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'


2. Triangulation

A triangulation network is a survey composed of primarily angle measurements. Before electronic distance measurement, most control surveys were done with triangulation since it was faster to measure angles than distances. Triangulation networks required periodic distances, called baselines, to help fix geometry. But a point position can be fixed with just angle measurements for an intersection like in Figure H-3.

Figure H-3
Intersection by Angles


The survey has three measurements and two unknowns so a least squares adjustment can be used to determine point B's coordinates.

Point B initial coordinates

There isn't sufficient information to easily compute coordinates for point B (try it, you'll see). But because the adjustment is iterative, we just need reasonable coordinate values to start with. The better the initial values, the fewer iterations needed.

We'll guesstimate coordinates based on point B's proximity to the others. Point B is approximately one third the distance north from point H to point F and approximately one quarter the distance east from point F to point G:


Matrix Structures

Because each angle has an uncertainty, a weight matrix will be used. 


Set up observations

General angle observation equation: F-From, A-At, T-To

Since point B is the At point for all three angles, only the third and fourth terms of the observation equation are computed.

Compute distances

Angle FBG

Angle GBH

Angle HBF

Set up matrices

[K] matrix values are large due to point B's guesstimated coordinates. If they are too far off, the solution might not converge. Let's see what happens.

Solve the matrix algorithm iteratively


First iteration

Coordinate corrections are relatively large, a reflection of our guesstimated point B coordinates.

Updated coordinates are: NB1 = 4,556.558     EB1 = 2,779.278

Second iteration

Recomputed matrices

Corrections are getting smaller - going in the right direction. All right!

Updated coordinates are: NB2 = 4,545.909     EB2 = 2,783.113

Third iteration

Now we're cooking - corrections are smaller. One more should do it.

Updated coordinates are: NB3 = 4,545.954     EB3 = 2,782.940

Fourth iteration

Bingo! The solution took four iterations because our initial coordinates for point B were pretty rough. A better guesstimate might have reduced the iterations to three or possibly two. Worse approximations might have increased the iterations or possibly caused the solution to diverge.

Final coordinates:  NB = 4,545.954     EB = 2,782.940

Adjusted Angles

Because the final corrections are 0.000, the angle residuals are the last computed [K] matrix values. 

Position uncertainties


Compute standard and 95% CI error ellipses 



The standard and 95% CI error ellipses are shown in Figure H-4.

Figure H-4
Standard and 95% CI Error Ellipses

Because the ellipses are much smaller than those in the trilateration example, they must be magnified considerably more (500x) in order to be seen.

Adjustment Summary

  NB = 4,545.954' ±0.018' EB = 2,782.940' ±0.014'
  Error Ellipse 
  AzU = 158°22'   
  SU =  ±0.018' SV =  ±0.014'
  SU 95% =  ±0.254' SV 95% = ±0.198'