Bottomley projection

Pseudoconical equal-area map projection
Bottomley projection of the world with standard parallel at 30°N.

The Bottomley map projection is a pseudoconical equal area map projection[1] defined as:

x = ρ sin E sin φ 1 , y = π 2 ρ cos E {\displaystyle x={\frac {\rho \sin E}{\sin \varphi _{1}}},\qquad y={\frac {\pi }{2}}-\rho \cos E\,}

where

ρ = π 2 φ , E = λ sin φ 1 sin ρ ρ {\displaystyle \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:

φ = π 2 ρ λ = E ρ sin φ 1 sin ρ {\displaystyle {\begin{aligned}\varphi &={\frac {\pi }{2}}-\rho \\\lambda &={\frac {E\rho }{\sin \varphi _{1}\sin \rho }}\end{aligned}}}

where

ρ = ( x sin φ 1 ) 2 + ( φ 1 y + cot φ 1 ) 2 , E = tan 1 ( x sin φ 1 φ 1 y + cot φ 1 ) . {\displaystyle \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. For larger values of φ1, it produces a heart shape.[2]

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.

See also

References

  1. ^ Maplesoft, List of Projections for Use with WorldMap
  2. ^ Compare Map Projections

External links

Wikimedia Commons has media related to Maps with Bottomley projection.
  • Cybergeo article: Between the Sinusoidal projection and the Werner: an alternative to the Bonne (Une alternative à la projection cartographique de Bonne), Henry Bottomley, 2003
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Map projection
Cylindrical
Mercator-conformal
Equal-area
Pseudocylindrical
Equal-area
Conical
Pseudoconical
Azimuthal
(planar)
General perspective
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Bonne
Bottomley
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Equidistant in
some aspect
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(Mecca or Qibla)
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