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If the observer’s location is (X0,Y0) and the observed points are (X1,Y1), (X2, Y2) and the measured distances to each is D1 and D2 respectively, the values of the location of the observer can be determined by simultaneously solving the equations D1^2= (X0-X1)^2 + (Y0-Y1)^2 and D2^2= (X0-X2)^2 + (Y0-Y2)^2. This will reduce to a quadratic equation with two possible solutions. Additional information is required to determine which of the two solutions is appropriate. The additional information usually takes the form of requiring input of which side of the line 1-2 the observer is on. Graphically, this is demonstrated by drawing a circle of radius D1 around (X1, Y1) and a circle of radius D2around (X2,Y2).
Now comes the warning!!!! The program assumes that the values of D1, D2, X1, Y1, X2, and Y2 are perfect. Of course they are not perfect. To demonstrate how a small variation in the actual values can lead to large uncertainties in the observer’s position try this using a cad program.
Draw a circle of radius 160’ centered on (85’,64’) and one of 130’ centered on (115’,344’). The point of intersection to the right of the line from (85’,64’) to (115’,344’) is (135.89’,215.69’). Vary the distances by 0.01’ + or – by offsetting the circles. Notice that the possible combination of intersection points covers an ellipse of 0.08’ by 0.02’!
Now draw a line from (0,120) to (220,375). Intersect a line bearing S42degrees E from (0,120) with a line S30degrees E from (220,375). If a circle is drawn through the points (0,102), (220,375) and the point of intersection it will form a circle with a radius of 809.93. Any line from a point on this circle drawn through (0,120) will form and angle of 12 degrees with a line from the same point on the circle through (220,375).
Therefore, if in addition to the distances from two points, if the angle formed between the lines of sight is also measured, an independent check of the point of observation can be performed. (I don’t know of a canned program that does this check). This same relationship can be used to determine the observer’s position from three points by measuring two angles without the distances being known.
In every case, however, the mathematics does not take into account errors in the positions of the points observed or the measurements made. Even in the case of a least squares adjustment only systematic errors in the measured values are evaluated. The “known points” are treated as if there were no error in their positions. Patently, this is not the case.
Positioning by resection is fun and useful for navigational applications. Surveyors must be wary of the weaknesses associated with measurement errors. Position derived by these processes can accumulate large errors quickly. Beware.
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