Special properties of the error figure for multiple resection with adjusted directions at the station point.
Abstract
The resection (Pothenot problem) is a very common method for determining the position of a new point. The presence of a multiple resection always requires adjustment. The simplest and most illustrative method of adjustment in this case is the graphical method, in which, as is known, lines of position are used. Assuming the principles of this adjustment as known, the present article examines, without unnecessary detail, the error figure formed by the lines of position of the resection, without delving into the method of its construction; a particular case of the error figure is discussed. At the station, we measure directions to reference points and then adjust them. As is known, with such a preliminary adjustment, the error figure has a number of double and triple intersection points. The essence of graphical adjustment consists in adjusting the error figure, and this naturally explains the research that has been aimed at simplifying this process. Such research has led us to an important conclusion. Namely, it turned out that the point of application of the resultant of the double intersection points and the point of application of the resultant of the triple intersection points coincide with each other, and the sum of the weights of the triple points is in a simple ratio to the sum of the weights of the double points. Furthermore, from the Pothenot problem with four points, a transition is made to any number of given directions to points. By means of simple reasoning, the validity of our theorem is proved also for these cases.
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