Linear Weingarten Surfaces Research Articles

Channel linear Weingarten surfaces in space forms

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.

Designing funicular grids with planar quads using isotropic Linear-Weingarten surfaces

The design of curved structural building envelopes such as gridshells or cable-net roofs is challenging as it requires to account for a wide variety of constraints. In particular, the chosen shape must be mechanically efficient, fabricable, and fit the site geometry. This article shows how a family of surfaces, called isotropic Linear Weingarten (iLW) surfaces, may fulfil all these constraints together, and be used as an intuitive design tool. We start by showing that these shapes are funicular for a uniform vertical load, and that principal projected stress lines form a conjugate net. This allows in particular for the design of gridshells with planar faces and low bending moments or for the design of self-stressed cable-nets cladded by planar glass panels. We then propose a discrete model of iLW surfaces based on recent advances in discrete differential geometry. We use this model to propose an optimization-based generation method, for which the inputs are the boundary curves and two control parameters. We demonstrate the shape potential on several examples.

Spheres and Tori as Elliptic Linear Weingarten Surfaces

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric.

Discrete Weierstrass-Type Representations

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the \(\varOmega \)-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.

Surfaces of prescribed linear Weingarten curvature in

Given $a,\,b\in \mathbb {R}$ and $\Phi \in C^{1}(\mathbb {S}^{2})$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb {R}^{3}$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi (N)$, where $N:\Sigma \rightarrow \mathbb {S}^{2}$ is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

Some Characterizations of Linear Weingarten Surfaces in 3-Dimensional Space Forms

Some Characterizations of Linear Weingarten Surfaces in 3-Dimensional Space Forms

Canal Surface Whose Center Curve is a Hyperbolic Curve with Hyperbolic Frame

In this paper, we obtain the parametrization of the canal surfaces whose center curves are the hyperbolic curves on the hyperbolic space $H^{2}$ in $%\mathbb{E}_{1}^{3}$. The parametrization of the canal surface is expressed according to the hyperbolic frame given in [10]. Then, the parallelsurface of this surface is studied. Also, we define the notion of the associated canal surface. Lastly, we give the geometric properties of thesesurfaces such that Weingarten surface, $\left( X,Y\right) $-Weingarten surface and linear Weingarten surface.

Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface

A method is presented for generating a discrete piecewise constant Gaussian curvature (CGC) surface. An energy functional is first formulated so that its stationary point is the linear Weingarten (LW) surface, which has a property such that the weighted sum of mean and Gaussian curvatures is constant. The CGC surface is obtained using the gradient derived from the first variation of a special type of the energy functional of the LW surface and updating the surface shape based on the Gaussian curvature flow. A filtering method is incorporated to prevent oscillation and divergence due to unstable property of the discretized Gaussian curvature flow. Two techniques are proposed to generate a discrete piecewise CGC surface with preassigned internal boundaries. The step length of Gaussian curvature flow is adjusted by introducing a line search algorithm to minimize the energy functional. The effectiveness of the proposed method is demonstrated through numerical examples of generating various shapes of CGC surfaces.

The complete classification of a class of new linear Weingarten surfaces in Minkowski space

We consider spacelike and timelike surfaces satisfying an interesting geometric property that the gradient of the mean curvature is a principal direction in a Minkowski 3-space. It is proved that these surfaces are linear Weingarten surfaces and contain all biconservative surfaces in a Minkowski 3-space. We call them generalized biconservative surfaces (or GB surfaces for short). We give complete explicit classifications of spacelike and timelike GB surfaces in Minkowski 3-space. Our results show that the non-constant-mean-curvature GB surfaces in Minkowski 3-space are locally either surfaces of revolution or null scrolls.

Semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In particular, we discuss singularities of (1) semi-discrete surfaces with non-zero constant Gaussian curvature, (2) parallel surfaces of semi-discrete minimal and maximal surfaces, and (3) semi-discrete constant mean curvature $1$ surfaces in de Sitter $3$-space. We include comparisons with different previously known definitions of such singularities.

A variational characterization of profile curves of invariant linear Weingarten surfaces

We show that the profile curve of any invariant linear Weingarten (LW) surface of a semi-Riemannian 3-space form, Mr3(ρ), is an extremal curve for a curvature energy. Moreover, we also give a construction procedure of LW surfaces from extremal curves of this energy, which allows us to understand invariant LW surfaces as binormal evolution surfaces with prescribed velocity. In particular, if the profile curve is planar, the variational problem depends on a total curvature type energy.We use this characterization to give a new approach to locally get all invariant constant Gaussian curvature surfaces whose profile curve is planar. Indeed, we also prove that the orbits of these surfaces are critical curves of the same energy acting on a different space of curves. Finally, restricting ourselves to the Riemannian case, M3(ρ), we study the existence of closed rotational surfaces with constant Gaussian curvature and we see that the planar profile curves of these rotational surfaces of M3(ρ) are just the horizontal projections of sub-Riemannian geodesics of semi-Riemannian unit tangent bundles.

Planar p-Elasticae and Rotational Linear Weingarten Surfaces

We variationally characterize the profile curves of rotational linear Weingarten surfaces as planar p-elastic curves. Moreover, by evolving these planar p-elasticae under the binormal flow with prescribed velocity, we describe a procedure to construct all rotational linear Weingarten surfaces, locally. Finally, we apply our findings to two well-known family of surfaces.

Polynomial conserved quantities of Lie applicable surfaces

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and L-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? By obtaining necessary and sucient conditions for the surfaces given by the two procedures to be congruent.The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation.

Generalized Weingarten surfaces of harmonic type in hyperbolic 3-space

In this paper we study a large class of Weingarten surfaces M with prescribed hyperbolic Gauss map in the Hyperbolic 3-space, which are the analogous to the Laguerre minimal surfaces in Euclidean space, these surfaces will be called Generalized Weingarten surfaces of harmonic type (HGW-surfaces), this class includes the surfaces of mean curvature one and the linear Weingarten surfaces of Bryant type (BLW-surfaces). We obtain a Weierstrass type representation for this surfaces which depend of three holomorphic functions. As applications we classify the HGW-surfaces of rotation and we obtain a Weierstrass type representation for surfaces of mean curvature one with prescribed hyperbolic Gauss map which depend of two holomorphic functions. Moreover, we classify a class of complete mean curvature one surfaces parametrized by lines of curvature whose coordinates curves has the same geodesic curvature up to sign.

On a Class of Linear Weingarten Surfaces

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional $k_{\mu}$ and parallel $k_{\pi}$, satisfy the relation $k_{\mu}=(n+1)k_{\pi}$, $n=0,\,1,\,2,\ldots\, .$ The first two members of this class of surfaces are the sphere $(n=0)$ and the Mylar balloon $(n=1)$. Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when $n=2$. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.

DISCRETE LINEAR WEINGARTEN SURFACES

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

A Liebmann type theorem for linear Weingarten surfaces

We deal with complete two-sided surfaces isometrically immersed in a Riemannian space form \(\mathbb Q^3_c\) of constant sectional curvature \(c\in \{-1,0,1\}\), satisfying a linear Weingarten condition of the type \(K=aH+b\), where a and b are constant and H and K denote the mean and Gaussian curvatures, respectively. In this setting, we show that these surfaces must be either totally umbilical or isometric to a hyperbolic cylinder \(\mathbb {H}^{1}(-\sqrt{1+r^{2}})\times \mathbb {S}^{1}(r)\), when \(c=-1\), a circular cylinder \(\mathbb {R}\times \mathbb {S}^{1}(r)\), when \(c=0\), or to a flat Clifford torus \(\mathbb {S}^{1}(\sqrt{1-r^{2}})\times \mathbb {S}^{1}(r)\), when \(c=1\). In particular, our result provides a new extension of Liebmann’s classical rigidity theorem.

ON THE OFFSETS OF RULED SURFACES IN EUCLIDEAN SPACE

In the present paper, we study evolute offsets of a non-developable ruled surface in Euclidean 3-space E and classify the evolute offset with constant Gaussian curvature and constant mean curvature. In last section, we investigate linear Weingarten evolute offsets in E . A linear Weingarten surface is the surface having a linear equation between the Gaussian curvature and the mean curvature of a surface. AMS Subject Classification: 53C30, 53B25

Classes of generalized Weingarten surfaces in the Euclidean 3-space

Abstract We present surfaces with prescribed normal Gaussmap. These surfaces are obtained as the envelope of a sphere congruencewhere the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coefficients are functions that depend on the support function and the distance function from a fixed point (in short, DSGW-surfaces). The linear Weingarten surfaces, Appell’s surfaces and Tzitzeica’s surfaces are all DSGW-surfaces. From them we obtain new classes of DSGWsurfaces applying inversions, dilatations and parallel surfaces. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSG Wsurfaces). As applications we classify the EDSGW-surfaces of rotation and present a 2-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes.