Editing Geographic coordinate conversion (section)

=== Helmert transformation ===

{{main|Helmert transformation}}

Use of the Helmert transform in the transformation from geodetic coordinates of datum <math>A</math> to geodetic coordinates of datum <math>B</math> occurs in the context of a three-step process:<ref name=HelmertNZ>{{cite web|title=Equations Used for Datum Transformations|url=http://www.linz.govt.nz/geodetic/conversion-coordinates/geodetic-datum-conversion/datum-transformation-equations/index.aspx|publisher=Land Information New Zealand (LINZ)|access-date=5 March 2014|archive-date=6 March 2014|archive-url=https://web.archive.org/web/20140306005832/http://www.linz.govt.nz/geodetic/conversion-coordinates/geodetic-datum-conversion/datum-transformation-equations/index.aspx|url-status=live}}</ref>

# Convert from geodetic coordinates to ECEF coordinates for datum <math>A</math>
# Apply the Helmert transform, with the appropriate <math>A\to B</math> transform parameters, to transform from datum <math>A</math> ECEF coordinates to datum <math>B</math> ECEF coordinates
# Convert from ECEF coordinates to geodetic coordinates for datum <math>B</math>

In terms of ECEF XYZ vectors, the Helmert transform has the form (position vector transformation convention and very small rotation angles simplification)<ref name=HelmertNZ/>

: <math>
  \begin{bmatrix} X_B \\ Y_B \\ Z_B \end{bmatrix} =
  \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix} + \left(1 + s \times 10^{-6}\right)
  \begin{bmatrix}
      1 & -r_z &  r_y \\
    r_z &    1 & -r_x \\
   -r_y &  r_x &    1
  \end{bmatrix} \begin{bmatrix} X_A \\ Y_A \\ Z_A \end{bmatrix}.
</math>

The Helmert transform is a seven-parameter transform with three translation (shift) parameters <math>c_x,\, c_y,\, c_z</math>, three rotation parameters <math>r_x,\, r_y,\, r_z</math> and one scaling (dilation) parameter <math>s</math>. The Helmert transform is an approximate method that is accurate when the transform parameters are small relative to the magnitudes of the ECEF vectors. Under these conditions, the transform is considered reversible.<ref name=OGP7_2>{{cite web|title=Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas|url=http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf|publisher=International Association of Oil and Gas Producers (OGP)|access-date=5 March 2014|url-status=dead|archive-url=https://web.archive.org/web/20140306005736/http://info.ogp.org.uk/geodesy/guides/docs/G7-2.pdf|archive-date=6 March 2014}}</ref>

A fourteen-parameter Helmert transform, with linear time dependence for each parameter,{{r|OGP7_2|page1=131-133}} can be used to capture the time evolution of geographic coordinates dues to [[geomorphic]] processes, such as continental drift<ref name=Bolstad>{{cite book|last=Bolstad|first=Paul|title=GIS Fundamentals, 4th Edition|year=2012 |publisher=Atlas books|isbn=978-0-9717647-3-6|page=93|url=http://www.paulbolstad.net/4thedition/samplechaps/GISFundChap3.pdf|url-status=dead|archive-url=https://web.archive.org/web/20160202201558/http://www.paulbolstad.net/4thedition/samplechaps/GISFundChap3.pdf|archive-date=2016-02-02}}</ref> and earthquakes.<ref name=addend_8350_2>{{cite web|title=Addendum to NIMA TR 8350.2: Implementation of the World Geodetic System 1984 (WGS 84) Reference Frame G1150|url=http://gis-lab.info/docs/nima-tr8350.2-addendum.pdf|publisher=National Geospatial-Intelligence Agency|access-date=6 March 2014|archive-date=11 May 2012|archive-url=https://web.archive.org/web/20120511090551/http://gis-lab.info/docs/nima-tr8350.2-addendum.pdf|url-status=live}}</ref> This has been incorporated into software, such as the Horizontal Time Dependent Positioning (HTDP) tool from the U.S. NGS.<ref name=HTDP>{{cite web|title=HTDP - Horizontal Time-Dependent Positioning|url=https://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml|publisher=U.S. National Geodetic Survey (NGS)|access-date=5 March 2014|archive-date=25 November 2019|archive-url=https://web.archive.org/web/20191125025630/https://www.ngs.noaa.gov/TOOLS/Htdp/Htdp.shtml|url-status=live}}</ref>