Surfaces In Three-dimensional Space Research Articles

Quantifying the Consequences of the Ill-Defined Nature of Neutral Surfaces

Abstract In the absence of diapycnal mixing processes, fluid parcels move in directions along which they do not encounter buoyant forces. These directions define the local neutral tangent plane. Because of the nonlinear nature of the equation of state of seawater, these neutral tangent planes cannot be connected globally to form a well-defined surface in three-dimensional space; that is, continuous “neutral surfaces” do not exist. This inability to form well-defined neutral surfaces implies that neutral trajectories are helical. Consequently, even in the absence of diapycnal mixing processes, fluid trajectories penetrate through any “density” surface. This process amounts to an extra mechanism that achieves mean vertical advection through any continuous surface such as surfaces of constant potential density or neutral density. That is, the helical nature of neutral trajectories causes this additional diasurface velocity. A water-mass analysis performed with respect to continuous density surfaces will have part of its diapycnal advection due to this diasurface advection process. Hence, this additional diasurface advection should be accounted for when attributing observed water-mass changes to mixing processes. Here, the authors quantify this component of the total diasurface velocity and show that locally it can be the same order of magnitude as diasurface velocities produced by other mixing processes, particularly in the Southern Ocean. The magnitude of this diasurface advection is proportional to the ocean’s neutral helicity, which is observed to be quite small in today’s ocean. The authors also use a perturbation experiment to show that the ocean rapidly readjusts to its present state of small neutral helicity, even if perturbed significantly. Additionally, the authors show how seasonal (rather than spatial) changes in the ocean’s hydrography can generate a similar vertical advection process. This process is described here for the first time; although the vertical advection due to this process is small, it helps to understand water-mass transformation on density surfaces.

Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a Z2 index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time-reversal or particle-hole conjugation. Finally, in the case of the sphere—mathematically speaking, three almost commuting Hermitians whose sum of square is near the identity—we give the first quantitative result, showing that this index is the only obstruction to finding commuting approximations. We review the known nonquantitative results for the torus.

A new approach to characterising aspheric surfaces

In this paper, a new approach to fitting aspheric surfaces in three-dimensional space is proposed, based on the nonlinear least-squares algorithm. For the conditions tested, there are good indications that this method provides better results than conventional solutions as all the surface parameters can be estimated simultaneously based on the design equation, thus allowing the result to be directly compared to design parameters. Conventionally, aspheric surfaces can be fitted with simplified surface models, such as a second order surface or polynomial model. Using this approach the estimated parameters cannot be compared with the design values, breaking the link between the designed and measured surface. The new method is developed here and tested on computer simulated aspheric surfaces. Both ideal surfaces and surfaces with random irregularities are considered. Issues regarding the application of the fitting method to real measured surfaces are discussed.

Differential Invariants of Conformal and Projective Surfaces

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.

An [formula omitted] integration scheme for three-dimensional surface scattering problems

We present an accurate method of O ( 1 ) -complexity with respect to frequency (i.e., a method that, to achieve a prescribed error tolerance, requires a bounded computational cost for arbitrarily high frequencies) for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three-dimensional space. Like the two-dimensional counterpart of this algorithm, which we introduced recently and which is applicable to scattering by curves in the plane, the present method is based on a combination of two main elements: (1) a high-frequency ansatz for the unknown density in a boundary integral formulation of the problem, and (2) an extension of the ideas of the method of stationary phase to allow for O ( 1 ) (high-order-accurate) integration of oscillatory functions. The techniques we introduce to implement an efficient O ( 1 ) integrator in the present three-dimensional context differ significantly from those used in the earlier two-dimensional algorithm. In particular, in the present text, we introduce an efficient “canonical” (hybrid analytic-numerical) algorithm which, in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase, can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two-dimensional scattering surface. We include numerical results illustrating the behavior of the integration algorithm on sound-soft spheres with diameters of up to 5000 wavelengths: in such cases, for a single integral, the algorithm yields accuracies of the order of three digits in computational times of less than two seconds. In a preliminary full scattering simulation we present, a solution with two digits of accuracy in the surface density was obtained in about three hours running time, in a single 1.5 GHz AMD Athlon processor, for a sphere of 500 wavelengths in diameter.

The pressure—temperature—concentration correlation for aqueous solutions of hydroperoxide

The results of analysis of available experimental data on the pressure dependence of temperature of nonequilibrium boiling of rapidly heated aqueous solutions of hydroperoxide are used to determine the pressure(P)—temperature(T)—concentration(x) correlation. This correlation is obtained for a solution under conditions of phase equilibrium (saturation surface in three-dimensional space of variables P, T, and x) in a wide range of states, including the critical line. The obtained surface is used to calculate the equilibrium composition of gas over the solution as a function of x and T by solving the Duhem equation.

Development of a New Interfacing Strategy for Fluid-Structure Interaction

Displacement and pressure transfer between fluid and structure meshes in fluidstructure interaction analysis is complicated substantially when the two meshes on the domain interface are different and therefore form two distinct surfaces in three-dimensional space. A new, approximately energy-conserving interfacing strategy with two distinct components - mapping and interpolation - is proposed. The mapping between meshes is based on the assumption of a common parametric-based description of the wetted surface. The interpolation strategy is based on the smoothing element analysis method developed to recover stresses in finite element analysis, with an additional term to impose an energy-conserving constraint. The method is evaluated numerically with several examples from linear hydrodynamics/hydroelasticity. The advantage of these examples is that the components can be evaluated separately, and the performance of the method can be examined more clearly. Numerical results show the method is quite promising.

A new analytical method for computing solvent‐accessible surface area of macromolecules and its gradients

In the calculation of thermodynamic properties and three-dimensional structures of macromolecules, such as proteins, it is important to have an efficient algorithm for computing the solvent-accessible surface area of macromolecules. Here, we propose a new analytical method for this purpose. In the proposed algorithm we consider the transformation that maps the spherical circles formed by intersection of the atomic surfaces in three-dimensional space onto the circles on a two-dimensional plane, and the problem of computing the solvent-accessible surface area is reduced to the problem of computing the corresponding curve integrals on the plane. This allows to consider only the integrals along the circular trajectories on the plane. The algorithm is suitable for parallelization. Testings on many proteins as well as the comparison to the other analogous algorithms have shown that our method is accurate and efficient.

A model for frictional slip in woven fabrics

A continuum model for frictional slip of the yarns of a plain-weave fabric is presented. The model is based on the assumption that the weave is composed of two families of continuously distributed yarns constrained at all times to occupy a common evolving surface in three-dimensional space. The two families may slide relative to one another on the surface, subject to their respective equations of motion, fiber constitutive equations, and frictional slip rules. The theory is intended for the quantitative analysis of deformation, slip and energy dissipation during a ballistic impact event. To cite this article: B. Nadler, D.J. Steigmann, C. R. Mecanique 331 (2003).

Method for aligning any-order polynomial surfaces in three-dimensional space

A theoretically sound and computationally inexpensive method is proposed for expressing in a canonical form not only quadratic, but any-degree polynomial surfaces, thus enabling the alignment of 3-D objects composed of such surfaces to a standard coordinate system. Furthermore, an analytical proof of the lower upper bound for convergence is presented, by which the efficiency of the proposed method is verified. Experiments are conducted with up to sixth-degree polynomial surfaces, thus demonstrating the method converges to the desired results within a few steps.

Surface scattering in three dimensions: an accelerated high–order solver

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in threedimensional space. This algorithm evaluates scattered fields through fast, highorder, ac...

A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order “two-face” equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds.

Stereological estimation of integral mixed curvature with application

This article addresses a bounded system of compact smooth surfaces in three-dimensional space. Recently, an unbiased estimator of the integral mixed curvature of the given system based on a vertical sampling design has been proposed. The aims of the present paper are: (i) to show that the proposed estimator may have an infinite variance; (ii) to suggest a modification that has better statistical properties; (iii) to extend the point estimator to a function that monitors the curvature along the gradient microstructures; (iv) to present an application of the method to real microscopic images.

Piecewise optimal triangulation for the approximation of scattered data in the plane

We present an efficient algorithm to obtain a triangulated graph surface for scattered points (x i y i) T , i=1,…, n, with associated function values f i . The maximal distance between the triangulated graph surface and the function values is measured in z-direction ( z= f( x, y)) and lies within a user-defined tolerance. The number of triangles is minimized locally by adapting their shapes to different second-degree least squares approximations of the underlying data. The method consists of three major steps: 1. subdividing the given discrete data set into clusters such that each cluster can be approximated by a quadratic polynomial within a prescribed tolerance; 2. optimally triangulating the graph surface of each quadratic polynomial; and 3. “stitching” the triangulations of all graph surfaces together. We also discuss modifications of the algorithm that are necessary to deal with discrete data points, without connectivity information, originating from a general two-manifold surface, i.e., a surface in three-dimensional space that is not necessarily a graph surface.

Spatial variability in the soil foodwebs in a contaminated grassland ecosystem

Spatial relationships among soil nematodes and soil microorganisms were investigated in a grassland ecosystem at the U.S. Army's Aberdeen Proving Ground, Maryland. This area was contaminated with a wide range of concentrations of heavy metals. The study quantified microbial biomass, the abundance of soil protozoa and nematode populations. Geostatistical techniques were used to evaluate cross-correlations among biotic and abiotic parameters and to determine spatial distributions of these parameters in soil. The graphical analysis technique displayed a surface in three-dimensional space, the topography of which was generated by an independent variable and the surface colored by a dependent variable. Both the visualization technique and the cross general relative semivariogram showed a negative correlation between the abundance of fungivore nematodes and fungal biomass, the abundance of bacterivore nematodes and bacterial biomass, and between the numbers of protozoa and microbial biomass in less-contaminated portions of the study site. Graphical analyses of the spatial distribution of soil biota showed that relationships among components of the soil foodweb were altered in the highly contaminated area compared to the less-contaminated local background area. This suggests that environmental conditions in the soil of the contaminated areas may have a detrimental effect on trophic interactions in the soil foodweb.

Shape Representation and Recognition from Multiscale Curvature

We present a technique for shape representation and the recognition of objects based on multiscale curvature information. It provides a single framework for both the decomposition and recognition of both planar curves as well as surfaces in three-dimensional space. The decomposition operation simultaneously performs data interpolation, data smoothing, and segmentation. The unification of these three stages results in a smoothing operation that is coupled with the primitives to be used in description. Each of the minimization operators, in addition to having a curvature tuning, also has a different spatial sensitivity function. As a result, the different possible descriptions capture information at multiple spatial scales. This allows a single region of an object to be described in more than one way, when appropriate. The practicality of the ensuing representation is demonstrated by the recognition of planar curves. A matching strategy based on dynamic programming is used. The results illustrate the manner in which a continuous spectrum of similar objects can be defined, ranging from those that are very similar to a target to those that are very different from it.

Particle modeling of liquid drop formation on a solid surface in 3-D

Using a molecular aggregate approach and classical Newtonian dynamics, we show how to simulate a liquid drop formation on a horizontal solid surface in three-dimensional space. We use sets of quasi-molecular particles which interact in accordance with classical molecular-type formulas. For application, the liquid is taken to be water while the solid surface is taken to be graphite. The resulting dynamical equations for the particles are large systems of second-order, nonlinear, ordinary differential equations which must be solved by a numerical method. Computer simulations of the results and related contact angle calculations are presented and discussed. The results are in complete agreement with experimentation and the method can be extended to nonflat surfaces.

Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing

The study of geometric flows for smoothing, multiscale representation, and analysis of two- and three-dimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heat-type flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three-dimensional images.

Numerical Finiteness Results for Minimal Surfaces in Three-Dimensional Space Forms

We present a new method, based on generalizations of Shiffman's variational principle [Nowak 1993; 1994], for the construction of minimal surfaceson Schwarzian chains in curved space forms. The main emphasis of our approach is on the computation of all minimal surfaces of genus zero (disks with holes) that span a given boundary configuration—even unstable ones. For many boundary configurations we derive numerical finiteness results on the number of minimal surfaces spanning a given boundary configuration. We use graphs of Shiffman's function to illustrate bifurcation phenomena and the Morse index of minimal surfaces. We also present some convergence results for the numerical method.

On integrable potential perturbations of the Jacobi problem for the geodesics on the ellipsoid

All integrable mechanical systems describing motion of a particle on an ellipsoid surface in three-dimensional space are described in the class of the Loran polynomial potentials. Two countable families of the basic solutions are obtained. Explicit formulae are given. The limit, when the considered system goes into the billiard system within an ellipse, is analysed, and the results are compared with those obtained previously, in relation to the billiard system.