Affine transformation
Added in version 6.0.0.
The affine transformation applies translation and scaling/rotation terms on the x,y,z coordinates, and translation and scaling on the temporal coordinate.
Alias |
affine |
Domain |
4D |
Input type |
XYZT |
output type |
XYZT |
By default, the parameters are set for an identity transforms. The transformation is reversible unless the determinant of the sji matrix is 0, or tscale is 0
This can be used to implement:
EPSG "Affine parametric transformation" of code 9624 by setting A0, A1, A2, B0, B1, B2 EPSG parameters to respectively xoff, s11, s12, yoff, s21, s22 PROJ parameters.
EPSG "Similarity transformation" of code 9621 by setting:
xoff to \(X_{T0}\)
yoff to \(Y_{T0}\)
s11 to \(M \cos \theta\)
s12 to \(M \sin \theta\)
s21 to \(-M \sin \theta\)
s22 to \(M \cos \theta\)
where:
\(X_{T0}\) is the first ordinate of the origin point of the source CRS expressed in the target CRS.
\(Y_{T0}\) is the second ordinate of the origin point of the source CRS expressed in the target CRS.
\(M\) is the multiplication factor applied to coordinates in the source CRS to obtain the correct scale of the target CRS.
\(\theta\) is the angle about which the axes of the source CRS need to be rotated to coincide with the axes of the target CRS, counter-clockwise being positive
EPSG "P6 I=J+90 seismic bin grid coordinate operation" of code 9666 and "P6 I=J-90 seismic bin grid coordinate operation" of code 1049, which convert from the easting/northing of a map grid to the I/J bin numbers of a seismic bin grid, by setting:
s11 to \(\pm s_I \cos \theta\)
s12 to \(\mp s_I \sin \theta\)
s21 to \(s_J \sin \theta\)
s22 to \(s_J \cos \theta\)
xoff to \(I_0 - (s11 \times E_0 + s12 \times N_0)\)
yoff to \(J_0 - (s21 \times E_0 + s22 \times N_0)\)
where, naming each quantity after the EPSG parameter that carries it:
\(I_0\) and \(J_0\) are "Bin grid origin I" (EPSG:8733) and "Bin grid origin J" (EPSG:8734), the bin grid coordinates of the origin point of the bin grid.
\(E_0\) and \(N_0\) are "Bin grid origin Easting" (EPSG:8735) and "Bin grid origin Northing" (EPSG:8736), the map grid coordinates of that same origin point.
\(SF\) is "Scale factor of bin grid" (EPSG:8737), that is the point scale factor of the map grid at a chosen reference point.
\(M_I\) and \(M_J\) are "Bin width on I-axis" (EPSG:8738) and "Bin width on J-axis" (EPSG:8739).
\(Inc_I\) and \(Inc_J\) are "Bin node increment on I-axis" (EPSG:8741) and "Bin node increment on J-axis" (EPSG:8742).
\(\theta\) is "Map grid bearing of bin grid J-axis" (EPSG:8740), clockwise being positive.
\(s_I = Inc_I / (SF \times M_I)\) and \(s_J = Inc_J / (SF \times M_J)\) are the resulting number of bin grid units per map grid unit, along the I and J axes respectively.
the upper sign of \(\pm\) and \(\mp\) applies to the right-handed method 9666 (I = J+90°) and the lower one to the left-handed method 1049 (I = J-90°).
\(E_0\), \(N_0\), \(M_I\) and \(M_J\) must all be expressed in the same linear unit as the easting and northing fed to the affine operation.
The resulting bin numbers are not rounded, so that the operation remains exactly invertible. EPSG defines bin grid coordinate systems that accept real values as well as ones restricted to integer values, so whether bin numbers are whole values is a property of the bin grid coordinate system, not of these methods.
Examples
The +proj=affine operation applies an affine transformation on coordinate values. Note that it does not perform any unit conversion and will accept coordinates whether they are angular or linear. Users must ensure that the values provided to the affine operation are expressed in the appropriate units for their intended purpose.
The affine operation can be used with the cct utility or as a step inside a PROJ pipeline.
Offsetting a the x-component of a coordinate by 1:
echo "1 2" | cct +proj=affine +xoff=1
Parameters
Optional
- +xoff=<value>
Offset in X. Default value: 0
- +yoff=<value>
Offset in Y. Default value: 0
- +zoff=<value>
Offset in Z. Default value: 0
- +toff=<value>
Offset in T. Default value: 0
- +s11=<value>
Rotation/scaling term. Default value: 1
- +s12=<value>
Rotation/scaling term. Default value: 0
- +s13=<value>
Rotation/scaling term. Default value: 0
- +s21=<value>
Rotation/scaling term. Default value: 0
- +s22=<value>
Rotation/scaling term. Default value: 1
- +s23=<value>
Rotation/scaling term. Default value: 0
- +s31=<value>
Rotation/scaling term. Default value: 0
- +s32=<value>
Rotation/scaling term. Default value: 0
- +s33=<value>
Rotation/scaling term. Default value: 1
- +tscale=<value>
Time scaling term. Default value: 1