[نور]

مرحبا بكم مرة اخري في موضوع اخر وهو معادلات مهمة في هندسة المساحة
وهي منقولة من الموقع للفائدة
المعادلة الاولي
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 new system) - (Bearing between same 2 points in old system)
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)