Quasi-geodesic line

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A curve on a surface on any segment of which the right and left rotations have the same sign (see Swerve of a curve). For example, the edge of a lens is a quasi-geodesic line.

The class of quasi-geodesic lines substantially amplifies the class of geodesic lines (cf. Geodesic line), making its families (bounded in length and position) compact. In a two-dimensional manifold $M$ of bounded curvature there passes through each point at least one quasi-geodesic in each direction; it can always be extended. Segments of quasi-geodesics (at the ends of which there are no points of curvature $2\pi$ on $M$) are limits of geodesics lying on smooth surfaces properly converging to $M$.


[1] A.D. Aleksandrov, Yu.D. Burago, "Quasigeodesics" Proc. Steklov Inst. Math. , 76 (1965) pp. 58–76 Trudy Mat. Inst. Steklov. , 76 (1965) pp. 49–63



[a1] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)
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This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article