2. Direction-distance

Figure xx-2 is a diagram of a base line and an intersection point.

Figure E-2
Direction-distance example

Coordinates of the base line end points are:

Point North (ft) East (ft)
J 1419.51 3511.69
K 2056.64 2341.36

The azimuth from point J to point T is 329°46'45"; the distance from point K to point T is 738.15 ft.

What are the coordinates of point T?

Step (1) Inverse the base line from point K to point J

a. Triangle-based method

To get point T's coordinates, we'll perform a forward computation from K.

Step (2) Compute the angle at J

Step (3) Use Law of Sines to compute angle at T

Remember that we have to check the angle returned by the Law of Sines because it will always be <90°.

From the diagram, T is greater than 90° so it must be subtracted from 180°

Step (4) Compute angle at K from the Angle Condition

Step (5) Compute azimuth from point K to point T

Step (6) Forward compute from point K

The math check is to compute the coordinates from point J

Step (1) Compute distance from point J using Law of Sines

Step (2) Forward compute from point J


b. Arc-based method

Because the direction can intersect the arc at two points, we'll extend the diagram a bit to visualize the possibilities, Figure XX-XX.

Figure E-3
Direction-distance arc example

Step (2) Set up and solve Equations D-12 through D-15.

Step (3) Pick the correct distance and compute the coordinates

From Figure E-3, the correct distance dQ is the shorter one: 878.915,. Using that distance, perform a forward computation

The coordinates of point T are the same as those from the triangle-based method.