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Existence, uniqueness, and angle computation for the loxodrome on an ellipsoid of revolution. (English) Zbl 1171.53005

In a Euclidean \(3\)-space a loxodrome on a surface of revolution is a curve that transverses all meridians along its way at a constant angle. The existence and uniqueness of a loxodrome on an ellipsoid \(\mathcal E\) of revolution and the formula for its angle are known results [cf. R. Williams, Geometry of navigation, Horwood Publ.,West Sussex (1998)] proved usually with the help of a one-to-one conformal map and of “infinitesimals”.
Avoiding these two tools the author presents a rigorous proof of the mentioned results; he uses a diffeomorphism of an open connected half of \(\mathcal E\) and transforms the task into an initial value problem of an ordinary differential equation.
Reviewer’s remark: Fig. 1 does not satisfy the laws of descriptive geometry.

MSC:

53A05 Surfaces in Euclidean and related spaces
53A04 Curves in Euclidean and related spaces
51N20 Euclidean analytic geometry
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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