Simple Short-Line Formulas for the Geodetic Direct and Indirect Problems

• Published in: Surveying and land information science Vol. 81; no. 1; pp. 5 - 18
• Main Authors: Meyer, Thomas H., Elaksher, Ahmed
• Format: Journal Article
• Language: English
• Published: Gaithersburg American Association for Geodetic Surveying 01.05.2022; American Congress on Surveying and Mapping
• Subjects: Accuracy; Earth models; Elliptic functions; Exact solutions; Geodesics; Geodesy; Geodetic coordinates; Geometric Geodesy; Infinite series; Inverse problems; Iterative methods; Spreadsheets
• ISSN: 1538-1242; 1559-7202

Sometimes, the land surveyor wants the geodetic coordinates for stations that were coordinated in a spreadsheet from a total station's observations. There is a large body of literature devoted to this subject, the direct problem of geodesy, along with its inverse problem, to determine the geodesic distance and directions between two stations given their geodetic coordinates. For a spheroidal Earth model, such as a reference ellipsoid, the exact solution of these problems involves determining the arc length of the geodesic among stations, which requires evaluating an elliptic integral. Some modern software ecosystems, like MathematicaTM, MatlabTM, and SciPy, provide built-in elliptic integrals, but spreadsheets still lack them. Approximate solutions abound, starting since the time of Bessel. The approximate solutions often require evaluating highly complicated truncated infinite series and/or iteration-which can be done in a spreadsheet-but our aim is to compare methods that are simple enough to implement comfortably in a single row along the top of a spreadsheet. This constraint allows surveyors to easily convert many stations' coordinates by just dragging the computational columns down the rows holding the observations. We review some classical approaches that were used before electronic calculators were available and, thus, were relatively simple, and we derive some new formulas. We provide their spreadsheet implementations, and tabulate their accuracies for the subpolar portion of the Earth. One finds the usual trade-off between simplicity and accuracy.