On geometric properties of spherical conics and generalization of π in navigation and mapping

Abstract

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.