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INTRODUCTION
The definition of the connecting edge or line between two vertices is a minor great elliptic arc in Microsoft SQL Server (2014), whereas the more sophisticated definitions are the geodesics in the main geography databases (Oracle® Spatial Developer's Guide, 2013; ArcGIS Resource, 2014; Hipparchus® Tutorial and Programmer's Guide, 2004; IBM DB2 Universal Database 9.1., 2012). Great ellipse sailing has been studied in our previous work (Tseng et al., 2012a; 2013) and considering that the producers of Geographic Information Systems (GIS) and Electronic Chart Display and Information Systems (ECDIS) may adopt those databases in their systems, the geodesic algorithms need to be reviewed and studied.
The geodesic is of interest because it is the shortest path between two points on the Earth. In most terrestrial applications, the Earth is treated as a spheroid by adopting the World Geodetic System (WGS) 84 datum. Geodesics can also be used in the application of the United Nations Convention on maritime boundaries at sea; other uses involve distance measuring in GIS and ECDIS and governing rules of the Federal Aviation Administration bounding areas (Sjöberg, 2007; 2012).
Usual algorithms for the geodesic can be roughly divided into two groups: (a) numerical integration schemes and (b) series expansion of elliptic integrals. Group (a) can be further divided into integration schemes based on simple differential relationships of the spheroid (Kivioja, 1971; Jank and Kivioja, 1980; Thomas and Featherstone, 2005), or by numerical integration of elliptic integrals that are usually functions of elements in the spheroid and its corresponding auxiliary sphere (Saito, 1970; 1979; Sjöberg, 2007; 2012). Group (b) includes the original method of Bessel (1826) that uses functions of elements in the spheroid related to a corresponding auxiliary sphere and various modifications to his method (Rainsford, 1955; Vincenty, 1975a; Bowring, 1983; 1984; Karney, 2013).
The inverse geodetic problem on the spheroid is to determine the geodesic arc length between two endpoints and the azimuths of the arc. The more complete solution for the Clairaut constant (or the vertex latitude) which is compared with the solution provided by Sjöberg (2007; 2012) is presented in this paper. If the two given points are not nearly antipodal, each azimuth and location of the geodesic is unique, while for...