Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

Calculation of the stress state of a noncircular helicoidal tube based on the tensor theory of shells

The purpose of the study was to develop an algorithm for calculating helical-symmetric shells with a closed contour in oblique Gaussian coordinates. The twist and length of the shell were taken unchanged. The method is based on the representation of the generating contour of the helicoidal surface by a discrete set of points with the replacement of differentiation along the angular coordinate by finite differences. The unknown were the displacement vectors at the indicated points of the contour. Due to the helicoidal symmetry, the differentiation of vector quantities with respect to the helical coordinate was replaced by vector multiplication. The tensor of deformations and the tensor of the parameters of the change in curvature were calculated using the nabla operator, represented in oblique Gaussian coordinates. Integration over the contour coordinate was replaced by summation over discrete points. The tensors found, which characterize the deformed state, were used to calculate the strain energy of one period of the helicoidal shell, and then the total potential of the mechanical system was compiled. The unknown displacements were determined by minimizing the total potential, taking into account the constraints that prohibit the displacement of the shell as a rigid whole. The study gives a numerical example of the application of the developed approach.

Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density

In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We get a result for the minimal case. Additionally, we give examples of a helicoidal surface with a weighted mean curvature and weighted extrinsic curvature.

Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density

In this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ${\Bbb R}_1^3$with a positive density function. We get a result for minimal case. Also, we give examples of a helicoidal surface with weighted mean curvature and Gaussian curvature.

Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace–Beltrami operator of the rotational surface of value ( 0 , 1 ) .

Timelike Tangent Developable Surfaces and Bonnet Surfaces

A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.

Differential Equation of the Loxodrome on a Helicoidal Surface

In nature, science and engineering, we often come across helicoidal surfaces. A curve on a helicoidal surface in Euclidean 3-space is called a loxodrome if the curve intersects all meridians at a constant azimuth angle. Thus loxodromes are important in navigation. In this paper, we find the differential equation of the loxodrome on a helicoidal surface in Euclidean 3-space. Also we give some examples and draw the corresponding pictures via the Mathematica computer program to aid understanding of the mathematics of navigation.