Bonne projection

Bonne projection of the world, standard parallel at 45°N.

World map by Bernard Sylvanus, 1511

A Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre,^[1]:104 modified Flamsteed,^[1]:104 or a Sylvanus projection.^[1]:92 Although named after Rigobert Bonne (1727–1795), the projection was in use prior to his birth, in 1511 by Sylvano, Honter in 1561, De l'Isle before 1700 and Coronelli in 1696. Both Sylvano and Honter’s usages were approximate, however, and it is not clear they intended to be the same projection.^[1]:60–62

The projection is:

${\displaystyle {\begin{aligned}x&=\rho \sin E\\y&=\cot \varphi _{1}-\rho \cos E\end{aligned}}}$

where

${\displaystyle {\begin{aligned}\rho &=\cot \varphi _{1}+\varphi _{1}-\varphi \\E&={\frac {(\lambda -\lambda _{0})\cos \varphi }{\rho }}\end{aligned}}}$

and φ is the latitude, λ is the longitude, λ₀ is the longitude of the central meridian, and φ₁ is the standard parallel of the projection.^[2]

Parallels of latitude are concentric circular arcs, and the scale is true along these arcs. On the central meridian and the standard latitude shapes are not distorted.

The inverse projection is given by:

${\displaystyle {\begin{aligned}\varphi &=\cot \varphi _{1}+\varphi _{1}-\rho \\\lambda &=\lambda _{0}+{\frac {\rho }{\cos \varphi }}\arctan \left({\frac {x}{\cot \varphi _{1}-y}}\right)\end{aligned}}}$

where

${\displaystyle \rho =\pm {\sqrt {x^{2}+\left(\cot \varphi _{1}-y\right)^{2}}}}$

taking the sign of φ₁.

Special cases of the Bonne projection include the sinusoidal projection, when φ₁ is zero (i.e. the Equator), and the Werner projection, when φ₁ is 90° (i.e. the 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^[3]

See also

• List of map projections

References

1. John Parr Snyder (1993). Flattening the Earth: Two Thousand Years of Map Projections. ISBN 0-226-76747-7.
2. Map Projections - A Working Manual, USGS Professional Paper 1395, John P. Snyder, 1987, pp. 138–140
3. Between the Sinusoidal projection and the Werner: an alternative to the Bonne, Henry Bottomley 2002

External links

• Cybergeo article
• Bonne Map Projection (manifold.net)
• Table of examples and properties of all common projections, from radicalcartography.net
• An interactive Java Applet to study the metric deformations of the Bonne Projection
• Bonne Projection (wolfram.com)