2. Coordinate Transformation
A coordinate transformation is used to change point positions from one coordinate system to another. Figure H5 illustrates the same two sets of points from two different coordinate systems:
Figure H5 Two Points Known in Both Systems 
The To system is final coordinate system. Points A_{T} and B_{T} are points correctly located in the To system. Points A_{F} and B_{F} are the same points but in a different coordinate system (From system). When plotted in the To system using their From coordinates they appear in a different location. Points C_{F} and D_{F} are also in the From system; we want their coordinates in the To system.
A transformation consists of three elements
 Rotation
 Scaling
 Translation
a. Rotation
The From coordinates must be rotated to make directions coincide with those of the To system. To determine the rotation angle we need the direction of a common line in both systems.
Figure H6 Direction of Common Line in Both Systems 
The difference in directions is the rotation angle, ρ, Figure H7.
Figure H7 Rotation Angle 
Rotation is computed from the direction difference of the same line in both systems. If using two control points, inversing between them in both systems will provide the requisite directions. From the directions the rotation is computed from:
Equation H1 
The rotation would be applied to the From coordinates creating a new set, subscripted 1 in Figure H8:
Figure H8 Rotate From System 
b. Scaling
Scaling is used to increase or decrease distances between the points in order to make them fit in the new system. The distances may be off due to:

 random errors
 systematic errors
 ground to grid distortion (in larger regional systems)
 combinations of the above.
To scale the transformation, we need the length of the same line in both systems, Figure H9. The length can be from direct measurement (and subsequent adjustment) or by inversing between the coordinates of two common points.
Figure H9 Length of Common Line in Both Systems 
Scale is the ratio between the two lengths of the same line in both systems.
Figure H10 Scale 
Scale is computed from:
Equation H2 
Scaling isn't always necessary depending on the circumstances and allowable error. For example, if a maximum error of 1/5000 is acceptable, then a scale in the range of 1±1/5000 (0.9998 to 1.0002) can be ignored.
When accounting for scale there are three approaches:
1. Unitary  Maintains a 1:1 relationship between the two coordinate systems.
2. Uniform  Enforces a uniform scale (other than 1) in all directions. Only a single scale is computed for the entire transformation.
Both Unitary and Uniform approaches are referred to as a Conformal Transformation because true shapes are maintained.
3. Differential  Scale in the N/S (Y) direction differs from scale in the E/W (X) direction. Two different scales (S_{N} and S_{E}) are computed for the transformation. This is referred to as an Affine Transformation.
Once scale is applied another set of intermediate coordinates, subscripted 2, are created, Figure H11:
Figure H11 Scaled From System 
c. Translation
The final element are two translations: shift positions in the N/S (Y) and E/W (X) directions. These are determined using coordinates from the previous step with the final coordinates of those points.
Figure H12 Common Point Displacement Between Systems 
These two final elements complete the transformation:
Figure H13 Translated From system 
Translations are computed from a control point known in both systems (From: N_{F} & E_{F}; To: N_{T} & E_{T}) along with the rotation angle and scale previously determined.
Equation H3 
A second control point, if available, can be used for a math check for the translation parameters.
d. Number of parameters
A coordinate transformation is sometimes referred to as either a three, four or fiveparameter transformation, Table H1. The difference between the number of parameters is how scale is applied. Either way, a certain amount of data (control) is needed in both systems.
Table H1 

Transformation type 

Threeparameter 
Fourparameter 
Fiveparameter 

Parameters 
T_{N} 
TN TE ρ 
TN 

Control needed 
One point's coordinates, 
One point's coordinates, 
One point's coordinates, 


or 
or 
or 


two points 
two points 
two points and 




or 




three points 
These are generally the minimal amounts of control needed to uniquely determine the transformation parameters. Any error in the control will be undiscovered and become part of the parameters affecting all points transformed.
Additional control data can be used for math checks. Due to the presence of random errors, different combinations of control will result in slightly different transformation parameters. To get a better model which propagates the error into the final positions, all control should be incorporated in a least squares transformation. Creating such a model manually is time consuming so we will not address least squares transformations here. Many surveying software packages usually include a least squares transformation as an option.
We will examine the fourparameter model as it works well for most groundbased survey applications. Including a uniform scale absorbs smaller random errors satisfactorily as long as good control is used.
e. Transformation Equations
Once the parameters are determined, they are assembled to create the transformation equations which are used to move other points to the To system.
Equation H4  
