From 0e2f5a69605a1327f5b31328dff03acd7d5f49e3 Mon Sep 17 00:00:00 2001 From: julien2512 Date: Fri, 30 Sep 2016 00:04:08 +0200 Subject: Cassini projection docs (first try) --- docs/source/projections/cass.rst | 134 ++++++++++++++++++++++++++++++++++++++- 1 file changed, 133 insertions(+), 1 deletion(-) (limited to 'docs/source') diff --git a/docs/source/projections/cass.rst b/docs/source/projections/cass.rst index b81c5dbf..1d6a7ccf 100644 --- a/docs/source/projections/cass.rst +++ b/docs/source/projections/cass.rst @@ -1,10 +1,142 @@ .. _cass: ******************************************************************************** -Cassini +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 Elliptical | ++---------------------+-------------------------------------------------------------------------+ +| **Defined area** | Global, but best used near the central meridian with long, narrow areas | ++---------------------+-------------------------------------------------------------------------+ +| **Implemented by** | Gerald I. Evenden | ++---------------------+-------------------------------------------------------------------------+ +| **Options** | ++---------------------+-------------------------------------------------------------------------+ +| `+lat_0` | Center of the map : latitude of origin | ++---------------------+-------------------------------------------------------------------------+ + .. image:: ./images/cass.png :scale: 50% :alt: Cassini +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 (Steers, 1970, p. 229). 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 +no_defs + 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 +datum=potsdam +units=m +no_defs + 31343.05 7932.76 + + +Mathematical definition +####################### + +The formulas describing the Cassini projection are all taken from proj4 sources. + +:math:`\phi_0` is the latitude of origin that match the center of the map. It can be set with ``+lat_0``. + + +Spherical form +============== + +Forward projection +------------------ + +.. math:: + + x = \arcsin(\cos(\phi) * \sin(\lambda)) + +.. math:: + + y = \arctan(\tan(\phi), \cos(\lambda)) - \phi_0 + +Inverse projection +------------------ + +.. math:: + + \phi = \arcsin(\sin(y+\phi_0) \cos(x)) + +.. math:: + + \DeclareMathOperator{\arctantwo}{arctan2} + \lambda = \arctantwo(\tan(x), \cos(y+\phi_0)) + +Elliptical form +=============== + +Forward projection +------------------ + +.. math:: + + N = (1 - e^2 \sin^2(\phi))^{-1/2} + +.. math:: + + T = \tan^2(\phi) + +.. math:: + + A = \lambda \cos(\phi) + +.. math:: + + C = \frac{e^2}{1-e^2} cos^2(\phi) + +.. math:: + + x = N ( A - T \frac{A^3}{6} - (8-T+8C)T\frac{A^5}{120}) + +.. math:: + + y = M(\phi) - M(\phi_0) + N \tan(\phi)(\frac{A^2}{2} + (5-T+6C)\frac{A^4}{24}) + +and M() is the meridinial distance function. + +Inverse projection +------------------ + +.. math:: + + \phi' = M^{-1}(M(\phi_0)+y) + +if :math:`\phi' = \frac{\pi}{2}` then :math:`\phi=\phi'` and :math:`\lambda=0` otherwise evaluate T and N above using :math:`\phi'` and + +.. math:: + + R = (1 - e^2)(1 - e^2 sin^2 \phi')^{-3/2} + +.. math:: + + D = x/N + +.. math:: + + \phi = \phi' - \tan \phi' \frac{N}{R}(\frac{D^2}{2}-(1+3T)\frac{D^4}{24}) + +.. math:: + + \lambda = (D - T\frac{D^3}{3} + (1+3T)T\frac{D^5}{15})/cos \phi' + + +Further reading +############### + +#. `Wikipedia `_ +#. `John P. Snyder, Map projections—a working manual `_ +#. `EPSG, POSC literature pertaining to Coordinate Conversions and Transformations including Formulas `_ + -- cgit v1.2.3 From 9349d19727a9233491bf8809f2974419eb95d551 Mon Sep 17 00:00:00 2001 From: julien2512 Date: Mon, 3 Oct 2016 22:40:03 +0200 Subject: Cass docs update @kbevers @QuLogic #422 --- docs/source/projections/cass.rst | 24 ++++++++++++++---------- 1 file changed, 14 insertions(+), 10 deletions(-) (limited to 'docs/source') diff --git a/docs/source/projections/cass.rst b/docs/source/projections/cass.rst index 1d6a7ccf..be2aa1e9 100644 --- a/docs/source/projections/cass.rst +++ b/docs/source/projections/cass.rst @@ -17,7 +17,7 @@ Although the Cassini projection has been largely replaced by the Transverse Merc +---------------------+-------------------------------------------------------------------------+ | **Options** | +---------------------+-------------------------------------------------------------------------+ -| `+lat_0` | Center of the map : latitude of origin | +| `+lat_0` | Latitude of origin (Default to 0) | +---------------------+-------------------------------------------------------------------------+ .. image:: ./images/cass.png @@ -25,9 +25,9 @@ Although the Cassini projection has been largely replaced by the Transverse Merc :alt: Cassini 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 (Steers, 1970, p. 229). 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. +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 :ref:`[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):: @@ -46,7 +46,7 @@ Mathematical definition The formulas describing the Cassini projection are all taken from proj4 sources. -:math:`\phi_0` is the latitude of origin that match the center of the map. It can be set with ``+lat_0``. +:math:`\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 @@ -57,11 +57,12 @@ Forward projection .. math:: - x = \arcsin(\cos(\phi) * \sin(\lambda)) + x = \arcsin(\cos(\phi) \sin(\lambda)) .. math:: - y = \arctan(\tan(\phi), \cos(\lambda)) - \phi_0 + \DeclareMathOperator{\arctantwo}{arctan2} + y = \arctantwo(\tan(\phi), \cos(\lambda)) - \phi_0 Inverse projection ------------------ @@ -105,7 +106,7 @@ Forward projection y = M(\phi) - M(\phi_0) + N \tan(\phi)(\frac{A^2}{2} + (5-T+6C)\frac{A^4}{24}) -and M() is the meridinial distance function. +and M() is the meridionial distance function. Inverse projection ------------------ @@ -114,7 +115,9 @@ Inverse projection \phi' = M^{-1}(M(\phi_0)+y) -if :math:`\phi' = \frac{\pi}{2}` then :math:`\phi=\phi'` and :math:`\lambda=0` otherwise evaluate T and N above using :math:`\phi'` and +if :math:`\phi' = \frac{\pi}{2}` then :math:`\phi=\phi'` and :math:`\lambda=0` + +otherwise evaluate T and N above using :math:`\phi'` and .. math:: @@ -130,8 +133,9 @@ if :math:`\phi' = \frac{\pi}{2}` then :math:`\phi=\phi'` and :math:`\lambda=0` o .. math:: - \lambda = (D - T\frac{D^3}{3} + (1+3T)T\frac{D^5}{15})/cos \phi' + \lambda = \frac{(D - T\frac{D^3}{3} + (1+3T)T\frac{D^5}{15})}{\cos \phi'} +.. _further-reading: Further reading ############### @@ -139,4 +143,4 @@ Further reading #. `Wikipedia `_ #. `John P. Snyder, Map projections—a working manual `_ #. `EPSG, POSC literature pertaining to Coordinate Conversions and Transformations including Formulas `_ - +#. `[Steers1970] Steers, J.A., 1970, An introduction to the study of map projections (15th ed.): London, Univ. London Press, p. 229` -- cgit v1.2.3 From 23f7e8d76f99e70775f4bbf98cb55c130689ef32 Mon Sep 17 00:00:00 2001 From: julien2512 Date: Mon, 3 Oct 2016 23:41:07 +0200 Subject: Cass docs Steers1970 reference @kbevers --- docs/source/projections/cass.rst | 3 +-- docs/source/references.rst | 1 + 2 files changed, 2 insertions(+), 2 deletions(-) (limited to 'docs/source') diff --git a/docs/source/projections/cass.rst b/docs/source/projections/cass.rst index be2aa1e9..48ec80cf 100644 --- a/docs/source/projections/cass.rst +++ b/docs/source/projections/cass.rst @@ -27,7 +27,7 @@ Although the Cassini projection has been largely replaced by the Transverse Merc 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 :ref:`[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. +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):: @@ -143,4 +143,3 @@ Further reading #. `Wikipedia `_ #. `John P. Snyder, Map projections—a working manual `_ #. `EPSG, POSC literature pertaining to Coordinate Conversions and Transformations including Formulas `_ -#. `[Steers1970] Steers, J.A., 1970, An introduction to the study of map projections (15th ed.): London, Univ. London Press, p. 229` diff --git a/docs/source/references.rst b/docs/source/references.rst index 3bd200df..22fd96ea 100644 --- a/docs/source/references.rst +++ b/docs/source/references.rst @@ -9,3 +9,4 @@ References .. [Evenden2005] Evenden, G. I., 2005, `libproj4: A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft) `_ +.. [Steers1970] Steers, J.A., 1970, An introduction to the study of map projections (15th ed.): London, Univ. London Press, p. 229 -- cgit v1.2.3