Theory, strict formula derivation and algorithm development for the computation of a geodesic polygon area

Abstract

The paper presents a coherent, original theory and an effective algorithm to calculate the surface area of a geodesic polygon, i.e. the area on an ellipsoid of revolution limited by geodesics, connecting pairs of subsequent vertices of known geographical coordinates. Using the cylindrical coordinate system and introducing an equatorial geodesic triangle, the novel strict recursive formulas were derived for the longitude on the ellipsoid and the distance along the geodesic and the area of a geodesic polygon. The area is determined by constant coefficients, not the values of finite series. The orthodromic properties of geodesics, which led to the detection of bifurcation lines composed of the double solutions of the inverse geodetic problem were analysed. The novel algorithm presented, integrating the problem with geodesic polygon area calculation, is based on the newly proposed parameterisation with two alternative working unknowns and a new formula to obtain the initial value of the geodesic C parameter. It allows solutions for all cases, including near-antipodal or near-vertices, and offers two solutions if bifurcation occurs. The algorithm presented was tested and verified using numerical examples from known publications. The numerical tests proved the accuracy of 1m2 of the computed geodesic area using standard IEEE 754 Double Floating Point PC arithmetic, for geodesic lengths up to ten thousand km.

Accurate and Robust Rotation-Invariant Estimation for High-Precision Outdoor AR Geo-Registration

“…∆longitude, ∆latitude, and ∆altitude represent the variations in longitude, latitude, and altitude between the current MAR device's valid RTK-GPS coordinates and the coordinates during the initialization phase. Long_CC and Lat_CC represent the unit conversion coefficients, expressed in meters/degree, which were obtained through the Vincenty formula-based geodesic method [49] to calculate the fixed length of longitude and latitude in the target research area [50]. By incorporating T into the transformation method described in Equation ( 17), the coordinates (X aw , Y aw , Z aw ) of the MAR device during motion can be calculated in the AR world coordinate system.…”

Enabling geosimulations for global scale: Spherical geographic automata

“…However, when equal‐area map projections are used, polygon segments are represented as straight lines instead of arcs of geodesics which can be an important source of error when computing surface area and distance at the global level (Santini et al, 2010). Moreover, many of these projections are designed for different regions and often exhibit large areal distortions in higher latitudes (Nowak & Nowak Da Costa, 2022).…”

Variable-rate spray system for unmanned aerial applications using lag compensation algorithm and pulse width modulation spray technology

“…Once a is obtained with sufficient accuracy, spherical triangulation method can be used to calculate the longitude and latitude of point B. For straight lines of any length with a longitudinal difference less than 180 degrees, Vincenty's formulae is capable of determining coordinates to an accuracy on the level of millimeters (Nowak and Nowak Da Costa, 2022). The spray volume (L ha -1 ) can be obtained from the prescription map according to the compensation coordinates B(fB,lB).…”