Abstract
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, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.
Highlights
It seems to be natural that when one deals with rotational surfaces, generated by the rotation of a plane curve around a coplanar fixed line, the geometric linear momentum of the generatrix with respect to the axis of revolution plays a predominant role to control the geometry of the rotational surface. We show it in Corollary 2, proving that any rotational surface is uniquely determined, up to translations along the axis of revolution, by the geometric linear momentum of the generatrix curve. This main result is confirmed when we study the geometry of a rotational surface since both its first and second fundamental forms can be expressed only in terms of the geometric linear momentum and, the distance from the surface to the axis of revolution
This allows our main contribution in the paper, which 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. We illustrate this procedure analyzing under our optics the two types of linear Weingarten surfaces one can find in the literature and we emphasize two new classification results involving a cubic and an hyperbola in the W-diagram of the rotational Weingarten surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids
As a consequence of Theorem 4, we provide simple new proofs of the above mentioned classical results concerning rotational surfaces such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones
Summary
This main result is confirmed when we study the geometry of a rotational surface since both its first and second fundamental forms can be expressed only in terms of the geometric linear momentum and, the (non-constant) distance from the surface to the axis of revolution This allows our main contribution in the paper, which 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 (see Section 3). As a consequence of Theorem 4, we provide simple new proofs of the above mentioned classical results concerning rotational surfaces such as Euler’s theorem about minimal ones (see Corollary 3), Delaunay’s theorem on constant mean curvature ones (see Corollary 4), and Darboux’s theorem about constant Gauss curvature ones (see Corollary 5)
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