Cassini projection

Cassini projection of the world

Cassini projection of the world modeled as a highly oblated ellipsoid with flattening 1:2 (= eccentricity ^√3⁄₂)

The Cassini projection (also sometimes known as the Cassini-Soldner projection or Soldner projection^[1]) is a map projection described by César-François Cassini de Thury in 1745.^[2] It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of these operations:

x=\arcsin(\cos \varphi \sin \lambda )\qquad y=\arctan \left({\frac {\tan \varphi }{\cos \lambda }}\right)

where λ is the longitude from the central meridian and φ is the latitude. When programming these equations, the inverse tangent function used is actually the atan2 function, with the first argument sin φ) and the second cos φ cos λ.

The reverse operation is composed of these operations :

\phi =\arcsin(\sin(y)\cos(x))\qquad \lambda =\arctan 2(\tan(x),\cos(y))

In practice, the projection has always been applied to models of the earth as an ellipsoid, which greatly complicates the mathematical development but is suitable for surveying. Nevertheless, the use of the Cassini projection has largely been superseded by the Transverse Mercator projection, at least with central mapping agencies.

Distortions

Areas along the central meridian, and at right angles to it, are not distorted. Elsewhere, the distortion is largely in a north-south direction, and varies by the square of the distance from the central meridian. As such, the greater the longitudinal extent of the area, the worse the distortion becomes.

Due to this, the Cassini projection works best on long, narrow areas, and worst on wide areas.

Elliptical form

Cassini is known as a spherical projection, but can be generalised as an elliptical form.

Considering the earth as an ellipse, the projection is composed of these operations:

N=(1-e^{2}\sin ^{2}(\phi ))^{-1/2}

T=\tan ^{2}(\phi )

A=\lambda \cos(\phi )

C={\frac {e^{2}}{1-e^{2}}}cos^{2}(\phi )

x=N(A-T{\frac {A^{3}}{6}}-(8-T+8C)T{\frac {A^{5}}{120}})

y=M(\phi )-M(\phi _{0})+N\tan(\phi )({\frac {A^{2}}{2}}+(5-T+6C){\frac {A^{4}}{24}})

and M() is the meridionial distance function.

The reverse operation is composed of these operations :

\phi '=M^{-1}(M(\phi _{0})+y)

If $\phi '={\frac {\pi }{2}}$ then $\phi =\phi '$ and $\lambda =0$

otherwise calculate T and N as above with $\phi '$ then

R=(1-e^{2})(1-e^{2}sin^{2}\phi ')^{-3/2}

D=x/N

\phi =\phi '-\tan \phi '{\frac {N}{R}}({\frac {D^{2}}{2}}-(1+3T){\frac {D^{4}}{24}})

\lambda ={\frac {(D-T{\frac {D^{3}}{3}}+(1+3T)T{\frac {D^{5}}{15}})}{\cos \phi '}}

References

↑ "Cassini-Soldner - Help". Environmental Systems Research Institute, Inc. Retrieved 9 June 2016.
↑ Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 74–76, ISBN 0-226-76747-7.

External links

Media related to Cassini projection at Wikimedia Commons
Table of examples and properties of all common projections, from radicalcartography.net
Ordnance Survey GeoFacts on the Cassini Projection
Cassini dans proj4

Map projection

History
List
Portal

By surface

Cylindrical

Mercator-conformal	Gauss–Krüger Transverse Mercator

Equal-area	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert Smyth equal-surface Trystan Edwards

Cassini Central Equirectangular Gall stereographic Miller Space-oblique Mercator Web Mercator

General perspective	Gnomonic Orthographic Stereographic

Equidistant Lambert equal-area

Bonne	Sinusoidal Werner

Bottomley	Sinusoidal Werner

Cylindrical	Balthasart Behrmann Gall–Peters Hobo–Dyer Lambert cylindrical equal-area Smyth equal-surface Trystan Edwards

Tobler hyperelliptical	Collignon Mollweide

Albers Briesemeister Eckert II Eckert IV Eckert VI Hammer Lambert azimuthal equal-area Quadrilateralized spherical cube

Equidistant in
some aspect

Gnomonic

Gnomonic

Loxodromic

Retroazimuthal
(Mecca or Qibla)

Planar	Gnomonic Orthographic Stereographic

Central cylindrical

Polyhedral

See also

Latitude Longitude Tissot's indicatrix

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Cassini projection

Distortions

Elliptical form

See also

References

External links