The Mathematica Journal
Volume 9, Issue 2


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Updating a Geographic Database
Leendert van Gastel
Harry Uitermark

Rubber-Sheet Adjustment

When adding a new measurement to the database, we discriminate between control points and free points.

Control points are points that already exist in the database; free points are new points to be added to the database. As we see in Figure 2, the points pt3, pt4, pt30, pt31, and pt504 are already in the database and therefore they are control points. The points pt20, pt21, pt25, and pt32 are from the new house and therefore free. Points pt1 and pt2 are the theodolite station points. These points are used in the computation but not stored in the database.

Figure 2. The control points (pt3, pt4, pt30, and pt504) and free points (pt20, pt21, pt25, and pt32) of the database.

The rubber-sheet transformation maps the measured coordinates of the control points exactly onto the coordinates of the control points in the database (this is the prime reason for choosing the rubber-sheet transformation).

There are two main steps in the operation.

  • The first step is to link the coordinate system of the measurement with the database. This is done by a similarity transformation. This transformation can only map the measured coordinates of two control points exactly onto the coordinates in the database. So, there are differences for the other control points.
  • The second step is to eliminate these differences.

The following formula is central to the second step:


  • being the corrections to the coordinates of a free point
  • representing the covariance matrix of a free point and the control points
  • representing the covariance matrix of the control points
  • being the vector of differences between the coordinates of the control points in the measurement (after the similarity transformation) and the database.

This formula expresses the correction for any point as a weighted linear combination of the corrections of the control points. The weights depend on the distance between the point and the control points. We explain below how the matrices and are constructed.

The most important property of this method is that it does not matter which basis you choose for the similarity transformation: after applying the above formula, you get the same result regardless of the basis. This is a form of invariance.

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