2. Transforming coordinates
A coordinate transformation changes point positions from one coordinate system to another. It does not require the original measurements used to create the coordinates. Figure J5 illustrates a situation similar to combining Jones' and Davis' surveys: the same two physical points (A abd B) have two different mathematical positions.
Figure J5 
The two coordinate systems are To and From. The To system is final coordinate system in which we'd like all positions expressed. Points A_{T} and B_{T} are the points correctly located in the To system. Points A_{F} and B_{F} are the same points but in a the From system. Points C_{F} and D_{F} are also in the From system and we would like to know their coordinates in the To system.
A transformation consists of three elements
 Scaling
 Rotation
 Translation
We'll discuss the elements in the order shown. To differentiate between the systems we'll used X, Y coordinates for the From system and N, E for the To.
3. Transformation elements
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
 unit conversion (eg, meters to feet)
 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 J16. The length can be from direct measurement (and subsequent adjustment) or by inversing between the coordinates of two common points.
Figure J6 
The scale is a ratio between the To and From system line lengths, Equation J1.
Equation J1  
L_{T }L_{F} 
To length 

Scaling isn't always necessary depending on the circumstances and allowable error. For example, if a maximum error of 1/20,000 is acceptable, then a scale in the range of 1±1/20,000 (0.99995 to 1.00005) can be treated as 1.00000 and this step skipped.
When accounting for scale there are three approaches:
 Unitary  Maintains a 1:1 relationship between the two coordinate systems.
 Uniform  Enforces the same consistent 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.  Differential  Scale in the NorthSouth direction differs from scale in the EastWest direction. Two different scales (S_{N} and S_{E}) are computed for the transformation. This is referred to as an Affine Transformation.
Scale is radially applied from the To system's origin. Once applied another set of updated From coordinates, subscripted 1, are created, Figure J7.
Figure J7 
b. Rotation
The From coordinates must be rotated to make its directions coincide with those of the To system. To determine the rotation angle we need the direction of a common line in both systems. In Figure J8, we need to make the direction of line A_{F}B_{F} the same as the direction of line A_{T}B_{T}.
Figure J8 
Beacuse a line doesn't physically rotate, we need to make the North meridian of the From system parallel with To's. For example, line JK has an assumed azimuth of 270°00'00"; the same line has a true azimuth pf 245°00'00". The line is the same in both systems, only the directions of North differ, Figure J9.
Figure J9 
The rotation angle, ρ, is the amount the Assumed North must be rotated to coincide with the True North. In this case, it must be rotated clockwise (to the right) by 270°00'00"245°00'00" = +25°00'00".
If the Assumed azimuth is 215°00'00" and True azimuth is 245°00'00", Figure J10, then Assumed North must be rotated counterclockwise (to the left) which means it must be a negative angle: 215°00'00"245°00'00" = 30°00'00".
Figure J10 
The rotation angle, ρ, is the From direction minus the To direction, Equation J2.
Equation J2  
Dir_{T} 
To direction  
Dir_{F} 
From direction 
If using two control points, inversing between them in both systems will provide the requisite directions.
The rotation point is the origin of the To coordinate system. The effect of rotating the From meridian creates another updated From coordinate set, subscripted 2 in Figure J11.
Figure J11 
c. Translation
The final element consists of two translations: shift positions in the NorthSouth and EastWest directions. These are the differences at one point between its To and updated From coordinates, Figure J12.
Figure J12 
Translations are computed from a control point known in both systems. Because the rotation and scale have already been applied, they must be incorporated to determine the translations. The translations are Equations J3 and J4 in terms of the original From system.
Equation J3 

Equation J4 
These two final elements complete the transformation, Figure J13.
Figure J13 
e. Transformation Equations
Once the parameters are determined, they are assembled to create the transformation equations, J5 and J6, which are used to move other points to the To system, Figure J14.


Equation J6  

Figure J14 
f. CAD equivalent
Consider how you would solve this problem in Figure J3 using CAD. You would use CAD Move, Rotate, and Scale operations, Figure J15, to put Lot 4 in Lot 3's system. What you've done is apply a coordinate transformation graphically.
Figure J15 
g. Number of parameters
A coordinate transformation is sometimes referred to as either a three, four or fiveparameter transformation. 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. Each parameter is an unknown which must be solved. This will dictate the minimum amount and type of control that is needed.
Table J1  
Transformation type  Three parameter (Conformal) 
Four Parameter (Conformal) 
FiveParameter (Affine) 
Parameters  T_{N} T_{E} ρ 
T_{N} T_{E} ρ S 
T_{N} T_{E} ρ S_{N} S_{E } 
Scale  Scale = 1  Single scale in all directions  N/S and E/W scales can differ 
Control needed in both systems  One point's coordinates, one line's directions 
One point's coordinates, one line's direction, one line's distance 
One point's coordinates, one line's directions, two line's distances 
or  or  or  
two points  two points  two points & one line's directions  
or  
three points 
These are generally the minimal amounts of control needed to uniquely determine the transformation parameters. For example, a fourparameter transformation has four unknowns. Two control points contribute four knowns: two coordinate pairs. Having only a single control point won't allow you to compute the transformation parameters. At best, you can translate the From coordinates, but there isn't sufficient information to determine a rotation or scale.
Using just the minimum control means errors 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 an optional least squares transformation.
We will concentrate on 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.
Note: Since this material was originally posted, I've written TransPack, a Windows program which computes Conformal, Affine, and Projective transformation parameters using least squares and transforms point files. The software is described and available for download at Support Files  Coordinate Transformations.