معادلات في هندسة المساحة

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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)

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