Complex polynomial (4th-order)


Data source: EPSG

Information source: EPSG guidance note #7.

Revision date: 2000-03-07

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 regular 4th-order polynomial.


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 third-order polynomial in complex numbers is used in Belgium.  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.

(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 = (XS - XS0).10-5
and     V = (YS - YS0).10-5

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

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.


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.
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