Abstract:In this paper, equations are established to solve problems of Rhumb Line Sailing (RLS) on an oblate spheroid. Solutions are provided for both the inverse problem and the direct problem, thereby providing a complete solution to RLS. Development of these solutions was achieved in part by means of computer based symbolic algebra. The inverse solution described attains a high degree of accuracy for distance and azimuth. The direct solution has been obtained from a solution for latitude in terms of distance derived… Show more
“…As the two points are very close to each other, the line is very short and the differences between the geodesic and GE are very small; both sailing methods may be applied here, even using rhumbline or plane sailing (Tseng et al, 2012b) gives near identical results (shown in Table 1).…”
Section: N U M E R I Ca L R E S U Lt S Fo R D I S Ta N C E a N D Lo Nmentioning
An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.
K E Y
“…As the two points are very close to each other, the line is very short and the differences between the geodesic and GE are very small; both sailing methods may be applied here, even using rhumbline or plane sailing (Tseng et al, 2012b) gives near identical results (shown in Table 1).…”
Section: N U M E R I Ca L R E S U Lt S Fo R D I S Ta N C E a N D Lo Nmentioning
An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.
K E Y
“…We also mention that several moons of the Solar System approximate prolate spheroids in shape; however, they are actually triaxial ellipsoids. Also, the algorithms for accurate and global navigational calculations correspond to spheroidal (oblate, ) geometric models; see, e.g., Earle (2006), Pallikaris and Latsas (2012), Tseng et al (2012) and Kopacz (2018a) in this regard. Figure 4.The single-heading (pseudoloxodromic) solutions of minimum time starting from different colatitudes on the prolate ellipsoid (), i.e., (solid blue), (solid black) and (solid red), among other time-minimal paths (dashed colours, respectively), under weak rotational wind given by Equation (24), with ; .…”
Introducing the notion of a pseudoloxodrome, we generalise a single-heading navigation to conformally flat Riemannian manifolds, under the action of a perturbing vector field (wind, current) of arbitrary force. The findings are applied to time-optimal navigation with the use of the Euler–Lagrange equations. We refer to the Zermelo navigation problem admitting space and time dependence of both a perturbation and a ship's speed. The necessary conditions for single-heading time-optimal navigation are obtained and the pseudoloxodromes of minimum and maximum time are discussed. Furthermore, we describe winds which yield the pseudoloxodromic and loxodromic time extremals. Our research is also illustrated with the examples in dimension two emphasising the single-heading solutions among the time-optimal trajectories in the presence of some space-dependent winds.
“…It is found that this approximation introduces a maximum error of 17 nm or 0·3% of the total meridional arc length. The exact treatment on the ellipsoid by classical methods is given by Williams (1998) and Tseng et al (2012).…”
Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.
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