View source for Bonne projection

[[File:Bonne projection SW.jpg|300px|thumb|Bonne projection of the world, standard parallel at 45°N.]]
[[File:Bonne with Tissot's Indicatrices of Distortion.svg|300px|thumb|Bonne projection with [[Tissot's indicatrix]] of deformation.]]
[[File:Sylvanus map 1511.jpg|300px|thumb|World map by Bernard Sylvanus, 1511]]
The '''Bonne projection''' is a pseudoconical  equal-area [[map projection]], sometimes called a '''dépôt de la guerre''',<ref name="Flattening">{{Cite book
| title = Flattening the Earth: Two Thousand Years of Map Projections
| author = John Parr Snyder
| year = 1993
| ISBN = 0-226-76747-7
}}</ref>{{rp|104}} '''modified Flamsteed''',<ref name="Flattening"/>{{rp|104}} or a '''Sylvanus''' projection.<ref name="Flattening"/>{{rp|92}} Although named after [[Rigobert Bonne]] (1727–1795), the projection was in use prior to his birth, by Sylvanus in 1511, [[Johannes_Honter|Honter]] in 1561, [[Guillaume_Delisle|De l'Isle]] before 1700 and [[Vincenzo_Coronelli|Coronelli]] in 1696. Both Sylvanus and Honter's usages were approximate, however, and it is not clear they intended to be the same projection.<ref name="Flattening"/>{{rp|60}}

The Bonne projection maintains accurate shapes of areas along the [[meridian (geography)|central meridian]] and the [[Map projection#Notable lines|standard parallel]], but progressively distorts away from those regions. Thus, it best maps "t"-shaped regions. It has been used extensively for maps of Europe and Asia.<ref name="Flattening" />{{rp|61}}

The projection is defined as:

:<math>\begin{align} x &= \rho \sin E \\ y &= \cot \varphi_1 - \rho \cos E\end{align}</math>

where

:<math>\begin{align}\rho &= \cot \varphi_1 + \varphi_1 - \varphi \\ E &= \frac {(\lambda - \lambda_0) \cos \varphi} {\rho}\end{align}</math>

and ''φ'' is the latitude, ''λ'' is the longitude, ''λ''<sub>0</sub> is the longitude of the central meridian, and ''φ''<sub>1</sub> is the standard parallel of the projection.<ref>[https://pubs.er.usgs.gov/usgspubs/pp/pp1395 ''Map Projections - A Working Manual''] {{Webarchive|url=https://web.archive.org/web/20100701103721/http://pubs.er.usgs.gov/usgspubs/pp/pp1395 |date=2010-07-01 }}, [[United States Geological Survey|USGS]] Professional Paper 1395, John P. Snyder, 1987, pp. 138–140</ref>

Parallels of latitude are concentric circular arcs, and the scale is true along these arcs.  On the [[meridian (geography)|central meridian]] and the standard latitude shapes are not distorted.

The inverse projection is given by:

:<math>\begin{align}\varphi &= \cot \varphi_1 + \varphi_1 - \rho \\

\lambda &= \lambda_0 + \frac{\rho} {\cos \varphi}  \arctan \left(\frac {x}{ \cot \varphi_1 - y} \right) \end{align}</math>

where

:<math> \rho = \pm \sqrt{ x^2 + \left( \cot \varphi_1 - y\right)^2 }</math>

taking the sign of ''φ''<sub>1</sub>.

Special cases of the Bonne projection include the [[sinusoidal projection]], when ''φ''<sub>1</sub> is zero (i.e. the [[Equator]]), and the [[Werner projection]], when ''φ''<sub>1</sub> is 90° (i.e. the [[North Pole|North]] or [[South Pole]]). The Bonne projection can be seen as an intermediate projection in the unwinding of a [[Werner projection]] into a [[Sinusoidal projection]]; an alternative intermediate would be a [[Bottomley projection]].<ref>[https://journals.openedition.org/cybergeo/3977 Between the Sinusoidal projection and the Werner: an alternative to the Bonne], Henry Bottomley 2002</ref>

==See also==

*[[List of map projections]]

==References==
{{Reflist}}

==External links==
{{Commons category|Bonne projection}}
*[http://www.radicalcartography.net/?projectionref Table of examples and properties of all common projections], from radicalcartography.net
*[https://web.archive.org/web/20070927225242/http://www.uff.br/mapprojections/Bonne_en.html An interactive Java Applet to study the metric deformations of the Bonne Projection]
*[http://mathworld.wolfram.com/BonneProjection.html Bonne Projection (wolfram.com)]

{{Map projections}}

{{DEFAULTSORT:Bonne Projection}}
[[Category:Equal-area projections]]


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