Isoparametric Surfaces Research Articles

Polyharmonic surfaces in 3-dimensional homogeneous spaces

In the first part of this paper we shall classify proper triharmonic isoparametric surfaces in 3-dimensional homogeneous spaces (Bianchi-Cartan-Vranceanu spaces, shortly BCV-spaces). We shall also prove that triharmonic Hopf cylinders are necessarily CMC. In the last section we shall determine a complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces, r≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\ge 3$$\\end{document}. This result ensures the existence, for suitable values of r, of an ample family of new examples of r-harmonic surfaces in BCV-spaces.

Structure-aware geometric optimization of hexahedral mesh

Current geometric optimization methods do not fully consider the overall structure of the model, which may make the effect of geometric optimization of the hex meshes for the models with concave curves unsatisfactory. To this end, this paper proposes an approach to structure-aware geometric optimization of hex meshes. In the approach, by relocating the position of the base complex of the hex mesh, the overall parameterized energy is reduced to obtain an optimized geometric embedding. First, a frame field which conforms to the base complex structure of the input hex mesh is generated. Then, a parameterization that satisfies the structure constraints is established based on the frame field. Finally, the positions of the singular lines of the hex mesh are optimized according to the gradient of the parameterization energy function, and the key isoparametric surfaces are used as the position constraints of the vertices of the patches of the base complex to optimize the whole hex mesh. The experimental results show that the scaled Jacobian value of hex meshes can be effectively improved by the proposed method.

Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi–Cartan–Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only nontrivial example of a totally biharmonic surface appears in the product space [Formula: see text].

Gauss Map and Local Approach of Isoparametric Surfaces in Lorentz and Euclidean Space

In this study, we determine the isoparametric surfaces and we give the Gauss map of these surfaces by semi symmetric matrix, in Lorentz space. Also we define any chord property and we show that the surfaces which have the chord property corresponds to isoparametric surfaces. Moreover, we consider the chord property locally and we give some examples in the Euclidean space.

Isoparametric surfaces in $E(\kappa,\tau)$-spaces

We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential.

A Characterization of Isoparametric Surfaces in Space Forms via Minimal Surfaces

We give a characterization of isoparametric surfaces in three dimensional space forms with the help of minimal surfaces. Let us consider a surface in a three dimensional space form with the following property: through each point of the surface pass three curves such that the ruled orthogonal surface determined by each of them is minimal. We prove that necessarily the initial surface is isoparametric. It is also shown, that the curves are necessarily geodesics. On other hand, using the classification of isoparametric surfaces it is possible to prove that they have the above property along every geodesic. So, we have a characterization. As a preliminary result we prove that a surface with two geodesics, through every point, which are helices of the ambient, has parallel shape operator.

W-Curves in Lorentz-Minkowski Space

In this paper, we investigate the chord properties of the non-null W-curves in Lorentz-Minkowski space. We give the general equation form for W-curves in (2n+1)-dimension. We define some special curves and give the relations between these curves and isoparametric surfaces. Finally we obtain the geodesics of the pseudospherical cylinder and pseudohyperbolic cylinder in 4-dimensional space.

Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

We study constant angle semi-Riemannian hypersurfaces M immersed in semi-Riemannian space forms, where the constant angle is defined in terms of a closed and conformal vector field Z in the ambient space form. We show that such hypersurfaces belong to the class of hypersurfaces with a canonical principal direction. This property is a type of rigidity. We further specialize to the case of constant mean curvature (CMC) hypersurfaces and characterize them in two relevant cases: when the hypersurface is orthogonal to Z then it is totally umbilical, whereas if Z is tangent to the hypersurface then it has zero Gauss–Kronecker curvature and either its mean curvature vanishes or the ambient is a semi-Euclidean space. We also treat in detail the surface case, giving a full characterization of the constant angle CMC surfaces immersed in all three dimensional space forms. They are isoparametric surfaces with constant principal curvatures when the ambient is flat. If the mean curvature of the surface is not ±2/3 they are either totally umbilic or totally geodesic. In particular when the surface has zero mean curvature it is totally geodesic.

A data-centric algorithm for automated detection and extraction of isoparametric surfaces

The traditional workflow for high-performance computing simulation is often to prepare input, run a simulation, and visualize the results as a post-processing step. In the biomedical and seismic industries, these results comprise uniform 3D arrays that can approach tens of petabytes depending on the domain. Visually exploring output data requires significant system resources and time, as data is moved between the simulation cluster and the visualization cluster. Resources and time can be conserved if the simulation and visualization can access the same system resources and data. End-to-end workflow time can be decreased if the simulation and visualization can be performed simultaneously. Data-centric visualization provides a platform in which the same high-performance server can execute both the simulation and visualization. In this paper, we discuss a visualization framework for exploring very-large data sets using both direct and isoparametric point extraction volume rendering techniques. Our design considers accelerators available in next-generation servers using IBM Power technology and GPUs (graphics processing units). GPUs can accelerate generation and compression of two-dimensional display images that can be transferred across a network to a variety of display devices. Users will be able to remotely steer visualization and simulation applications. In this paper, we discuss an early implementation and additional challenges for future work.

Filtering and Gradient Estimation for Distance Fields by Quadratic Regression

Distance fields show up in many problems of 3D vision and rendering, for example, a volumetric fusion of depth images results in such a field. Distance fields obtained from measured values are inherently noisy, so its filtering is needed before isoparametric surfaces are extracted from them, and robust normal vector estimation also requires a local smoothing since differentiation is especially sensitive to noise. In this paper, we use regression to find a quadratic function that approximates the zero level surface of the distance field, and apply this both for filtering and normal vector estimation. We also present a computationally efficient method that exploits the regular structure of samples, the symmetry and separability of the weighting functions, and thus avoids the solution of larger linear equations, which otherwise would become necessary when regression is generally attacked. The algorithm is a part of a real-time volumetric fusion application running on the Graphics Processing Units (GPU).

Electromagnetic modeling of composite metallic and dielectric structures

A new, general, and very efficient method for analysis of arbitrary composite metallic and dielectric structures, based on the PMCHW formulation and Galerkin method, is presented in this paper. Flexible geometrical modeling is performed by isoparametric surfaces (i.e., by bilinear surfaces in the particular case). Efficient approximation of currents is achieved by using polynomial entire-domain expansions (i.e., rooftop subdomain expansions in the particular case) that automatically satisfy the continuity equation, assuming that there are no line charges along surface edges. Special care is devoted to the treatment of arbitrary multiple metallic and/or dielectric junctions. Numerical results for different structures, obtained by using an extremely small numbers of unknowns, show very good agreement with other available data.

An improved method for contouring on isoparametric surfaces

AbstractA general non‐linear interpolation procedure for contouring on any isoparametric surface was presented in a previous article. The method used standard isoparametric interpolation functions and a predictor method for tracing element contour lines. The method presented here extends the previous work by using a predictor‐corrector method to trace element contour lines, thereby making the contouring algorithm more accurate.