Abstract
A new geometric condition necessary for regularity of a curved three-web is found. A class of three-webs from circles generalizing the regular three-web of Blaschke from three elliptic pencils of circles with pairwise coinciding vertices is considered, and it is shown that only webs equivalent to the Blaschke web are regular in this class.
This is a preview of subscription content, log in via an institution to check access.
Notes
Although the parameter space can be considered Riemannian, by equipping it with some kind of metric, in particular, invariantly related to the three-web [1].
REFERENCES
W. Blaschke, Einführung in die Geometrie der Waben, Elemente der Mathematik vom höheren Standpunkt aus, Vol. 4 (Birkhäuser, Basel, 1955). https://doi.org/10.1007/978-3-0348-6952-2
W. Blaschke and G. Bol, Geometrie der Gewebe, topologische Fragen der Differentialgeometrie, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 49 (Julius Springer, Berlin, 1938).
H. Pottmann, L. Shi, and M. Skopenkov, “Darboux cyclides and webs from circles,” Comput. Aided Geometric Des. 29, 77–97 (2012). https://doi.org/10.1016/j.cagd.2011.10.002
K. Strubecker, “Über eine Klasse spezieller Dreiecksnetze aus Kreisen,” Monatsh. Mathematik und Phys. 39, 395–398 (1932). https://doi.org/10.1007/bf01699080
W. Wunderlich, “Über ein besonderes Dreiecksnetz aus Kreisen,” Sitzungsber, Akad. Wiss. Wien 147, 385–399 (1938).
F. Nilov, “New examples of hexagonal webs from circles,” arXiv Preprint (2013). https://doi.org/10.48550/arXiv.1309.5029
F. K. Nilov, “On new constructions in the Blaschke–Bol problem,” Sb. Math. 205 (11), 1650–1667 (2014). https://doi.org/10.1070/SM2014v205n11ABEH004432
V. B. Lazareva, “Parallelized three-webs generated by pencils of circles,” in Webs and Quasigroups (Kalininskii Gos. Univ., Kalinin, 1988), pp. 74–77.
V. B. Lazareva and A. M. Shelekhov, “On triangulations of the plane by pencils of conics,” Sb.: Math. 198, 1637–1663 (2007). https://doi.org/10.1070/SM2007v198n11ABEH003899
V. B. Lazareva, “Classification of regular circle three-webs up to circular transformations,” J. Math. Sci. 177, 579–588 (2011). https://doi.org/10.1007/s10958-011-0483-7
A. M. Shelekhov, V. B. Lazareva, and A. A. Utkin, Curvilinear Three-Webs (Tver. Gos. Univ., Tver, 2013).
V. B. Lazareva and A. M. Shelekhov, “On triangulations of the plane by pencils of conics. II,” Sb.: Math. 204, 869–909 (2013). https://doi.org/10.1070/SM2013v204n06ABEH004323
A. M. Shelekhov, “On three-webs generated by families of circles,” Tr. Mezhdunarodnogo Geom. Tsentra 5 (2), 6–16 (2012).
A. M. Shelekhov, “On hexagonal three-webs formed by the Cartesian net and a family of circles,” Russ. Math. 62, 68–81 (2018). https://doi.org/10.3103/S1066369X18100080
A. M. Shelekhov, “A generalized theorem on curvilinear three-web boundaries and its applications,” Sb.: Math. 211, 422–454 (2020). https://doi.org/10.1070/SM9167
V. B. Lazareva and A. M. Shelekhov, “On the geometric interpretation of the invariant equipment of a point correspondence between three lines,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 9, 43–47 (1984).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Publisher’s Note.
Allerton Press remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Shelekhov, A.M. Three-Webs from Circles. Russ Math. 67, 64–81 (2023). https://doi.org/10.3103/S1066369X23120083
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X23120083