1. Distance Intersection
Distance intersection, also known as Trilateration, is used to determine the position of an unknown point from two or more control points. Two control points with a distance from each intersect at two possible locations. Adding a third control point and distance eliminates one of the intersections. The presence of random errors, however, means the three distances will not intersect perfectly.
Figure H-1 is shows three control points with distances that intersect at point P. Use a least squares adjustment to determine the best coordinates of point P, its uncertainty, and error ellipse.
Figure H-1 Distance Intersections |
Initial coordinate approximations for point P
Distance and azimuth of line BA:
Law of Cosines to solve angle at B from A to P
Compute azimuth from B to P
Forward computation from B to P
Calculate initial distances using point P's initial approximate coordinates
Because point B was used to compute point C, the computed distance is the measured distance.
Perform inverse computation to obtain distances from points A and C
Matrix structures
Set up the observation equations
Dist AP
Dist BP
Dist CP
Set up matrices
Solve the matrix algorithm iteratively
First iteration
Invert the [C^{T}C] matrix using Determinant Method
Compute coordinate updates
Update point P's coordinates
Second iteration
Update [C] and [K] matrices using the observation equations and new coordinates of point P.
Recompute updates
Solution converged.
Adjusted distances
Compute distance residuals from
Since the last updates were zero, [C x U] is a column matrix with all elements equal to zero. Therefore:
Adjusted distances
Position uncertainties
Compute error ellipse
The error ellipses in Figure H-2 are magnified 25 times since they would not be visible at the drawing scale.
Figure H-2 Standard and 95% CI Error Ellipses |
Adjustment Summary
N_{P} = 2530.024' ±0.574' | E_{P} = 1109.762' ±0.519' | |
Error Ellipse | ||
Az_{U} = 170°04' | ||
S_{U} = ±0.576' | S_{V} = ±0.517' | |
S_{U} 95% = ±8.14' | S_{V} 95% = ±7.30' |