From 66ab646cbf39c7ac8a88f75a1d7f48530803b9dd Mon Sep 17 00:00:00 2001 From: julien2512 Date: Mon, 17 Oct 2016 23:09:22 +0200 Subject: [PATCH 1/3] Transverse Mercator first projection doc shot --- docs/source/projections/tmerc.rst | 109 +++++++++++++++++++++++++++++- 1 file changed, 108 insertions(+), 1 deletion(-) diff --git a/docs/source/projections/tmerc.rst b/docs/source/projections/tmerc.rst index 239134850e..387a834b5e 100644 --- a/docs/source/projections/tmerc.rst +++ b/docs/source/projections/tmerc.rst @@ -1,10 +1,117 @@ .. _tmerc: ******************************************************************************** -Transverse Mercator +Transverse Mercator (Gauss-Kruger) ******************************************************************************** +The transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. + ++---------------------+--------------------------------------------------------------------------------+ +| **Classification** | Transverse and oblique cylindrical | ++---------------------+--------------------------------------------------------------------------------+ +| **Available forms** | Forward and inverse, Spherical and Elliptical | ++---------------------+--------------------------------------------------------------------------------+ +| **Defined area** | Global, but reasonably accurate only within 15 degrees of the central meridian | ++---------------------+--------------------------------------------------------------------------------+ +| **Implemented by** | Gerald I. Evenden | ++---------------------+--------------------------------------------------------------------------------+ +| **Options** | ++---------------------+--------------------------------------------------------------------------------+ +| `+lat_0` | Latitude of origin (Default to 0) | ++---------------------+--------------------------------------------------------------------------------+ +| `+k0` | Scale factor at natural origin (Default to 1) | ++---------------------+--------------------------------------------------------------------------------+ + + .. image:: ./images/tmerc.png :scale: 50% :alt: Transverse Mercator +Usage +##### + + +Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the interwar period. Calculating the distance between two points on these maps could be performed more easily in the field (using the Pythagorean theorem) than was possible using the trigonometric formulas required under the graticule-based system of latitude and longitude. In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps. + +The following table gives special cases of the Transverse Mercator projection. + ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| Projection Name | Areas | Central meridian | Zone width | Scale Factor | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| Transverse Mercator | World wide | Various | less than 6° | Various | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| Transverse Mercator south oriented | Southern Africa | 2° intervals E of 11°E | 2° | 1.000 | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| UTM North hemisphere | World wide equator to 84°N | 6° intervals E & W of 3° E & W | Always 6° | 0.9996 | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| UTM South hemisphere | World wide north of 80°S to equator | 6° intervals E & W of 3° E & W | Always 6° | 0.9996 | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| Gauss-Kruger | Former USSR, Yugoslavia, Germany, S. America, China | Various, according to area | Usually less than 6°, often less than 4° | 1.0000 | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ +| Gauss Boaga | Italy | Various, according to area | 6° | 0.9996 | ++-------------------------------------+-----------------------------------------------------+--------------------------------+------------------------------------------+--------------+ + + + +Example using Gauss-Kruger on Germany area (aka EPSG:31467) :: + + $ echo 9 51 | proj +proj=tmerc +lat_0=0 +lon_0=9 +k=1 +x_0=3500000 +y_0=0 +ellps=bessel +datum=potsdam +units=m +no_defs + 3500000.00 5651505.56 + +Example using Gauss Boaga on Italy area (EPSG:3004) :: + + $ echo 15 42 | proj +proj=tmerc +lat_0=0 +lon_0=15 +k=0.9996 +x_0=2520000 +y_0=0 +ellps=intl +units=m +no_defs + 2520000.00 4649858.60 + +Mathematical definition +####################### + +The formulas describing the Transverse Mercator 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:`k_0` is the scale factor at the natural origin (on the central meridian). It can be set with ``+k_0``. + +Spherical form +************** + +Forward projection +================== + +.. math:: + + B = \cos \phi \sin \lambda + +.. math:: + + x = \frac{k_0}{2} \ln(\frac{1+B}{1-B}) + +.. math:: + + y = k_0 ( \arctan(\frac{\tan(\phi)}{\cos \lambda}) - \phi_0) + + +Inverse projection +================== + +.. math:: + + D = \frac{y}{k_0} + \phi_0 + +.. math:: + + \phi = \arcsin(\frac{\sin D}{\cosh x'}) + +.. math:: + + \lambda = \arctan(\frac{\sinh x'}{\cos D}) + +.. math:: + + x' = \frac{x}{k_0} + +Further reading +############### + +#. `Wikipedia `_ +#. `EPSG, POSC literature pertaining to Coordinate Conversions and Transformations including Formulas `_ From 737c9c06b08a1f6da0c4d339ad0e8118e06f6755 Mon Sep 17 00:00:00 2001 From: julien2512 Date: Sat, 22 Oct 2016 22:53:00 +0200 Subject: [PATCH 2/3] Add Elliptical maths for tmerc for Gauss Kruber only --- docs/source/projections/tmerc.rst | 80 ++++++++++++++++++++++++++++++- 1 file changed, 79 insertions(+), 1 deletion(-) diff --git a/docs/source/projections/tmerc.rst b/docs/source/projections/tmerc.rst index 387a834b5e..ff0c8a788b 100644 --- a/docs/source/projections/tmerc.rst +++ b/docs/source/projections/tmerc.rst @@ -72,6 +72,8 @@ The formulas describing the Transverse Mercator are all taken from proj4 sources :math:`k_0` is the scale factor at the natural origin (on the central meridian). It can be set with ``+k_0``. +:math:`M(\phi)` is the meridianal distance. + Spherical form ************** @@ -98,6 +100,10 @@ Inverse projection D = \frac{y}{k_0} + \phi_0 +.. math:: + + x' = \frac{x}{k_0} + .. math:: \phi = \arcsin(\frac{\sin D}{\cosh x'}) @@ -106,9 +112,81 @@ Inverse projection \lambda = \arctan(\frac{\sinh x'}{\cos D}) + +Elliptical form +*************** + +Forward projection +================== + .. math:: - x' = \frac{x}{k_0} + N = \frac{k_0}{(1 - e^2 \sin^2\phi)^{1/2}} + +.. math:: + + R = \frac{k_0(1-e^2)}{(1-e^2 \sin^2\phi)^{3/2}} + +.. math:: + + t = \tan(\phi) + +.. math:: + + \eta = \frac{e^2}{1-e^2}cos^2\phi + +.. math:: + + x &= k_0 \lambda \cos \phi \\ + &+ \frac{k_0 \lambda^3 \cos^3\phi}{3!}(1-t^2+\eta^2) \\ + &+ \frac{k_0 \lambda^5 \cos^5\phi}{5!}(5-18t^2+t^4+14\eta^2-58t^2\eta^2) \\ + &+\frac{k_0 \lambda^7 \cos^7\phi}{7!}(61-479t^2+179t^4-t^6) + +.. math:: + + y &= M(\phi) \\ + &+ \frac{k_0 \lambda^2 \sin(\phi) \cos \phi}{2!} \\ + &+ \frac{k_0 \lambda^4 \sin(\phi) \cos^3\phi}{4!}(5-t^2+9\eta^2+4\eta^4) \\ + &+ \frac{k_0 \lambda^6 \sin(\phi) \cos^5\phi}{6!}(61-58t^2+t^4+270\eta^2-330t^2\eta^2) \\ + &+ \frac{k_0 \lambda^8 \sin(\phi) \cos^7\phi}{8!}(1385-3111t^2+543t^4-t^6) + +Inverse projection +================== + +.. math:: + + \phi_1 = M^-1(y) + +.. math:: + + N_1 = \frac{k_0}{1 - e^2 \sin^2\phi_1)^{1/2}} + +.. math:: + + R_1 = \frac{k_0(1-e^2)}{(1-e^2 \sin^2\phi_1)^{3/2}} + +.. math:: + + t_1 = \tan(\phi_1) + +.. math:: + + \eta_1 = \frac{e^2}{1-e^2}cos^2\phi_1 + +.. math:: + + \phi &= \phi_1 \\ + &- \frac{t_1 x^2}{2! R_1 N_1} \\ + &+ \frac{t_1 x^4}{4! R_1 N_1^3}(5+3t_1^2+\eta_1^2-4\eta_1^4-9\eta_1^2t_1^2) \\ + &- \frac{t_1 x^6}{6! R_1 N_1^5}(61+90t_1^2+46\eta_1^2+45t_1^4-252t_1^2\eta_1^2) \\ + &+ \frac{t_1 x^8}{8! R_1 N_1^7}(1385+3633t_1^2+4095t_1^4+1575t_1^6) + +.. math:: + + \lambda &= \frac{x}{\cos \phi N_1} \\ + &- \frac{x^3}{3! \cos \phi N_1^3}(1+2t_1^2+\eta_1^2) \\ + &+ \frac{x^5}{5! \cos \phi N_1^5}(5+6\eta_1^2+28t_1^2-3\eta_1^2+8t_1^2\eta_1^2) \\ + &- \frac{x^7}{7! \cos \phi N_1^7}(61+662t_1^2+1320t_1^4+720t_1^6) Further reading ############### From 7aee40c1b7f714a7b25c6de51ab77bffebd8f447 Mon Sep 17 00:00:00 2001 From: julien2512 Date: Wed, 26 Oct 2016 23:03:24 +0200 Subject: [PATCH 3/3] @kbevers remarks on tmerc projection doc --- docs/source/projections/tmerc.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/source/projections/tmerc.rst b/docs/source/projections/tmerc.rst index ff0c8a788b..b3d540fa86 100644 --- a/docs/source/projections/tmerc.rst +++ b/docs/source/projections/tmerc.rst @@ -1,7 +1,7 @@ .. _tmerc: ******************************************************************************** -Transverse Mercator (Gauss-Kruger) +Transverse Mercator ******************************************************************************** The transverse Mercator projection in its various forms is the most widely used projected coordinate system for world topographical and offshore mapping. @@ -66,7 +66,7 @@ Example using Gauss Boaga on Italy area (EPSG:3004) :: Mathematical definition ####################### -The formulas describing the Transverse Mercator are all taken from proj4 sources. +The formulas describing the Transverse Mercator are all taken from Evenden's formula. :math:`\phi_0` is the latitude of origin that match the center of the map. It can be set with ``+lat_0``.