1. A typical scenario
Surveyor Jones traversing Lot 3 in a subdivision assumes the coordinates of one corner and the direction of a line:
Figure H1 Lot 3 Survey 
Later, Surveyor Davis is hired to map Lot 4 next door to the east. He assumes his own coordinates and line direction:
Figure H2 Lot 4 Survey 
Both surveyors have good closure and compute coordinates for their lot corners.
Today, you are hired to do a survey in the vicinity of the two lots. Realizing that Jones' line CD and Davis' line TU are the same line, you decide to plot both their surveys in CAD. The result? Figure H3.
Figure H3 Lots 3 and 4 Combined 
Something doesn't look right  the two lots are supposed to be next to each other.
What happened?
The two surveys are in separate coordinate systems. Since each surveyor assumed his own starting coordinate and direction independent of each other, they have different coordinate origins and meridian direction. To resolve this problem we need to move Lot 3 into Lot 4's system or Lot 4 into Lot 3's system, Figure H4 (we also have the option of moving both to a third system).
Figure H4 
There are a few ways to do this, including a coordinate transformation. Regardless of the method, we must know something about how the systems are related to each other.
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  

3. Variations
While a full coordinate transformation includes rotation, scaling, and translation, there are situations where only one or two of the elements are necessary.
a. Rotation only
An example where only a rotation is needed is converting magnetic to true directions. The traverse in Figure H14 must be rotated through the declination angle to make the True North meridian coincide with the Magnetic North meridian.
Figure H14 Rotation Without Scale or Translation 
A single traverse point can be used as a pivot; there is no scaling or translation.
b. Translation only
When dealing with regional coordinate systems, the corners of a small survey may have large coordinates as shown in Figure H15.
Figure H15 Large Coordinates in Formal System 
To work with smaller coordinates, the surveyor may subtract one constant from all the North coordinates and another constant from the East coordinates. For traverse in Figure H15 we could subtract 384,000.00 from the North coordinates and 2,307,000.00 from the East coordinates. The result would be Figure H16.
Figure H16 Constants Subtracted From North and East Coordinates 
This effectively creates a local origin:
Figure H17 Local Origin 
Coordinate differences are still the same since each coordinate pair has been changed the same amount. Inverse and area computations are similarly unaffected outside the fact that the magnitude of the computations are somewhat simplified.
c. Translation and rotation
A typical field situation could be collecting data referenced to a base line without first knowing the base line's location. The data is relative to the base line so later fixing the base line fixes the data.
A field crew sets up on one control station and uses another as a backsight for a topo survey. Not knowing coordinates of either point they assume the coordinates of A, their total station (TSI) location, and direction of the backsight line. From there they collect and reduce their topo data.
Later in the office, they are able to obtain point A's coordinates and the correct direction to point B, Figure H18.
Figure H18 Data Collected Reference to Assumed Control 
Using this information, they are able to translate and rotate the topo data to its correct location, Figure H19.
Figure H19 Data Transformed Along with Control 
4. Example
A crew surveyed the exterior of the parcel shown. They assumed the coordinates of the southernmost corner (1001) and used the record bearing for the southwesterly line (10011002).


Figure H20 Parcel Survey 
Later, a second survey crew was sent out to locate the existing drainfield. They did not have the first crew's coordinates so they occupied the northernmost corner (NE), for which they assumed coordinates, and backsighted the westernmost corner (NW), for which they assumed a direction. They then shot the drainfield corners. As a check, they also shot the distance from NE to NW.


Figure H21 Drainfield Survey 
When the coordinates were brought together and plotted the results were as shwon in Figure H22:
Figure H22 Parcel and Drainfield Combined 
Using the boundary survey as the reference system, translate the drainfield data.
Determine transformation parameters
Step (1) Rotation angle
Line NENW coincides with line 10031002; line NENW must be rotated into line 10031002.
Convert the bearings to azimuths and compute the rotation angle.
Line NENW  
Line 10001002 
From Equation H1:
Step (2) Scale
Length of line NENW is 476.41', line 10031002 is 476.37'.
From Equation H2:
This represents a distortion of 1/(10.999916) = 1/11,900. If this is within tolerance, then the scale factor can be ignored. We will apply it in this example.
Step (3) Translations
Use the coordinates of points 1003 and NE in Equation H3:
Step (4) Math check
For a parameters math check, convert NW to see if it matches the coordinates of 1002.
In the From system, NW's coordinates are:
Transform and check:
Step (5) Transform points 501504
Use Equation H14 with the parameters computed in Steps (1)(3)
Point 501
Point 502
Point 503
Point 504
5. TwoDimensions vs ThreeDimensions
While the discussion here is limited to horizontal (two dimensional) coordinate systems, it is relatively easy to extend this into the third dimension by incorporating elevations or Z coordinates. An additional translation parameter would be needed for the elevation component, T_{Elev}.
The rotation angle, ρ, for a two dimensional transformation is a horizontal angle rotated about the vertical, or Z, axis. In a three dimensional transformation there would three rotation angles, one about each axis (XYZ, or ENElev).
Figure H23 Three Dimensional Transformation 
Scale could again be Unitary, Uniform, or Differential. In the latter case, there would be three scale factors.
The total number of parameters in a three dimensional transformation could be:
 Six: Three translations, three rotations, 1:1 scale
 Seven: Three translations, three rotations, uniform scale
 Eight: Three translations, three rotations, one scale for horizontal, one scale for vertical
 Nine: Three translations, three rotations, three scales
The eightparameter transformation would be used where there is a quality difference between horizontal and vertical positions in one system or the other.
6. Spreadsheet
An Excel spreadsheet to perfom basic twodimensional coordinate transformations can be downloaded here. It uses Visual Basic for Applications script so Excel must have macros enabled when loading the sheet.
7. Summary
A coordinate transformation gives a the surveyor the ability to move position information from one coordinate reference system to another. The amount of error introduced into the new positions is a function of the mathematical model used to perform the transformation and the control quality. Examples, as shown in this section, which use minimal control to determine unique model parameters have the effect of spreading a mistake or systematic error through all the data points. Additional control provides some math checking although it will not be perfect due to the presence of random errors. The best way to determine parameters for a coordinate transformation is by least squares which not only better models those random errors, but propagates them into the new positions.