In our work, it may sometimes be necessary to transform a set of co-ordinates from one cartesian system to another. The following formulae may be used to transform a set of (e, n) co-ordinates into a set of (e', n') co-ordinates.
A simple scale change, for example changing feet to metres or applying a meteorological scale factor, may be applied thus:
e' = k e
n' = k n
where e, n = original (old) co-ordinates: k = scale factor: e', n' = new co-ordinates
For a rotation of axis about an angle θ, which may be given or derived from known co-ordinates in both systems:
e' = e cos θ - n sin θ
n' = e sin θ + n cos θ
where e',n' = new co-ordinates: e, n = original co-ordinates: θ = angle of rotation
For a change of origin by factors E and N:
e' = e + E
n' = n + N
where e',n' = new co-ordinates: e, n = original co-ordinates: E & N = shift factors
Scale, Rotation and Translation
If the transformation parameters are known
(i) e' = k (e cos θ) - k (n sin θ) + E
(ii) n' = k (e sin θ) + k (n cos θ) + N
These formulae work for all cases.
If no scale factor is required, substitute k = 1.
If no rotation is needed then substitute θ = 0.
Similarly, if no Translations are required E & N = 0 as required.
If the transformation parameters are NOT known
In this case, two points in each system must be known (preferably as far apart as possible).
The following parameters may be calculated:
k = (Distance between 2 points in
new system) / (Distance between 2 points in old system)
θ = (Bearing between 2 points in
old system) - (Bearing between same 2 points in new system)
Note the "sign" of
this value. This is used in the following
If (e, n) = 1 point in old co-ordinate system and (e', n') = same point in new system:
E = e' - k
(e cos θ) + k (n sin θ)
N = n' - k
(e sin θ) - k (n cos θ)
points may now be transformed by applying these
parameters into the above formulae (i) and (ii).