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OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010
To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org.
Page 1 of 132
Surveying and Positioning
Guidance Note Number 7, part 2
Coordinate Conversions and Transformations including Formulas
Revised - November 2010
OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010
To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org.
Page 2 of 132
Index
Preface 4
Revision history 5
1 MAP PROJECTIONS AND THEIR COORDINATE CONVERSION FORMULAS 9
1.1 INTRODUCTION 9
1.2 MAP PROJECTION PARAMETERS 10
1.3 MAP PROJECTION FORMULAS 16
1.3.1 LAMBERT CONIC CONFORMAL 17
1.3.1.1 Lambert Conic Conformal (2SP) 19
1.3.1.2 Lambert Conic Conformal (1SP) 20
1.3.1.3 Lambert Conic Conformal (1SP West Orientated) 22
1.3.1.4 Lambert Conic Conformal (2SP Belgium) 22
1.3.1.5 Lambert Conic Near-Conformal 23
1.3.2 KROVAK 25
1.3.2.1 Krovak 26
1.3.2.2 Krovak (North Orientated) 29
1.3.2.3 Krovak Modified 30
1.3.2.4 Krovak Modified (North Orientated) 32
1.3.3 MERCATOR 33
1.3.3.1 Mercator (Spherical) 38
1.3.3.2 Popular Visualisation Pseudo Mercator 39
1.3.4 CASSINI-SOLDNER 41
1.3.4.1 Hyperbolic Cassini-Soldner 43
1.3.5 TRANSVERSE MERCATOR 44
1.3.5.1 General Case 44
1.3.5.2 Transverse Mercator Zoned Grid System 52
1.3.5.3 Transverse Mercator (South Orientated) 53
1.3.6 OBLIQUE MERCATOR 54
1.3.6.1 Hotine Oblique Mercator 56
1.3.6.2 Laborde Oblique Mercator 59
1.3.7 STEREOGRAPHIC 62
1.3.7.1 Oblique and Equatorial Stereographic 62
1.3.7.2 Polar Stereographic 65
1.3.8 NEW ZEALAND MAP GRID 70
1.3.9 TUNISIA MINING GRID 71
1.3.10 AMERICAN POLYCONIC 72
1.3.11 LAMBERT AZIMUTHAL EQUAL AREA 73
1.3.11.1 Lambert Azimuthal Equal Area (Spherical) 75
1.3.12 LAMBERT CYLINDRICAL EQUAL AREA 75
1.3.12.1 Lambert Cylindrical Equal Area (Spherical) 75
1.3.13 ALBERS EQUAL AREA 76
1.3.14 EQUIDISTANT CYLINDRICAL 76
1.3.14.1 Equidistant Cylindrical (Spherical) 78
1.3.14.2 Pseudo Plate Carrée 79
1.3.15 BONNE 79
1.3.15.1 Bonne (South Orientated) 80
1.3.16 AZIMUTHAL EQUIDISTANT 80
1.3.16.1 Modified Azimuthal Equidistant 80
1.3.16.2 Guam Projection 82
1.3.17 PERSPECTIVES 83
OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010
To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org.
Page 3 of 132
1.3.17.1
Intoduction 83
1.3.17.2 Vertical Perspective 85
1.3.17.3 Vertical Perspective (orthographic case) 86
1.3.18 ORTHOGRAPHIC 87
2 FORMULAS FOR COORDINATE OPERATIONS OTHER THAN MAP PROJECTIONS 90
2.1 INTRODUCTION 90
2.2 COORDINATE CONVERSIONS OTHER THAN MAP PROJECTIONS 91
2.2.1 GEOGRAPHIC/GEOCENTRIC CONVERSIONS 91
2.2.2 GEOCENTRIC/TOPOCENTRIC CONVERSIONS 92
2.2.3 GEOGRAPHIC/TOPOCENTRIC CONVERSIONS 95
2.2.4 GEOGRAPHIC 3D TO 2D CONVERSIONS 97
2.3 COORDINATE OPERATION METHODS THAT CAN BE CONVERSIONS OR TRANSFORMATIONS 98
2.3.1 POLYNOMIAL TRANSFORMATIONS 98
2.3.1.1 General case 98
2.3.1.2 Polynomial transformation with complex numbers 103
2.3.1.3 Polynomial transformation for Spain 105
2.3.2 MISCELLANEOUS LINEAR COORDINATE OPERATIONS 106
2.3.2.1 Affine Parametric Transformation 106
2.3.2.2 Affine General Geometric Transformation 107
2.3.2.3 Similarity Transformation 110
2.3.2.4 UKOOA P6 Seismic Bin Grid Transformation 113
2.4 COORDINATE TRANSFORMATIONS 117
2.4.1 OFFSETS - GENERAL 117
2.4.1.1 Cartesian Grid Offsets from Form Function 117
2.4.2 TRANSFORMATIONS BETWEEN VERTICAL COORDINATE REFERENCE SYSTEMS 117
2.4.2.1 Vertical Offset 117
2.4.2.2 Vertical Offset by Interpolation of Gridded Data 119
2.4.2.3 Vertical Offset and Slope 119
2.4.3 TRANSFORMATIONS BETWEEN GEOCENTRIC COORDINATE REFERENCE SYSTEMS 120
2.4.3.1 Geocentric Translations (geocentric domain) 121
2.4.3.2 Helmert 7-parameter transformations 121
2.4.3.3 Molodensky-Badekas transformation (geocentric domain) 124
2.4.4 TRANSFORMATIONS BETWEEN GEOGRAPHIC COORDINATE REFERENCE SYSTEMS 125
2.4.4.1 Transformations using geocentric methods 125
2.4.4.2 Abridged Molodensky transformation 128
2.4.4.3 Geographic Offsets 129
2.4.4.4 Geographic Offset by Interpolation of Gridded Data 130
2.4.5 GEOID AND HEIGHT CORRECTION MODELS 131
2.4.5.1 Geographic3D to GravityRelatedHeight 131
2.4.5.2 Geographic3D to Geographic2D+GravityRelatedHeight 131
2.4.5.3 Geographic2D with Height Offsets 132
OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010
To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org.
Page 4 of 132
Preface
The EPSG Geodetic Parameter Dataset, abbreviated to the EPSG Dataset, is a repository of parameters
required to:
• define a coordinate reference system (CRS) which ensures that coordinates describe position
unambiguously.
• define transformations and conversions that allow coordinates to be changed from one CRS to
another CRS. Transformations and conversions are collectively called coordinate operations.
The EPSG Dataset is maintained by the OGP Surveying and Positioning Committee’s Geodetic
Subcommittee. It conforms to ISO 19111 – Spatial referencing by coordinates. It is distributed in three
ways:
• the EPSG Registry, in full the EPSG Geodetic Parameter Registry, a web-based delivery platform
in which the data is held in GML using the CRS entities described in ISO 19136.
• the EPSG Database, in full the EPSG Geodetic Parameter Database, a relational database structure
where the entities which form the components of CRSs and coordinate operations are in separate
tables, distributed as an MS Access database;
• in a relational data model as SQL scripts which enable a user to create an Oracle, MySQL,
PostgreSQL or other relational database and populate that database with the EPSG Dataset;
OGP Surveying and Positioning Guidance Note 7 is a multi-part document for users of the EPSG Dataset.
• Part 0, Quick Start Guide, gives a basic overview of the Dataset and its use.
• Part 1, Using the Dataset, sets out detailed information about the Dataset and its content,
maintenance and terms of use.
• Part 2, Formulas, (this document), provides a detailed explanation of formulas necessary for
executing coordinate conversions and transformations using the coordinate operation methods
supported in the EPSG dataset. Geodetic parameters in the Dataset are consistent with these
formulas.
• Part 3, Registry Developer Guide, is primarily intended to assist computer application developers
who wish to use the API of the Registry to query and retrieve entities and attributes from the dataset.
• Part 4, Database Developer Guide, is primarily intended to assist computer application developers
who wish to use the Database or its relational data model to query and retrieve entities and attributes
from the dataset.
The complete text may be found at http://www.epsg.org/guides/index.html. The terms of use of the dataset
are also available at http://www.epsg.org/CurrentDB.html.
In addition to these documents, the Registry user interface contains online help and the Database user
interface includes context-sensitive help accessed by left-clicking on any label.
This Part 2 of the multipart Guidance Note is primarily intended to assist computer application developers in
using the coordinate operation methods supported by the EPSG Dataset. It may also be useful to other users
of the data.
OGP Publication 373-7-2 – Surveying and Positioning Guidance Note number 7, part 2 – November 2010
To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org.
Page 5 of 132
Further test data to supplement the examples in this document may be found in OGP Publication 430,
Guidelines for Geospatial Integrity of Geoscience Software (GIGS), part 3, Test Dataset.
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