EPSG Dataset :: v12.003

Complex polynomial of degree 4

Coordinate Operation Method Details [VALID]

Name:

Complex polynomial of degree 4

Code:

9653

Operation is Reversible:

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 relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the ‘A’ and ‘B’ coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.

mT.(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3 + (A7 + i.A8).(U + i.V)^4

where U = mS.(XS - XS0)
V = mS.(YS - YS0)
and mS, mT are the scaling factors for the coordinate differences in the source and target coordinate reference systems.

The polynomial to degree 4 can alternatively be expressed in matrix form.

Then
XT = XS - XS0 + XT0 + dX
YT = YS - YS0 + YT0 + dY

where
XT , YT are coordinates in the target coordinate reference system,
XS , YS are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0 are coordinates of the evaluation point in the target coordinate reference system.

Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared. In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.

The transformation parameter values (the coefficients) are not reversible. For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction.

Example:

For transformation RD / Netherlands New to ED50 / UTM zone 31N,

Eo1 = 155000
No1 = 463000
Eo2 = 663395.607
No2 = 5781194.380
A1 = -51.681
A2 = 3290.525
A3 = 20.172
A4 = 1.133
A5 = 2.075
A6 = 0.251
A7 = 0.075
A8 = -0.012

For source coordinate system E1=200000 N1=500000, then
E2 =707155.557 N2 = 5819663.128.

Method
Parameters:

Parameter Name	Parameter Code	Parameter Description
Ordinate 1 of evaluation point in source CRS	8619	The value of the first ordinate of the evaluation point expressed in the source coordinate reference system.
Ordinate 2 of evaluation point in source CRS	8620	The value of the second ordinate of the evaluation point expressed in the source coordinate reference system.
Ordinate 1 of evaluation point in target CRS	8621	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	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.
Scaling factor for source CRS coord differences	8694	Used in general polynomial transformations to normalise coordinate differences to an acceptable numerical range.
Scaling factor for target CRS coord differences	8695	Used in general polynomial transformations to normalise coordinate differences to an acceptable numerical range.
A1	8624	Coefficient used in affine (general parametric) and polynomial transformations.
A2	8625	Coefficient used in affine (general parametric) and polynomial transformations.
A3	8626	Coefficient used in polynomial transformations.
A4	8627	Coefficient used in polynomial transformations.
A5	8628	Coefficient used in polynomial transformations.
A6	8629	Coefficient used in polynomial transformations.
A7	8630	Coefficient used in polynomial transformations.
A8	8631	Coefficient used in polynomial transformations.

Meta Data

Remarks:

Coordinate pairs treated as complex numbers. This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the general polynomial of degree 4.

Information Source:

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

Data Source:

EPSG

Revision Date:

June 13, 2017

Change ID:

[2017.018]

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.
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