Coordinate Conversions and Transformations …

1 EPSG Guidance Note number 7, part 2 October 2004 Page 1 of 85 EPSGE uropean Petroleum Survey Group Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas Revised October 2004 Revision history: Version Date Amendments 1 December 1993 First release POSC Epicentre 10 May 1998 Additionally issued as an EPSG guidance note. 11 November 1998 Polynomial for Spain and Tunisia Mining Grid methods added. 12 February 1999 Abridged Molodensky formulas corrected. 13 July 1999 Lambert Conic Near Conformal and American Polyconic methods added. 14 December 1999 Stereographic and Tunisia Mining Grid formulas corrected. Krovak method added. 15 June 2000 General Polynomial and Affine methods added 16 December 2000 Lambert Conformal (Belgium) remarks revised; Oblique Mercator methods consolidated and formulas added. Similarity transformation reversibility remarks amended.

2 17 June 2001 Lambert Conformal, Mercator and Helmert formulas corrected. 18 August 2002 Revised to include ISO 19111 terminology. Section numbering revised. Added Preface. Lambert Conformal (West Orientated), Lambert Azimuthal Equal Area, Albers, Equidistant Cylindrical (Plate Carr e), TM zoned, Bonne, Molodensky-Badedas methods added. Errors in Transverse Mercator (South Orientated) formula corrected. 19 December 2002 Polynomial formulas amended. Formula for spherical radius in Equidistant Cylindrical projection amended. Formula for Krovak projection amended. Degree representation Conversions added. Editorial amendments made to subscripts and superscripts. 20 May 2003 Font for Greek symbols in Albers section amended. 21 October 2003 Typographic errors in example for Lambert Conic (Belgium) corrected. Polar Stereographic formulae extended for secant variants. General polynomial extended to degree 13. Added Abridged Molodensky and Lambert Azimuthal Equal Area examples and Reversible polynomial formulae.

3 22 December 2003 Errors in FE and FN values in example for Lambert Azimuthal Equal Area corrected. 23 January 2004 Database codes for Polar Stereographic variants corrected. Degree representation Conversions withdrawn. 24 October 2004 From this revision, published as part 2 of a two-part set. Corrected equation for u in Oblique Mercsator. Added Guam projection, Geographic 3D to 2D conversion , vertical offset and gradient method, geoid models, bilinear interpolation methods. Added tables giving projection parameter definitions. Amended Molodensky-Badekas method name and added example. Added section on reversibility to Helmert 7-parameter Transformations . transformation section 2 reordered. Section 3 (concatenated operations) added. Problem in reverse formulas for Lambert Conic Near-Conformal to be resolved. EPSG Guidance Note number 7, part 2 October 2004 Page 2 of 85 Index to part 2 PREFACE Coordinate Conversions and Coordinate Transformations 1 MAP PROJECTIONS AND THEIR Coordinate conversion FORMULAS 5 INTRODUCTION 5 IDENTIFICATION OF MAP PROJECTION METHOD 6 MAP PROJECTION PARAMETERS 7 MAP PROJECTION FORMULAS 18 LAMBERT CONIC CONFORMAL

4 18 Lambert Conic Conformal (2SP) 19 Lambert Conic Conformal (1SP) 20 Lambert Conic Conformal (West Orientated) 21 Lambert Conic Conformal (2 SP Belgium) 22 Lambert Conic Near-Conformal 23 KROVAK OBLIQUE CONFORMAL CONIC 25 MERCATOR 27 CASSINI-SOLDNER 29 TRANSVERSE MERCATOR 31 General Case 31 Transverse Mercator Zoned Grid System 34 Transverse Mercator

5 (South Orientated) 35 OBLIQUE MERCATOR AND HOTINE OBLIQUE MERCATOR 35 STEREOGRAPHIC 40 Oblique and Equatorial Stereographic cases 40 Polar Stereographic 43 NEW ZEALAND MAP GRID 48 TUNISIA MINING GRID 49

6 AMERICAN POLYCONIC 50 LAMBERT AZIMUTHAL EQUAL AREA 50 Lambert Azimuthal Equal Area (Spherical) 51 ALBERS EQUAL AREA 51 EQUIDISTANT CYLINDRICAL 52 Pseudo Plate Carr e 53 BONNE 53 Bonne (South Orientated)

7 54 AZIMUTHAL EQUIDISTANT 54 Guam Projection 54 2 FORMULAS FOR Coordinate OPERATIONS OTHER THAN MAP PROJECTIONS 56 INTRODUCTION 56 Coordinate Conversions OTHER THAN MAP PROJECTIONS 56 GEOGRAPHIC/GEOCENTRIC Conversions 56 GEOGRAPHIC 3D TO 2D Conversions 58 Coordinate OPERATION METHODS THAT CAN BE Conversions OR Transformations 59 POLYNOMIAL Transformations 59 General case 59 Polynomial transformation with complex numbers 64 Polynomial transformation for Spain 66 MISCELLANEOUS LINEAR Coordinate OPERATIONS 67 Affine Parametric transformation 67 EPSG

8 Guidance Note number 7, part 2 October 2004 Page 3 of 85 Affine General Geometric transformation 68 Affine Orthogonal Geometric transformation 70 Similarity transformation 71 Coordinate Transformations 74 OFFSETS - GENERAL 74 Transformations BETWEEN VERTICAL Coordinate REFERENCE SYSTEMS 75 Vertical Offset 75 Vertical Offset by Interpolation of Gridded Data 75 Vertical Offset and Slope 75

9 Transformations BETWEEN GEOCENTRIC Coordinate REFERENCE SYSTEMS 76 Geocentric Translations 77 Helmert 7-parameter Transformations 77 Molodensky-Badekas 10-parameter transformation 80 Transformations BETWEEN GEOGRAPHIC Coordinate REFERENCE SYSTEMS 81 Transformations using geocentric methods 81 Abridged Molodensky transformation 83 Geographic Offsets 84 Geographic Offset by Interpolation of Gridded Data 85 GEOID MODELS 85 Geographic3D to GravityRelatedHeight 85 Geographic3D to Geographic2D+GravityRelatedHeight 86 Geographic2D with Height Offsets 86 EPSG Guidance Note number 7, part 2 October 2004 Page 4 of 85 Preface This document, EPSG Guidance note 7 part 2, provides a detailed explanation of formulas necessary for executing Coordinate Conversions and Transformations .

10 Most of these Conversions and Transformations require paramters as arguments. The values for these parameters may be obtained from the EPSG geodetic parameter dataset. EPSG Guidance note 7 part 1 provides a conceptual overview of Coordinate reference systems and Coordinate operations, details of how these are dealt with in the EPSG dataset, details of how the EPSG dataset is managed, and guidance on how to extract data from the dataset and on how to use it. A Coordinate system is a set of mathematical rules for specifying how coordinates are to be assigned to points. It includes the definition of the Coordinate axes, the units to be used and the geometry of the axes. The Coordinate system is unrelated to the Earth. A Coordinate reference system (CRS) is a Coordinate system related to the Earth through a datum. Colloquially the term Coordinate system has historically been used to mean Coordinate reference system.