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The Geometric Algorithm of Inverse and Direct Problems with an Area Solution for the Great Elliptic Arcs

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The Geometric Algorithm of Inverse and Direct Problems with an Area Solution for the Great Elliptic Arcs Journal of Marine Science and Technology, Vol. 23, No. 4, pp. 481-490 (2015) 481 DOI: 10.6119/JMST-014-1120-1 THE GEOMETRIC ALGORITHM OF INVERSE AND DIRECT PROBLEMS WITH AN AREA SOLUTION FOR THE GREAT ELLIPTIC ARCS Wei-Kuo Tseng, Jiunn-Liang Guo, and Chung-Ping Liu Key words: great ellipse, edge, geography type, computer algebra The result of any computation, e.g. the length of a path or system. the intersection of polygons, depends on the definition of the edges that connect these points. On a planar map, the edge between two points is obviously the line segment that connects ABSTRACT them, but on an ellipsoidal earth model the choice is not ob- The definition of the connecting edge(s) between two ver- vious, and it varies between different software products. While tices in a geography type is the great elliptic minor arc in differences in accuracy and performance are to be expected, it Microsoft SQL Server’s Geography Type. Similar to the way is a sad state of affairs when different software packages dis- an edge is defined by Microsoft SQL Server, an edge in agree on the theoretical results of their computations. The Geodyssey’s Hipparchus library is also defined as a great paper (Kallay, 2007) presents the definition of edges in Mi- circle arc on a reference sphere. Using Hipparchus for their crosoft’s SQL Server’s Geography Type, proposing it as an computations, IBM’s DB2 Geodetic Extender and Informix industry standard. It stands to reason that a round earth edge Geodetic Datablade share this definition. Compact formulae should satisfy the following requirements: are given for the great elliptic sailing on a spheroid providing solutions to both the forward and inverse problems with ex- 1. Locally, an edge should be experienced as straight. ceptional accuracy, and latitude in terms of longitude. The 2. A pair of points should define a unique edge between them. solution incorporates a closed form for the azimuth and the 3. An edge should admit a differentiable parameterization, derivation of the algorithm is presented and illustrated. In which assigns a point on the edge to every real number addition, the area of polygon bounded by the elliptic arcs is between 0 and 1. treated. This paper also shows that a computer algebra system is a powerful tool to solve mathematical derivations in navi- The geodesic is the curve on the surface of an ellipsoid de- gation, geodesy, and cartography. fining the shortest distance between two points. Kallay (2007) points out that geodesic curves score poorly on the require- ment 2, 3, and even on 1 they are not the obvious choice: I. INTRODUCTION Lines, polygonal paths and polygons are widely used in the 1. While the geodesic curve is the shortest path that is con- description of geospatial data, and they are usually defined in fined to the surface, most human activities take us beyond terms of their endpoints and vertices. The definition of the the surface, for example, surveyors measure along straight connecting edge between two vertices is the shorter great el- lines of sight and airplanes fly miles above the surface of liptic arc in Microsoft SQL Server’s Geography Type (Kallay, the globe. 2007; Microsoft, 2013). Similar to the way an edge is defined 2. There are numerous (not necessarily antipodal) pairs of by Microsoft SQL Server, an edge in Geodyssey’s Hipparchus points on an ellipsoid between which there are more than library is also defined as a great circle arc on a reference sphere one short geodesic (Rapp, 1991). (Geodyssey, 2013). Using Hipparchus for their computations, 3. Computing points along geodesic curves is notoriously IBM’s DB2 Geodetic Extender and Informix Geodetic Datab- difficult and expensive. An exact differentiable parame- lade share this definition (IBM, 2013). terization is not known, and approximate ones are also dif- ficult and expensive to compute. Paper submitted 03/05/14; revised 08/07/14; accepted 11/20/14. Author for Classical surveying suggests the definition of an edge as a correspondence: Wei-Kuo Tseng (e-mail: [email protected]). 1Department of Merchant Marine, National Taiwan Ocean University, Kee- normal section (Rapp, 1991), which is a plane curve created lung, Taiwan, R.O.C. by intersecting a plane containing the normal to the spheroid 482 Journal of Marine Science and Technology, Vol. 23, No. 4 (2015) However, the existed formulae need cumbersome algorithms Great Ellipse and their accuracies are not very high. In addition, the mathe- matical derivations in those literatures are a bit tedious, and abstruse, hardly suited for coding (Bowring, 1984; Pallikaris Great Circle and Latsas, 2009). The direct solution was also not completely a provided in those articles (Earle, 2011). For these reasons, in b P2 P Auxiliary Spere this paper we revisit the solution for the great elliptic arc and P1 Spheroid provide a more straightforward and compact mathematical φ a derivation of the spherical trigonometric solutions. This paper λ a a also gives a general inverse and direct solution attaining any accuracy requirement for the calculation of the great ellipse R(φ) ()cosφ cosλ, cosφ sinλ, sinφ sailing. In the mathematical derivation, we consider the direct and Fig. 1. A great ellipse on a spheroid. inverse scenarios to produce solutions determining the great ellipse from one point and its azimuth or between two points. with the surface of the spheroid. Alas, this definition is rarely Furthermore, the interpolation for latitude in terms of longi- unique. The surveyor’s plane at the other endpoint may define tude between end points of a great ellipse on the spheroid has a different normal section. not yet been found in the literature. As a consequence of these The paper (Kallay, 2007) evaluates the definition of the observations, the complete solution to the great ellipse pre- great elliptic arc against the above stated requirements: sented here will include a method to determine latitude for any specified longitude along the ellipse. Because the calculation 1. Edges are experienced as straight or approximately straight of the area of polygon bounded by the geodesics needs cum- in several senses: As a great elliptic arc, an edge is planar. bersome algorithms (Sjöberg, 2006), the alternative calcula- The angular deviation of its plane from a surveyor’s planes tion of the area of polygon bounded by the great elliptic arc is at either endpoint is no more than 12’. This translates to also provided here. The accuracies attained can satisfy the about 2.8 cm for an edge whose length is 10 km. An edge is requirement of ECDIS and GIS environments. Finally, we approximately the shortest path between its endpoints – the give the full formulae of spherical trigonometry that can easily length of no edge exceeds the geodesic distance by more be coded in a programming language so that readers should than 0.02%. In the space of direction, the edge is a line comprehensively grasp the meaning of the geometry. segment, and so are its gnomonic projections. 2. Every pair of non-antipodal points defines a unique edge. II. PARAMETERS OF THE GREAT ELLIPSE 3. The parameterization is simple and differentiable. The parameterization as a quadratic rational Bezier curve may Using geodetic and geocentric latitudes, a point P on the be slightly more expensive to set up but very efficient for surface of a spheroid such as Earth can be represented as a generating multiple points along the edge. vector function of longitude , geodetic latitude , or geo- centric latitude . The great elliptic arc on spheroid has been investigated in (Bowring, 1984; Walwyn, 1990; Williams, 1998; Earle, 2000, P(,) xyz 2008; Kally, 2007; Tseng and Lee 2010), but is rarely men- 2 tioned elsewhere. The great elliptic arc between two points P1 Ne()coscos, cossin, (1 )sin (1) and P2 on a spheroid, centered at O, is the minor arc of the ellipse of the intersection between the spheroid and the plane OP1P2 (Fig. 1). If the two points are antipodal, the collinear and points P1, O, and P2 do not determine a unique plane, in such a case it would be reasonable to choose the route passing through PxyzR( , ) ( ) cos cos , cos sin , sin the two poles of the spheroid. The azimuth at the point P1 is (2) the angle that the tangent at P1 to great ellipse P1P2 makes with 221/2 the meridian through P1, and is measured from the clockwise where e is the eccentricity, and Nae( ) /(1 sin ) is direction northerly. The azimuth at arbitrary points on the the radius of curvature of the prime vertical, and great ellipse would be similarly defined (Bowring, 1984). Some approximate formulae, the great elliptic equation and 1/2 Rae( ) (1222 ) /(1 e cos ) . (3) great circle equations have been provided in a number of pa- pers (Earle, 2000; Pallikaris and Latsas, 2009; Tseng and Lee 2007a, 2007b, 2010, 2012, 2013) that have studied this prob- The following equation can transform Cartesian coordi- lem of the great ellipse sailing and achieved remarkable results. nates of a point on the spheroid to geodetic coordinates. W.-K. Tseng et al.: Geometric Algorithm of Inverse and Direct Problems with an Area Solution for the Great Elliptic Arcs 483 ' ' TTNNN=−⋅+−⋅cos(ϕφ ) sin( ϕφ ) TTN cos( ) sin( ) (8) N N NP α N ϕ − φ T p TN v The normal to spheroid at point P and the unit velocity γ vector at point P are orthogonal, so the inner product of the T P E two vectors equals to 0. Substitute Eq. (8) into Eq. (7) to obtain (Bowring, 1984; Earle, 2008): O ψ σ λ atan2TTVE , cos( ) TT VN (9) λe E Therefore the following relations exist: Fig.
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