EPSG Dataset :: v12.003

Reversible polynomial of degree 13

Coordinate Operation Method Details [VALID]

Name:

Reversible polynomial of degree 13

Code:

9654

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.

See method code 9648 for description of general polynomial formula.

A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.

Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - In the reverse transformation the roles of the source and target coordinate reference systems are reversed. Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units. That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant. The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary. It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.

An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea. The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).

The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = ( output valu/ input value) which is approximately 10E-6

As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.

Example:

See Reversible polynomial of degree 4, code 9651, for general methodology.

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Ordinate 1 of evaluation point	8617	No	The value of the first ordinate value of the evaluation point.
Ordinate 2 of evaluation point	8618	No	The value of the second ordinate of the evaluation point.
Scaling factor for coord differences	8696	No	Used in reversible polynomial transformations to normalise coordinate differences to an acceptable numerical range. For the reverse transformation the forward target CRS becomes the reverse source CRS and forward source CRS becomes the reverse target CRS.
A0	8623	Yes	Coefficient used in affine (general parametric) and polynomial transformations.
Au1v0	8716	Yes	Coefficient used in polynomial transformations.
Au0v1	8717	Yes	Coefficient used in polynomial transformations.
Au2v0	8718	Yes	Coefficient used in polynomial transformations.
Au1v1	8719	Yes	Coefficient used in polynomial transformations.
Au3v0	8721	Yes	Coefficient used in polynomial transformations.
Au2v1	8722	Yes	Coefficient used in polynomial transformations.
Au4v0	8633	Yes	Coefficient used in polynomial transformations.
Au4v1	8669	Yes	Coefficient used in polynomial transformations.
Au5v2	8697	Yes	Coefficient used in polynomial transformations.
Au0v8	8698	Yes	Coefficient used in polynomial transformations.
Au9v0	8699	Yes	Coefficient used in polynomial transformations.
Au2v7	8700	Yes	Coefficient used in polynomial transformations.
Au1v9	8701	Yes	Coefficient used in polynomial transformations.
Au3v9	8702	Yes	Coefficient used in polynomial transformations.
B0	8639	Yes	Coefficient used in affine (general parametric) and polynomial transformations.
Bu1v0	8724	Yes	Coefficient used in polynomial transformations.
Bu0v1	8725	Yes	Coefficient used in polynomial transformations.
Bu2v0	8726	Yes	Coefficient used in polynomial transformations.
Bu1v1	8643	Yes	Coefficient used in polynomial transformations.
Bu0v2	8644	Yes	Coefficient used in polynomial transformations.
Bu3v0	8645	Yes	Coefficient used in polynomial transformations.
Bu4v0	8649	Yes	Coefficient used in polynomial transformations.
Bu1v3	8652	Yes	Coefficient used in polynomial transformations.
Bu5v0	8681	Yes	Coefficient used in polynomial transformations.
Bu2v3	8684	Yes	Coefficient used in polynomial transformations.
Bu1v4	8685	Yes	Coefficient used in polynomial transformations.
Bu0v5	8686	Yes	Coefficient used in polynomial transformations.
Bu6v0	8687	Yes	Coefficient used in polynomial transformations.
Bu3v3	8690	Yes	Coefficient used in polynomial transformations.
Bu2v4	8691	Yes	Coefficient used in polynomial transformations.
Bu1v5	8692	Yes	Coefficient used in polynomial transformations.
Bu7v0	8703	Yes	Coefficient used in polynomial transformations.
Bu6v1	8704	Yes	Coefficient used in polynomial transformations.
Bu4v4	8705	Yes	Coefficient used in polynomial transformations.
Bu8v1	8706	Yes	Coefficient used in polynomial transformations.
Bu7v2	8707	Yes	Coefficient used in polynomial transformations.
Bu2v7	8708	Yes	Coefficient used in polynomial transformations.
Bu0v9	8709	Yes	Coefficient used in polynomial transformations.
Bu4v6	8710	Yes	Coefficient used in polynomial transformations.
Bu9v2	8711	Yes	Coefficient used in polynomial transformations.
Bu8v3	8712	Yes	Coefficient used in polynomial transformations.
Bu5v7	8713	Yes	Coefficient used in polynomial transformations.
Bu9v4	8714	Yes	Coefficient used in polynomial transformations.
Bu4v9	8715	Yes	Coefficient used in polynomial transformations.

Meta Data

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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