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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 H-14 must be rotated through the declination angle to make the True North meridian coincide with the Magnetic North meridian.

Figure H-14
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 H-15. 

Figure H-15
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 H-15 we could subtract 384,000.00 from the North coordinates and 2,307,000.00 from the East coordinates. The result would be Figure H-16. 

Figure H-16
Constants Subtracted From North and East Coordinates


 This effectively creates a local origin: 

Figure H-17
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 H-18.

Figure H-18
Data Collected Reference to Assumed Control


Using this information, they are able to translate and rotate the topo data to its correct location, Figure H-19. 

Figure H-19
Data Transformed Along with Control


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