https://link.springer.com/article/10.1007/BF02521618 |journal=Bulletin Géodésique |language=en |volume=2 |issue=1 |pages=74–76 |doi=10.1007/BF02521618 |bibcode=1946BGeod..20...74C |issn=0007-4632|url-access=subscription }}</ref>

The projection has been used for all maps of the Asian continent by the National Geographic Society atlases since 1959,<ref name="Snyder CC"> {{Cite book | last1 = Snyder | first1 = J.P. | year = 1993 | title = Flattening the Earth: 2,000 years of map projections | pages = 234–235 | publisher = University of Chicago Press | isbn = 0226767469 }}</ref> though its purpose in that case was to reduce distortion throughout Asia rather than to measure from the two loci.<ref> {{Citation | magazine = National Geographic Magazine | year = 1959 | title = Portrait of Earth's largest continent | volume = 116 | issue = 6 | pages = 751 }}</ref> The projection sometimes appears in maps of air routes. The Chamberlin trimetric projection 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 minimization techniques rather than using closed-form formulae.<ref>{{Cite journal|last=Tobler|first=Waldo|date=April 1986|title=Measuring the Similarity of Map Projections|url=https://www.researchgate.net/publication/233489999|journal=Cartography and Geographic Information Science|volume=13|issue=2|pages=135–139|doi=10.1559/152304086783900103|via=ResearchGate}}</ref>

The projection can be generalized to an ellipsoid of revolution by using geodesic distance.<ref>{{cite arXiv |last=Karney |first=Charles F. F. |title=Geodesics on an ellipsoid of revolution |date=2011-02-07 |class=physics.geo-ph |eprint=1102.1215 }}</ref>

==References== {{reflist}}

* 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 (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.

Category:Equidistant projections