# Bottomley projection

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The Bottomley map projection is an equal area map projection defined as:

$x={\frac {\rho \sin E}{\sin \varphi _{1}}},\qquad y={\frac {\pi }{2}}-\rho \cos E\,$ where

$\rho ={\frac {\pi }{2}}-\varphi ,\qquad E={\frac {\lambda \sin \varphi _{1}\sin \rho }{\rho }}$ and φ is the latitude, λ is the longitude from the central meridian, and φ1 is the given parallel of the projection which determines its shape, all in radians.

The inverse projection is then given by:

{\begin{aligned}\varphi &={\frac {\pi }{2}}-\rho \\\lambda &={\frac {E\rho }{\sin \varphi _{1}\sin \rho }}\end{aligned}} where

$\rho ={\sqrt {(x\sin \varphi _{1})^{2}+\left(\varphi _{1}-y+\cot \varphi _{1}\right)^{2}}},\qquad E=\tan ^{-1}\left({\frac {x\sin \varphi _{1}}{\varphi _{1}-y+\cot \varphi _{1}}}\right).$ Parallels (i.e. lines of latitude) are concentric elliptical arcs of constant eccentricity equal to cos φ1, centred on the north pole. On the central meridian, shapes are not distorted, but elsewhere they are. Different projections can be produced by altering the eccentricity of the arcs, making it vary between the sinusoidal projection and the Werner projection. In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 to e = 1. In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. A (geographic) meridian is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km.

It was introduced by Henry Bottomley as an alternative to the Bonne projection to reduce the extent of extreme distortion at the edges and give a more satisfying overall shape. The Bonne projection is a pseudoconical equal-area map projection, sometimes called a dépôt de la guerre, modified Flamsteed, or a Sylvanus projection. 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.