Discrete channel surfaces. (English) Zbl 1437.53010
The authors present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data are associated with this notion that may be used to reconstruct the underlying particular discrete Legendre map. As an application, they investigate isothermic discrete channel surfaces and prove a discrete version of Vessiot’s theorem (Theorem 4.5).
The paper is organized in four sections containing the following aspects: Introduction (smooth channel surfaces and problems with the discretization, preliminaries, discrete Legendre maps), Discrete channel surfaces (Lie geometric characterization of discrete channel surfaces, Möbius geometric properties of a discrete channel surface, a discrete channel surface as the envelope of a sphere curve, discrete Dupin cyclides), A channel surface from two prescribed curvature lines, Discrete isothermic channel surfaces (discrete surfaces of revolution, cylinders and cones, Vessiot’s theorem, Examples).
Other works of the auhtors directly connected to this topic are: the first author’s book [Introduction to Möbius differential geometry. Cambridge: Cambridge University Press (2003; Zbl 1040.53002)] and the papers [first author et al., Beitr. Algebra Geom. 60, No. 1, 39–55 (2019; Zbl 1433.53010); F. Burstall et al., Nagoya Math. J. 231, 55–88 (2018; Zbl 1411.53007); M. Pember and G. Szewieczek, Beitr. Algebra Geom. 59, No. 4, 779–796 (2018; Zbl 1408.53014)].
The paper is organized in four sections containing the following aspects: Introduction (smooth channel surfaces and problems with the discretization, preliminaries, discrete Legendre maps), Discrete channel surfaces (Lie geometric characterization of discrete channel surfaces, Möbius geometric properties of a discrete channel surface, a discrete channel surface as the envelope of a sphere curve, discrete Dupin cyclides), A channel surface from two prescribed curvature lines, Discrete isothermic channel surfaces (discrete surfaces of revolution, cylinders and cones, Vessiot’s theorem, Examples).
Other works of the auhtors directly connected to this topic are: the first author’s book [Introduction to Möbius differential geometry. Cambridge: Cambridge University Press (2003; Zbl 1040.53002)] and the papers [first author et al., Beitr. Algebra Geom. 60, No. 1, 39–55 (2019; Zbl 1433.53010); F. Burstall et al., Nagoya Math. J. 231, 55–88 (2018; Zbl 1411.53007); M. Pember and G. Szewieczek, Beitr. Algebra Geom. 59, No. 4, 779–796 (2018; Zbl 1408.53014)].
Reviewer: Dorin Andrica (Riyadh)
MSC:
| 53A40 | Other special differential geometries |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 53A70 | Discrete differential geometry |
Keywords:
channel surface; Dupin cyclide; Lie sphere geometry; discrete Legendre map; Lie cyclide; blending surface; Vessiot’s theorem; discrete isothermic net; Ribaucour transformationReferences:
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