Geodesic
On new constructions in the Blaschke-Bol problem
Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1650-1667

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We find several essentially new constructions of hexagonal $3$-webs based on a combination of quadratic and linear families of circles. They are used to construct $5$ new types of hexagonal $3$-webs, which is an advance in the solution of the Blaschke-Bol problem (1938) on the classification of such webs. Unlike many known examples, in our proofs we give an explicit parallelizing diffeomorphism. We give a brief survey of all known examples of hexagonal $3$-webs and their properties. In conclusion, we formulate several conjectures and open problems. Bibliography: 13 titles.
Keywords: webs, webs of circles, hexagonal closure condition, pencil of circles, quadratic family of circles. 
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     author = {F. K. Nilov},
     title = {On new constructions in the {Blaschke-Bol} problem},
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F. K. Nilov. On new constructions in the Blaschke-Bol problem. Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1650-1667. http://geodesic.mathdoc.fr/item/SM_2014_205_11_a4/