Cassini (Cassini-Soldner)

Although the Cassini projection has been largely replaced by the Transverse Mercator, it is still in limited use outside the United States and was one of the major topographic mapping projections until the early 20th century.

Classification

Transverse and oblique cylindrical

Available forms

Forward and inverse, Spherical and ellipsoidal

Defined area

Global, but best used near the central meridian with long, narrow areas

Alias

cass

Domain

2D

Input type

Geodetic coordinates

Output type

Projected coordinates

Cassini

proj-string: +proj=cass

Usage

There has been little usage of the spherical version of the Cassini, but the ellipsoidal Cassini-Soldner version was adopted by the Ordnance Survey for the official survey of Great Britain during the second half of the 19th century [Steers1970]. Many of these maps were prepared at a scale of 1:2,500. The Cassini-Soldner was also used for the detailed mapping of many German states during the same period.

Example using EPSG 30200 (Trinidad 1903, units in clarke’s links):

$ echo 0.17453293 -1.08210414 | proj +proj=cass +lat_0=10.44166666666667 +lon_0=-61.33333333333334 +x_0=86501.46392051999 +y_0=65379.0134283 +a=6378293.645208759 +b=6356617.987679838 +to_meter=0.201166195164
66644.94    82536.22

Example using EPSG 3068 (Soldner Berlin):

$ echo 13.5 52.4 | proj +proj=cass +lat_0=52.41864827777778 +lon_0=13.62720366666667 +x_0=40000 +y_0=10000 +ellps=bessel +units=m
31343.05    7932.76

Options

Note

All options are optional for the Cassini projection.

+lat_0=<value>

Latitude of projection center.

Defaults to 0.0.

+lon_0=<value>

Longitude of projection center.

Defaults to 0.0.

+x_0=<value>

False easting.

Defaults to 0.0.

+y_0=<value>

False northing.

Defaults to 0.0.

+ellps=<value>

See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.

+R=<value>

Radius of the sphere given in meters. If used in conjunction with +ellps +R takes precedence.

Mathematical definition

The formulas describing the Cassini projection are taken from [Snyder1987].

\(\phi_0\) is the latitude of origin that match the center of the map (default to 0). It can be set with +lat_0.

Spherical form

Forward projection

\[x = \arcsin(\cos(\phi) \sin(\lambda))\]
\[y = \arctan2(\tan(\phi), \cos(\lambda)) - \phi_0\]

Inverse projection

\[\phi = \arcsin(\sin(y+\phi_0) \cos(x))\]
\[\lambda = \arctan2(\tan(x), \cos(y+\phi_0))\]

Ellipsoidal form

Forward projection

\[N = (1 - e^2 \sin^2(\phi))^{-1/2}\]
\[T = \tan^2(\phi)\]
\[A = \lambda \cos(\phi)\]
\[C = \frac{e^2}{1-e^2} cos^2(\phi)\]
\[x = N ( A - T \frac{A^3}{6} - (8-T+8C)T\frac{A^5}{120})\]
\[y = M(\phi) - M(\phi_0) + N \tan(\phi)(\frac{A^2}{2} + (5-T+6C)\frac{A^4}{24})\]

and M() is the meridional distance function.

Inverse projection

\[\phi' = M^{-1}(M(\phi_0)+y)\]

if \(\phi' = \frac{\pi}{2}\) then \(\phi=\phi'\) and \(\lambda=0\)

otherwise evaluate T and N above using \(\phi'\) and

\[R = (1 - e^2)(1 - e^2 sin^2 \phi')^{-3/2}\]
\[D = x/N\]
\[\phi = \phi' - \tan \phi' \frac{N}{R}(\frac{D^2}{2}-(1+3T)\frac{D^4}{24})\]
\[\lambda = \frac{(D - T\frac{D^3}{3} + (1+3T)T\frac{D^5}{15})}{\cos \phi'}\]