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H. Von Webern, H. Kahmen:
"Gridline methods for detection of objects desplacements and rigid body movements";
Journal of Applied Geodesy, 2 (2008), 2; 111 - 123.

If measurements are performed without artificial targets, deformation analysis cannot refer to the points on the objects defined in an initial or prior measurement epoch. This paper describes how an arrangement of virtual lines, so called grid lines (GL), offers a solution. A total station, measuring distances without a reflector, or a theodolite measurement system can be used to make the mneasurements.. The GLs should intersect the object to be measured, as far as possible, at right angles and they can be arranged in such a way that each of them runs through one of the object points chosen for the initial measurements. Subsequently deformations of the object can be described along the GLs.

The general idea behind this method is to track the virtual lines with the optical axis of a theodolite, until it aims at the points where the GLs intersect the surface. Tracking starts where the GLs intersected the surface of the object during the last measurement epoch. However, if the surface moved this point will no longer be an intersection point and the optical axis hits the surface at another point. Consequently there will be a separation of these two points, and if the theodolite starts tracking the GL, a distance function is obtained. This can be used to determine the new intersection point of the GL if that line is tracked until the value of the distance function is zero..

The following sections describe how this process can be iteratively controlled by algorithms for an one-dimensional search space. The algorithms are on the binary-search-, so-called parabola-, and regula-falsi method. Simulations and practical tests are used to evaluate the different search algorithms. The accuracy of the intersection points and the number of necessary iteration steps, describing the speed of the algorithms, are used as quality assessment. parameters. Both quality characteristics depend on truncation criteria: a distance and a length of an interval.