EPSG guidance note #7-2, http://www.epsg.org
Defined for two-dimensional coordinate systems.
Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided. The similarity transformation in algebraic form is: XT = XT0 + XS * M * cos q + YS * M * sin q YT = YT0 – XS * M * sin q + YS * M * cos q where: XT0 , YT0 = the coordinates of the origin point of the source coordinate reference system expressed in the target coordinate reference system; M = the length of one unit in the source coordinate reference system expressed in units of the target coordinate reference system; q = the angle about which the axes of the source coordinate reference system need to be rotated to coincide with the axes of the target coordinate reference system, counter-clockwise being positive. Alternatively, the bearing of the source coordinate reference system Y-axis measured relative to target coordinate reference system north. The similarity transformation can also be described as a special case of the parametric affine transformation where coefficients A1 = B2 and A2 = - B1. Reversibility The reverse formula for the Similarity Transformation is: XS = [(XT – XTO) * cos q – (YT – YTO) * sin q ] / [M ] YS = [(XT – XTO) * sin q + (YT – YTO) * cos q] / [M ] An alternative approach for the reverse operation is to use the same formula as for the forward computation but with different parameter values: XT = XTO' + XS * M' * cos theta' + YS * M' * sin theta' YT = YTO' – XS * M' * sin theta' + YS * M' * cos theta' The reverse parameter values, indicated by a prime ('), can be calculated from those of the forward operation as follows: XTO' = (YTO sin theta – XTO cos theta) / M YTO' = –(YTO cos theta + XTO sin theta) / M M' = 1/M theta' = –theta
ED50 / UTM zone 31N to ETRS89 / UTM zone 31N Parameters of the Similarity Transformation: XTO = -129.549 metres YTO = -208.185 metres M = 1.00000155 theta = 1.56504" = 0.000007588 rad Forward computation for source coordinates 300000m E, 4500000m N: E(ETRS89) = –129.549 + 300000.465 + 34.144 = 299905.060 m E N(ETRS89) = –208.185 –2.276 + 4500006.977 = 4499796.515m N Reverse computation of ETRS89 / UTM 31N coordinates 299905.060m E, 4499796.515m N: E(ED50) = (300034.609 – 34.144) / 1.00000155 = 300000.000m E N(ED50) = (2.276 + 4500004.700) / 1.00000155 = 4500000.000m N Alternative reverse computation: First calculate new parameter values: XTO' = 129.5472 m YTO' = 208.1857 m M' = 0.99999845 theta' = –0.000007588 rad Then apply these values to forward formula: E(ED50) = 129.547 + 299904.595 + (–34.142) = 300000.000 m N(ED50) = 208.186 – (–2.276) + 4499789.539 = 4500000.000 m