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"Transformations"

 

Introduction

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.

Scale

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

Rotation

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

Translation

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:

Scale Factor

k = (Distance between 2 points in new system) / (Distance between 2 points in old system)

Rotation Angle

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 formulae...

Translation

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)

Further points may now be transformed by applying these parameters into the above formulae (i) and (ii).

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