Geodesic
Locally Euclidean Metrics with a~Given Geodesic Curvature of the Boundary
Informatics and Automation, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 218-226

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The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature. 
@article{TRSPY_2009_266_a11,
     author = {I. Kh. Sabitov},
     title = {Locally {Euclidean} {Metrics} with {a~Given} {Geodesic} {Curvature} of the {Boundary}},
     journal = {Informatics and Automation},
     pages = {218--226},
     publisher = {mathdoc},
     volume = {266},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2009_266_a11/}
}
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I. Kh. Sabitov. Locally Euclidean Metrics with a~Given Geodesic Curvature of the Boundary. Informatics and Automation, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 218-226. http://geodesic.mathdoc.fr/item/TRSPY_2009_266_a11/