## 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

Dist AP

Dist BP

Dist CP

#### 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