Guidance Note Number 7, part 2

Coordinate Conversions and Transformation including Formulas

Revision history:






June 2013

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Added Lambert Conic Conformal 2SP (Michigan), Colombia Urban and time-dependent Helmert 7-parameter transformation methods and Affine Parametric Transformation example. Amended Tunisia Mining Grid description.


July 2012


Added left-handed bin grid method.


April 2012


Added missing equations for reverse case of American Polyconic. Amended
Transverse Mercator JHS formula to allow for origin at pole. Corrected Modified
Krovak formula and example. Corrected equation for u' in Hotine Oblique


July 2011


Corrected formula for ξO in Transverse Mercator. Corrected coefficient B1 and longitude example in Polynomial transformation for Spain.


November 2010


Corrected formula for t in Mercator variant C reverse case. Changed name of Krovak defining parameter.


October 2010


Augmented Transverse Mercator for wide area. Amended TM(SO) example. Amended Mercator and Oblique Mercator variant names, added Mercator variant C method. corrected Oblique Mercator (Hotine variant B) formulae for azimuth = 90º case. Added Krovak variants. Corrected Polar Stereographic equations for calculation of longitude. Augmented reverse case for Similarity Transformation. References to OGP publication 430 (GIGS test data) added.


November 2009


For geographic/geocentric conversions, corrected equation for j. For Tranverse Mercator (South Oriented), added example. Corrected equation for computation of radius of authalic sphere (optionally referenced by equal area methods developed on a sphere). Augmented description of geocentric methods to clearly discriminate the coordinate domain to which they are applied.


April 2009


Preface revised to be consistent with other parts of GN7. For Lambert Azimuth Equal Area, in example corrected symbol for β'. For Krovak, corrected formulas. For Equidistant Cylidrical (spherical) corrected formula for R; comments on R added to all sperical methods. For Equidistant Cylindrical updated formula to harmoise parameters and symbols with similar methods.


January 2009


For Mercator (1SP), clarified use of φO. For Molodensky-Badekas, augmented example.Added Popular Visualisation Pseudo Mercator method, added formulas and examples for Mercator (Spherical) and formulas for American Polyconic


August 2008


Corrected general polynomial example


July 2008


For Lambert Conic Near Conformal, corrected equations for φO.


April 2008


Longitude wrap-around note clarified. For Oblique Mercator, corrected symbol in formula for longitude. For Krovak, clarified defining parameters. Amended Vertical Offset description and formula. Added geographic/topocentric conversions, geocentric/topocentric conversions, Vertical Perspective, Orthographic, Lambert Cylindrical Equal Area, ellipsoidal development of Equidistant Cylindrical. Removed section on identification of map projection method.


July 2007


Note on longitude wrap-around added prior to preample to map projection formulas, section 1.4. For Laborde, corrected formula for q'. For Albers Equal Area, corrected formulae for α and β'.


March 2007


For Krovak example, corrected axis names.


February 2007


Descriptive text for Oblique Mercator amended; formula for Laborde projection for Madagascar added. Added polar aspect equations for Lambert Azimuthal Equal Area. Corrected example in polynomial transformation for Spain. For Lambert 1SP, corrected equation for r'.


August 2006


Corrected sign of value for G in Modified Azimuthal Equidistant example.


August 2006


Added Hyperbolic Cassini-Soldner. Corrected FE and FN values in example for Modified Azimuthal Equidistant. Added note to Krovak. Amended Abridged Molodensky description, corrected example.


April 2006


Typographic errors corrected. (For oblique stereographic, corrected formula for w. For Lambert azimuthal equal area, changed example. For Albers equal area, corrected formulae for alpha. For modified azimuthal equidistant, corrected formula for c. For Krovak, corrected formula for theta', clarified formulae for tO and lat. For Cassini, in example corrected radian value of longitude of natural origin). References to EPSG updated.


November 2005


Corrected error in formula for t and false grid coordinates of 2SP example in Mercator projection


September 2005


Miscellaneous linear coordinate operations paragraphs re-written to include reversibility and UKOOA P6. Improved formula for r' in Lambert Conic Near-Conformal.


August 2005


Further correction to Lambert Azimuthal Equal Area formulae. Correction to Moldenski-Badekas example.


May 2005

Amended reverse formulas for Lambert Conic Near-Conformal. Corrected Lambert Azimuthal Equal Area formulae. Symbol for l atitude of pseudo standard parallel parameter made consistent. Corrected Affine Orthogonal Geometric transformation reverse example. Added Modified Azimuthal Equidistant projection.


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.


January 2004

Database codes for Polar Stereographic variants corrected. Degree representation conversions withdrawn.


December 2003

Errors in FE and FN values in example for Lambert Azimuthal Equal Area corrected.


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.


May 2003

Font for Greek symbols in Albers section amended.


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.


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 Carrée), TM zoned, Bonne, Molodensky-Badedas methods added. Errors in Transverse Mercator (South Orientated) formula corrected.


June 2001

Lambert Conformal, Mercator and Helmert formulas corrected.


December 2000

Lambert Conformal (Belgium) remarks revised; Oblique Mercator methods consolidated and formulas added. Similarity Transformation reversibility remarks amended.


June 2000

General Polynomial and Affine methods added


December 1999

Stereographic and Tunisia Mining Grid formulas corrected. Krovacmethod added.


July 1999

Lambert ConicNear Conformal and American Polyconic methods added


February 1999

Abridged Molodensky formulas corrected


November 1998

Polynomial for Spain and Tunisia Mining Grid methods added


May 1998

Additionally issued as an EPSG guidance note


December 1993

First release - POSC Epicentre