Three-Dimensional Forward and Inverse Computations

A system of equations is derived using the three-dimensional methods of vector and tensor analysis, and a computational routine is outlined, which can be used to compute the length, forward vertical angle and forward azimuth of a line connecting two points having known geodetic coordinates. Equations are similarly derived, and a computational routine is outlined, which can be used to compute the geodetic coordinates of a point which lies at the terminus of a vector of given length and direction, originating at a point having known geodetic coordinates. All data consists of field measurements in the form of slope distances, and horizontal and vertical angles, corrected only for refraction. The equations are exact with respect to a reference ellipsoid, but corrections are required wherever the actual topographic surface differs significantly from the surface of the reference ellipsoid.