Computation of Horizontal/Level Distances

With increasing precision in distance measurements made by surveyors and engineers, the definition of horizontal warrants reconsideration. For many applications, horizontal distance is obtained as a right‐triangle component of slope distance using a vertical angle. Since plumb lines converge near the earth's center, and because horizontal and level surfaces are different, a precisely measured slope distance loses its integrity if not reduced correctly. This paper looks at the geometry of a slope distance reduction and describes how a horizontal/level distance is obtained from two measurement configurations; slope distance combined with elevation of endpoints and slope distance combined with zenith (vertical) angle observations. Vertical refraction is considered. Tables show the similarity of horizontal and level distances and how they differ from the right triangle component of the same slope distance. Two graphs are included to help the reader decide if systematic error caused by ignoring convergence of the plumb lines is significant. Proper reduction of horizontal distance to sea level completes the discussion.