Geodesic
Cycles of the hyperbolic plane of positive curvature
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 137-162

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Properties of hyperbolic and elliptic cycles of the hyperbolic plane $\widehat H$ of positive curvature are investigated. An analog of Pythagorean theorem for a right trivertex with a parabolic hypotenuse is proved. For each type of straight lines, formulas expressing the length of a chord of a hyperbolic cycle in terms of the cycle radius, the measure of the central angle corresponding to the chord, and the radius of curvature of $\widehat H$ are obtained. The plane $\widehat H$ is considered in projective interpretation. 
@article{ZNSL_2013_415_a14,
     author = {L. N. Romakina},
     title = {Cycles of the hyperbolic plane of positive curvature},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--162},
     publisher = {mathdoc},
     volume = {415},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a14/}
}
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L. N. Romakina. Cycles of the hyperbolic plane of positive curvature. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 137-162. http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a14/