Similarity transformation

Definition: OGP XML

<div class="syntax"><pre><?xml version="1.0" encoding="UTF-8"?> <gml:OperationMethod xmlns:epsg="urn:x-ogp:spec:schema-xsd:EPSG:1.0:dataset" xmlns:gml="http://www.opengis.net/gml/3.2" xmlns:xlink="http://www.w3.org/1999/xlink" gml:id="ogp-method-9621"> <gml:metaDataProperty> <epsg:CommonMetaData> <epsg:informationSource>EPSG guidance note #7-2, http://www.epsg.org</epsg:informationSource> <epsg:revisionDate>2010-06-29</epsg:revisionDate> <epsg:changes> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2000.830" /> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2004.670" /> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2010.053" /> </epsg:changes> <epsg:show>true</epsg:show> <epsg:isDeprecated>false</epsg:isDeprecated> </epsg:CommonMetaData> </gml:metaDataProperty> <gml:metaDataProperty> <epsg:CoordinateOperationMethodMetaData> <epsg:isOperationReversible>true</epsg:isOperationReversible> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8621" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8622" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8611" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8614" /> <epsg: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</epsg:example> </epsg:CoordinateOperationMethodMetaData> </gml:metaDataProperty> <gml:identifier codeSpace="OGP">urn:ogc:def:method:EPSG::9621</gml:identifier> <gml:name>Similarity transformation</gml:name> <gml:remarks>Defined for two-dimensional coordinate systems.</gml:remarks> <gml: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</gml:formula> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8621" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8622" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8611" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8614" /> </gml:OperationMethod> </pre></div>

<?xml version="1.0" encoding="UTF-8"?>
 <gml:OperationMethod xmlns:epsg="urn:x-ogp:spec:schema-xsd:EPSG:1.0:dataset" xmlns:gml="http://www.opengis.net/gml/3.2" xmlns:xlink="http://www.w3.org/1999/xlink" gml:id="ogp-method-9621">
   <gml:metaDataProperty>
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         <epsg:revisionDate>2010-06-29</epsg:revisionDate>
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            <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2000.830" />
            <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2004.670" />
            <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2010.053" />
         </epsg:changes>
         <epsg:show>true</epsg:show>
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   <gml:metaDataProperty>
      <epsg:CoordinateOperationMethodMetaData>
         <epsg:isOperationReversible>true</epsg:isOperationReversible>
         <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8621" />
         <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8622" />
         <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8611" />
         <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8614" />
         <epsg: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</epsg:example>
      </epsg:CoordinateOperationMethodMetaData>
   </gml:metaDataProperty>
   <gml:identifier codeSpace="OGP">urn:ogc:def:method:EPSG::9621</gml:identifier>
   <gml:name>Similarity transformation</gml:name>
   <gml:remarks>Defined for two-dimensional coordinate systems.</gml:remarks>
   <gml: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</gml:formula>
   <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8621" />
   <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8622" />
   <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8611" />
   <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8614" />
</gml:OperationMethod>

EPSG:9621

Similarity transformation

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