## 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 [CTC] matrix using Determinant Method  Update point P's coordinates Second iteration

Update [C] and [K] matrices using the observation equations and new coordinates of point P.  Solution converged.

Compute distance residuals from Since the last updates were zero, [C x U] is a column matrix with all elements equal to zero. Therefore:  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