Coordinate transformations are quite common in surveying. They range from simple translations between coordinates and setting-out grids on a construction site, to transformation between global systems. Whilst the mathematical procedures are well defined for many types of transformation, problems can arise due to varying scale throughout the network used to establish position. Thus in a local system, there may be a variety of parameters, established empirically, to be used in different areas of the system.
From Figure it can be seen that the basic parameters in a conventional transformation between similar XYZ systems would be:
(1) Translation of the origin 0, which would involve shifts in X, Y and Z, i.e. X, Y, Z.
(2) Rotation about the three axes, θx, θy and θz, in order to render the axes of the systems involved parallel. θx and θy would change the polar axes, and θz the zero meridian.
(3) One scale parameter (1 S) would account for the difference of scale between different coordinate systems.
In addition to the above, the size (a) of the ellipsoid and its shape (f ) may also need to be included. However, not all the parameters are generally used in practice. The most common transformation is the translation in X, Y and Z only (three parameters). Also common is the four-parameter.
(X, Y, Z scale) and the five-parameter (X, Y, Z scale θz). A full transformation would entail seven parameters.
A simple illustration of the process can be made by considering the transformation of the coordinates of P(X, Y, Z) to (X, Y, Z) due to rotation θx about axis OX .
where Rθ = rotational matrix for angle θ
x = the vector of original coordinates
Similar rotation matrices can be produced for rotations about axes OY(α) and OZ(β), giving
x = RθRαRβx
If a scale change and translation of the origin by X, Y, Z is made, the coordinates of P would be
The a coefficients of the rotation matrix would involve the sines and cosines of the angles of rotation, obtained from the matrix multiplication of Rθ , Rα and Rβ.
For the small angles of rotation the sines of the angles may be taken as their radian measure (sin θ = θ) and the cosines are unity, with sufficient accuracy. In which case the equation simplifies to
Equation is referred to in surveying as the Helmert transformation and describes the full transformation between the two geodetic datums. Whilst the X, Y, Z coordinates of three points would be sufficient to determine the seven parameters, in practice as many points as possible are used in a least squares solution. Ellipsoidal coordinates (φ, λ, h) would need to be transformed to X, Y and Z for use in the transformations. As a translation of the origin of the reference system is the most common, a Molodenskii transform permits the transformation of ellipsoidal coordinates from one system to another in a single operation, i.e.
In the above formulae:
, h = ellipsoidal coordinates in the first system
φ, λ, h = ellipsoidal coordinates in the required system
a, f = ellipsoidal parameters of the first system
a,f = difference between the parameters in each system
X,Y,Z = origin translation values
v = radius of curvature in the prime vertical
ρ = radius of curvature in the meridian