Projections from surfaces of revolution in the Euclidean plane
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Projections from surfaces of revolution in the Euclidean plane C. Charitos1 · P. Dospra2 Received: 23 May 2020 / Accepted: 8 October 2020 © The Managing Editors 2020
Abstract In this paper, we determine the class of surfaces of revolution S for which there exists a smooth map from a neighbourhood U of S to the Euclidean plane E 2 preserving distances infinitesimally along the meridians and the parallels of S and sending the meridional arcs of U ∩ S to straight lines of E 2 . Keywords Surfaces of revolution · Meridians · Parallels · Projections Mathematics Subject Classification 53A05 · 34A05
1 Introduction In [4] (see [5] for the translation of [4] in English), Euler proved that there does not exist a perfect map from the sphere S 2 or, from a part of S 2 , to the Euclidean plane E 2 . Recall that a smooth map f from S 2 (or, from a part of S 2 ) to E 2 is called perfect if for each p ∈ S 2 there is a neighbourhood U ( p) of p in S 2 such that the restriction of f on U ( p) preserves distances infinitesimally along the meridians and the parallels of S 2 and f also preserves angles between meridians and parallels Charitos and Papadoperakis (2019). In modern geometric language, a perfect map is a local isometry from S 2 to E 2 and, thus, Euler’s theorem follows from the Gauss Egregium Theorem which was proved many years later. However, Euler’s method of proof is very fruitful and can be applied to similar problems, see for instance
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P. Dospra [email protected] C. Charitos [email protected]
1
Department of Natural Resources Management and Agricultural Engineering, Agricultural University Athens, Iera Odos 55, 11855 Athens, Greece
2
Department of Electrical and Computer Engineering, University of Western Makedonia, 50100 Kozani, Greece
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Beitr Algebra Geom
Proposition 5 of Charitos and Papadoperakis (2019). Very briefly, Euler’s basic idea for the non-existence of a perfect map from S 2 to E 2 , is to translate geometrical conditions to a system of differential equations and prove that this system does not have a solution. Using Euler’s method, the non-existence of a smooth map from a neighbourhood U of S 2 to E 2 which preserves distances infinitesimally along the meridians and the parallels of S 2 and which sends the meridional arcs of U ∩ S 2 to straight lines of E 2 , can be shown as well Charitos and Papadoperakis (2019). The origin of all these problems lies in the ancient problem of cartography, that is, the problem of constructing geographical maps from S 2 (or from a subset of S 2 ) to E 2 which satisfy certain specific requirements. This problem can also be considered as part of a more general question concerning the existence of coordinate transformations that preserve certain geometrical properties from one coordinate system to another. Several prominent mathematicians have studied this problem from antiquity to our days and in the course of this study, S 2 was replaced gradually by surfaces of revolution or by surfaces in E 3 in general (see Papadopoulos (2018) for an excellent historic
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