H. More Horizontal Examples

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

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

  NP = 2530.024' ±0.574' EP = 1109.762' ±0.519'
  Error Ellipse 
  AzU = 170°04'   
  SU = ±0.576' SV = ±0.517'
  SU 95% =  ±8.14' SV 95% = ±7.30'