Spherical Hyperbolae and Ellipses

Freiesleben, H. C.

doi:10.1017/s0373463300030186

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“…They often create the foundations for the methods and the following computations based on them. We recall the contributions presented in Freiesleben (1976) which also affects only the spherical model so included equations are valid only for the sphere and not for the ellipsoid. In this article our aim is to place the special emphasis on the geometric approach to the subject instead of the analytic computations.…”

Section: Pf Pfmentioning

confidence: 99%

“…The lines of altitude difference originally applied to astronomical position lines are hyperbolic position lines, as are also the position lines of equal difference in distance from two transmitters used to be adapted in such electronic fixing systems as Decca, Loran and Omega. For instance, we refer the reader to Freiesleben (1976) which deals with geometric construction of hyperbolae on a spherical surface for navigational application. 2-dimensional differentiable manifolds of positive curvature are mostly used to model the Earth's surface globally.…”

Section: Introductionmentioning

confidence: 99%

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On Geometric Properties of Spherical Conics and Generalization of Π in Navigation and Mapping

Kopacz

2012

Geodesy and Cartography

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First, we cover the conical curves on 2-dimensional modeling sphere S 2 showing their geometric properties affecting the hyperbolic navigation. We place emphasis on the geometric definition of spherical parabola and relate it to the notions of spherical ellipse and hyperbola and give simple geometric proofs for relations between conical curves on the sphere. In the second part of the paper function representing the ratio of the circle's circumference to its diameter has been defined and researched to analyze the potential discrepancies in the spherical and conical projective models on which the navigational computations are based on. We compare some non-Euclidean geometric properties of curved surfaces and its Euclidean plane model in reference to the local and global approximation. As a working tool we use function for geometric comparison analysis in the theory of long-range navigation and cartographic projection. We state the existence of the infinite number of the circles having the same radius but different circumference on the conical surface. Finally, we survey the exemplary proposals of generalization of function . In particular, we focus on the geometric structure of applied model treated as a metric space showing the differences in the outputting computations if the changes in a metric are made. We also relate the function to Tissot's indicatrix of distortion.

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Section: Pf Pfmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%