The focal geometry of circular and conical meshes

Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the three-dimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.      

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.     

Oriented Mixed Area and Discrete Minimal Surfaces

Recently a curvature theory for polyhedral surfaces has been established, which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed area of a derived polygon pair which has only the half number of vertices. Thus we are able to recursively characterize vanishing mixed area for hexagons and other n-gons in an incidence-geometric way. We use these geometric results for the construction of discrete minimal surfaces and a study of equilibrium forces in their edges, especially those with the combinatorics of a hexagonal mesh.     

General curvature estimates for stable H-surfaces immersed into a space form

In this paper, we give general curvature estimates for constant mean curvature surfaces immersed into a simply-connected 3-dimensional space form. We obtain bounds on the norm of the traceless second fundamental form and on the Gaussian curvature at the center of a relatively compact stable geodesic ball (and, more generally, of a relatively compact geodesic ball with stability operator bounded from below). As a by-product, we show that the notions of weak and strong Morse indices coincide for complete non-compact constant mean curvature surfaces. We also derive a geometric proof of the fact that a complete stable surface with constant mean curvature 1 in the usual hyperbolic space must be a horosphere.     

Affine translation surfaces with constant sectional curvature

In this paper we characterize affine translation surfaces with constant Gaussian curvature. We show that such surfaces must be flat and that one of the defining curves must be planar.     

On semidiscrete constant mean curvature surfaces and their associated families

The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete \(\sinh \)-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.     

Twisted Surfaces with Null Rotation Axis in Minkowski 3-Space

We examine curvature properties of twisted surfaces with null rotation axis in Minkowski 3-space. That is, we study surfaces that arise when a planar curve is subject to two synchronized rotations, possibly at different speeds, one in its supporting plane and one of this supporting plane about an axis in the plane. Moreover, at least one of the two rotation axes is a null axis. As is clear from its construction, a twisted surface generalizes the concept of a surface of revolution. We classify flat, constant Gaussian curvature, minimal and constant mean curvature twisted surfaces with a null rotation axis. Aside from pseudospheres, pseudohyperbolic spaces and cones, we encounter B-scrolls in these classifications. The appearance of B-scrolls in these classifications is of course the result of the rotation about a null axis. As for the cones in the classification of flat twisted surfaces, introducing proper coordinates, we prove that they are determined by so-called Clelia curves. With a Clelia curve we mean a curve that has linear dependent spherical coordinates.     

On the semidiscrete differential geometry of A-surfaces and K-surfaces

In the category of semidiscrete surfaces with one discrete and one smooth parameter we discuss the asymptotic parametrizations, their Lelieuvre vector fields, and especially the case of constant negative Gaussian curvature. In many aspects these considerations are analogous to the well known purely smooth and purely discrete cases, while in other aspects the semidiscrete case exhibits a different behaviour. One particular example is the derived T-surface, the possibility to define Gaussian curvature via the Lelieuvre normal vector field, and the use of the T-surface??s regression curves in the proof that constant Gaussian curvature is characterized by the Chebyshev property. We further identify an integral of curvatures which satisfies a semidiscrete Hirota equation.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

Constant Gaussian Curvature Surfaces with Parallel Mean Curvature Vector in Two-Dimensional Complex Space Forms

We classify constant Gaussian curvature surfaces with nonzero parallel mean curvature vector in two-dimensional complex space forms. As a result, we find new examples of such surfaces.     

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

A Laplace Operator on Semi-Discrete Surfaces

This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.     

Darboux transforms and spectral curves Darboux transforms and spectral curves of constant mean curvature surfaces revisited

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Translation surfaces of constant curvatures in a simply isotropic space

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.     

Generalized surfaces with constant <Emphasis Type="Italic">H</Emphasis>/<Emphasis Type="Italic">K</Emphasis> in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Björling formula to construct such surfaces with prescribed point or curve singularities.     

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001