Accurate Algorithms for Spatial Operations on the Spheroid in a Spatial Database Management System

Some of the most powerful spatial analysis software solutions (Oracle, Google Earth Engine, PostgreSQL + PostGIS, etc.) are currently performing geometric calculations directly on the ellipsoid (a quadratic surface that models the earth shape), with a double purpose: to attain a high degree of accuracy and to allow the full management of large areas of territory (countries or even continents). It is well known that both objectives are impossible to achieve by means of the traditional approach using local mathematical projections and Cartesian coordinates. This paper demonstrates in a quantitative methodological way that most of the spatial analysis software products make important deviations in calculations regarding to geodesics, being the users unaware of the magnitude of these inaccuracies, which can easily reach meters depending on the distance. This is due to the use of ellipsoid calculations in an approximate way (e.g., using a sphere instead of an ellipsoid). This paper presents the implementation of two algorithms that solve with high accuracy (less than 100 nm) and efficiently (few iterations) two basic geometric calculations on the ellipsoid that are essential to build more complex spatial operators: the intersection of two geodesics and the minimum distance from a point to a geodesic.