EPSG guidance note #7.
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 3rd-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 where U = (XS - XS0).10-5 and V = (YS - YS0).10-5 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