IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – September 2019 To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org. Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas Revised - September 2019 Page 1 of 162 IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – September 2019 To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org. Table of Contents Preface ............................................................................................................................................................5 1 IMPLEMENTATION NOTES ................................................................................................................. 6 1.1 ELLIPSOID PARAMETERS ......................................................................................................................... 6 1.2 ARCTANGENT FUNCTION ......................................................................................................................... 6 1.3 ANGULAR UNITS ....................................................................................................................................... 7 1.4 LONGITUDE 'WRAP-AROUND' .................................................................................................................. 7 1.5 OFFSETS .................................................................................................................................................... 7 1.6 COORDINATE TRANSFORMATION DIRECTION ........................................................................................ 7 2 COORDINATE CONVERSIONS AND TRANSFORMATIONS - INTRODUCTION ..................... 8 2.1 BASIC CONCEPTS ...................................................................................................................................... 8 2.2 MAP PROJECTION METHODS ................................................................................................................... 9 2.3 MAP PROJECTION PARAMETERS .......................................................................................................... 10 3 MAP PROJECTION FORMULAS ........................................................................................................ 17 3.1 CONIC MAP PROJECTIONS .................................................................................................................... 17 3.1.1 LAMBERT CONIC CONFORMAL ............................................................................................................. 17 3.1.1.1 Lambert Conic Conformal (2SP)....................................................................................................... 19 3.1.1.2 Lambert Conic Conformal (1SP)....................................................................................................... 20 3.1.1.3 Lambert Conic Conformal (1SP West Orientated)............................................................................ 21 3.1.1.4 Lambert Conic Conformal (2SP Belgium) ........................................................................................ 22 3.1.1.5 Lambert Conic Conformal (2SP Michigan) ...................................................................................... 23 3.1.1.6 Lambert Conic Near-Conformal........................................................................................................ 24 3.1.2 KROVAK ............................................................................................................................................... 26 3.1.2.1 Krovak ............................................................................................................................................... 27 3.1.2.2 Krovak (North Orientated) ................................................................................................................ 30 3.1.2.3 Krovak Modified ............................................................................................................................... 31 3.1.2.4 Krovak Modified (North Orientated) ................................................................................................ 33 3.1.3 ALBERS EQUAL AREA .......................................................................................................................... 34 3.1.4 BONNE .................................................................................................................................................. 35 3.1.4.1 Bonne (South Orientated) .................................................................................................................. 36 3.1.5 AMERICAN POLYCONIC ........................................................................................................................ 36 3.2 CYLINDRICAL MAP PROJECTIONS ........................................................................................................ 37 3.2.1 MERCATOR ........................................................................................................................................... 37 3.2.1.1 Mercator (Spherical) .......................................................................................................................... 43 3.2.1.2 Popular Visualisation Pseudo-Mercator ("Web Mercator") .............................................................. 44 3.2.2 CASSINI-SOLDNER ................................................................................................................................ 45 3.2.2.1 Hyperbolic Cassini-Soldner .............................................................................................................. 47 3.2.3 TRANSVERSE MERCATOR ..................................................................................................................... 49 3.2.3.1 General Case...................................................................................................................................... 49 3.2.3.2 Transverse Mercator Zoned Grid System .......................................................................................... 56 3.2.3.3 Transverse Mercator (South Orientated) ........................................................................................... 57 3.2.4 OBLIQUE MERCATOR ........................................................................................................................... 59 3.2.4.1 Hotine Oblique Mercator ................................................................................................................... 60 3.2.4.2 Laborde Oblique Mercator ................................................................................................................ 64 Page 2 of 162 IOGP Publication 373-7-2 – Geomatics Guidance Note number 7, part 2 – September 2019 To facilitate improvement, this document is subject to revision. The current version is available at www.epsg.org. 3.2.5 EQUIDISTANT CYLINDRICAL ................................................................................................................ 66 3.2.5.1 Equidistant Cylindrical (Spherical) ................................................................................................... 68 3.2.5.2 Pseudo Plate Carrée ........................................................................................................................... 69 3.2.6 LAMBERT CYLINDRICAL EQUAL AREA ................................................................................................ 69 3.2.6.1 Lambert Cylindrical Equal Area (Spherical) ..................................................................................... 69 3.3 AZIMUTHAL MAP PROJECTIONS........................................................................................................... 70 3.3.1 STEREOGRAPHIC ................................................................................................................................... 70 3.3.1.1 Oblique and Equatorial Stereographic ............................................................................................... 70 3.3.1.2 Polar Stereographic ........................................................................................................................... 73 3.3.2 LAMBERT AZIMUTHAL EQUAL AREA ................................................................................................... 78 3.3.2.1 Lambert Azimuthal Equal Area (Spherical) ...................................................................................... 80 3.3.3 AZIMUTHAL EQUIDISTANT ................................................................................................................... 81 3.3.3.1 Modified Azimuthal Equidistant ....................................................................................................... 81 3.3.3.2 Guam Projection ................................................................................................................................ 82 3.3.4 PERSPECTIVES ...................................................................................................................................... 84 3.3.4.1 Intoduction ........................................................................................................................................ 84 3.3.4.2 Vertical Perspective ........................................................................................................................... 85 3.3.4.3 Vertical Perspective