A new approach to rotational Weingarten surfaces
| Autor: | Paula Carretero, Ildefonso Castro |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
Differential Geometry (math.DG) General Mathematics Weingarten surfaces rotational surfaces principal curvatures quadric surfaces of revolution elasticoids Computer Science (miscellaneous) FOS: Mathematics Mathematics::Differential Geometry 53A05 (Primary) 53A04 74B20 (Secondary) Engineering (miscellaneous) |
| Zdroj: | Mathematics; Volume 10; Issue 4; Pages: 578 |
| Popis: | Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, like Euler's theorem about minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's theorem about constant Gauss curvature ones. 23 pages, 13 figures |
| Databáze: | OpenAIRE |
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