EPSG Dataset :: v12.003

Similarity transformation

Coordinate Operation Method Details [VALID]

Name:

Similarity transformation

Code:

9621

Operation is Reversible:

Yes

Formula:

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

Example:

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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Ordinate 1 of evaluation point in target CRS	8621	No	The value of the first ordinate of the evaluation point expressed in the target coordinate reference system. In the case of an affine transformation the evaluation point is the origin of the source coordinate reference system.
Ordinate 2 of evaluation point in target CRS	8622	No	The value of the second ordinate of the evaluation point expressed in the target coordinate reference system. In the case of an affine transformation the evaluation point is the origin of the source coordinate reference system.
Scale factor for source CRS axes	1061	No	Multiplication factor applied to coordinates in the source coordinate reference system to obtain the correct scale in the target coordinate reference system.
Rotation angle of source CRS axes	8614	No	Angle (counter-clockwise positive) through which both of the source coordinate reference system axes need to be rotated to coincide with the corresponding target coordinate reference system axes. Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.

Meta Data

Remarks:

Defined for two-dimensional coordinate systems.

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

November 9, 2019

Change ID:

[2000.83] [2004.67] [2010.053] [2019.005]

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.