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Compared to the concept of geodesics the concept of quasi-geodesics seems to be substantially harder to grasp and digest. I was given a promising hint to the concept of quasi-geodesics here but the usual references didn't reveal something like an "easy access" to the concept:

So I am still looking for something like a "gentle introduction to quasi-geodesics".

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The Wikipedia and EoM articles to which you linked discuss two unrelated concepts that go by the same name "quasi-geodesic". One, defined in Wikipedia, is useful in negatively curved (hyperbolic) spaces. The other, described in EoM, is mostly used in positively curved (Alexandrov) spaces. –  user53153 Feb 2 '13 at 2:45
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This is rather self-serving, but I do discuss quasigeodesics in my book with Erik Demaine, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, pp.372-380. I spent some time absorbing the Russian geometry literature before writing that section. Some of the same material is revisited in a more elementary presentation in my book with Satyan Devadoss, Discrete and Computational Geometry, pp.200-205:
Figure 6.35
I would be interested myself if there is a more "gentle introduction." I don't think so. And at least the way I have presented it in these books, it is, I think, rather easy to understand.

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:Thank you very much! I'll try to get your book. –  Hans Stricker Feb 2 '13 at 10:28
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I read from Vandereycken's PhD thesis (http://web.math.princeton.edu/~bartv) that quasi-geodesics are first order approximations, also called retractions (used in optimization). It is a cheap approximation to a geodesic.

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