Parametric representation of a surface pencil with a common line of curvature

Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed.     

Parametric representation of a surface pencil with a common asymptotic curve

In this paper, we study the problem of finding a surface pencil from a given spatial asymptotic curve. We obtain the parametric representation for a surface pencil whose members have the same curve as a given asymptotic curve. Using the Frenet frame of the given asymptotic curve, we present the surface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be asymptotic. We illustrate this method by presenting some examples.     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

Reduce the stretch in surface flattening by finding cutting paths to the surface boundary

This paper presents a method for finding cutting paths on a 3D triangular mesh surface to reduce the stretch in the flattened surface. The cutting paths link the surface boundary and the nodes where the Gaussian curvature is high, and their total length is minimized. First, a linear algorithm for computing an approximate boundary geodesic distance map is introduced; the map encapsulates the undirected geodesic distance from every triangular node to the surface boundary approximately. This is followed by determining the undirected shortest paths passing through all the nodes where the Gaussian curvature is larger than a threshold. The cutting paths walk along the triangular edges of the given surface. Compared with other similar approaches, our method reaches a faster speed, and can deal with surfaces with widely distributed curvatures.     

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.     

On the geodesic voronoi diagram of point sites in a simple polygon

Given a simple polygon withn sides in the plane and a set ofk point sites in its interior or on the boundary, compute the Voronoi diagram of the set of sites using the internal geodesic distance inside the polygon as the metric. We describe anO((n + k) log(n + k) logn)-time algorithm for solving this problem and sketch a fasterO((n + k) log(n + k)) algorithm for the case when the set of sites includes all reflex vertices of the polygon in question.Work on this paper was performed while the author held an AT&T Bell Laboratories Ph.D. Scholarship at New York University.     

Freeform surface flattening based on fitting a woven mesh model

This paper presents a robust and efficient surface flattening approach based on fitting a woven-like mesh model on a 3D freeform surface. The fitting algorithm is based on tendon node mapping (TNM) and diagonal node mapping (DNM), where TNM determines the position of a new node on the surface along the warp or weft direction and DNM locates a node along the diagonal direction. During the 3D fitting process, strain energy of the woven model is released by a diffusion process that minimizes the deformation between the resultant 2D pattern and the given surface. Nodes mapping and movement in the proposed approach are based on the discrete geodesic curve generation algorithm, so no parametric surface or pre-parameterization is required. After fitting the woven model onto the given surface, a continuous planar coordinate mapping is established between the 3D surface and its counterpart in the plane, based on the idea of geodesic interpolation of the mappings of the nodes in the woven model. The proposed approach accommodates surfaces with darts, which are commonly utilized in clothing industry to reduce the stretch of surface forming and flattening. Both isotropic and anisotropic materials are supported.     

WireWarping: A fast surface flattening approach with length-preserved feature curves

This paper presents a novel approach — WireWarping for computing a flattened planar piece with length-preserved feature curves from a 3D piecewise linear surface patch. The property of length-preservation on feature curves is very important to industrial applications for controlling the shape and dimension of products fabricated from planar pieces. WireWarping simulates warping a given 3D surface patch onto plane with the feature curves as tendon wires to preserve the length of their edges. During warping, the surface-angle variations between edges on wires are minimized so that the shape of a planar piece is similar to its corresponding 3D patch. Two schemes — the progressive warping and the global warping schemes are developed, where the progressive scheme is flexible for local shape control and the global scheme gives good performance on highly distorted patches. Experimental results show that WireWarping can successfully flatten surface patches into planar pieces while preserving the length of edges on feature curves.     

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.     

Matrix pencil method for simultaneously estimating azimuth and elevation angles of arrival along with the frequency of the incoming signals

In this paper we describe a method for simultaneously estimating the direction of arrival (DOA) of the signal along with its unknown frequency. In a typical DOA estimation problem it is often assumed that all the signals are arriving at the antenna array at the same frequency which is assumed to be known. The antenna elements in the array are then placed half wavelength apart at the frequency of operation. However, in practice seldom all the signals arrive at the antenna array at a single pre-specified frequency, but at different frequencies. The question then is what to do when there are signals at multiple frequencies, which are unknown. This paper presents an extension of the matrix pencil method to simultaneously estimate the DOA along with the operating frequency of each of the signals. This novel approach involves approximating the voltages that are induced in a three-dimensional antenna array, by a sum of complex exponentials by jointly estimating the direction of arrival (both azimuth and elevation angles) along with the carrier frequencies of multiple far-field sources impinging on the array by using the three-dimensional matrix pencil method. The matrix pencil method is a direct data domain method for approximating a function by a sum of complex exponentials in the presence of noise. The variances of the estimates computed by the matrix pencil method are quite close to the Cramer–Rao bound. Finally, we illustrate how to carry out the broadband DOA estimation procedure using realistic antenna elements located in a conformal array. Some numerical examples are presented to illustrate the applicability of this methodology in the presence of noise. It is shown that the variance decreases as the SNR increases. The Cramer–Rao bound for the estimators are also provided to illustrate the accuracy and the computational efficiency of this new methodology.     

Interactive surface representation system using a B-spline formulation with interpolation capability

An interactive surface representation system is described which uses a parametric uniform bicubic B-spline formulation which can describe a surface initially defined to interpolate a specified network of points.     

A plot drainage network as a conceptual tool for the spatial representation of surface flow pathways in agricultural catchments

The drainage network must take the farming systems and the landscape structure into consideration to describe flow pathways in the agricultural catchment. A new approach is proposed to build the drainage network which is based on the identification of the inlets and outlets for surface water flow on each farmers’ field (or plot), estimating the relative areas contributing to the surface yield. The delineation of these areas and their links in terms of surface flow pathways provides us with a pattern of relationships between individual plots, i.e. going from each plot to the other plots over the entire catchment. In this approach, flow directions are firstly calculated in the usual way by taking account of slope direction. Plot outlets are defined from the digital elevation model (DEM) and then linked together using a tree structure. If present, linear networks such as hedges modify both the flow directions and the location of plot outlets, hence modify this tree structure. In a final step, the plots are themselves linked together using a graph structure illustrated by an arrow diagram. This drainage network based on plot outlets is applied to a 15-km² catchment area represented by 38,300 pixels and 2000 plots. This new drainage network takes into consideration 5300 plot outlets, which greatly reduces the number of objects in comparison with a drainage network made up of pixels or DEM cells. This method leads to a simple and functional representation of surface flow pathways in an agricultural catchment. It allows us to identify the key plots controlling stream water pollution where converging flow pathways are coming from numerous or large-sized plots. Finally, it produces a functional representation for decision support.     

Singularity formation in a model for the vortex sheet with surface tension

In a recent analytical study, the author has proved well-posedness of the vortex sheet with surface tension. This work included using a formulation of the problem introduced by Hou, Lowengrub, and Shelley for a numerical study of the same problem. The analytical study required identification of a term in the evolution equations which can be viewed as being responsible for the Kelvin–Helmholtz instability; this term is of lower order than the surface tension term. In the present work, the author introduces a simple model for the vortex sheet with surface tension which maintains the same dispersion relation and the same destabilizing force as in the vortex sheet with surface tension. For the model problem, it is found that finite-time singularities can form when the initial data is taken from a certain class. For the vortex sheet with surface tension, the only observed singularities thus far in numerical work have coincided with self-intersection of the fluid interface. There is no analogue of self-intersection in the model problem, and thus the singularities observed in the present work may well be related to a previously unobserved singularity for the full vortex sheet problem.     

3D Surface-Tracking with a robot manipulator

This paper presents incremental research in Surface Tracking with a robot manipulator. Surface tracking is an important operation in self-teaching and exploratory tasks in unknown environments, and to cope with inaccurate workspace and environment modeling. The paper addresses the problem of 3D Surface-Tracking in contact, where the main concern is the angle formed between the end-effector and the surface. This study constitutes a first approach to the more general and important problem of surface following in contact. The new contribution is the 3D tracking operation based on a new alignment control algorithm using real-time contact force feedback. Simulations, and experimental results using the PUMA 560 manipulator demonstrate the feasibility of the proposed algorithm. This paper also presents a method for tool-tip contact frame calibration using forces/moments data.     

Dynamic Delaunay tetrahedralisation of a deforming surface

Reconstruction algorithms make it possible to retrieve a surface from the Delaunay tetrahedralisation (DT) of a point sampling, whose density reflects the surface local geometry and thickness. Most of these algorithms are static and some work remains to be done to handle deforming surfaces. In such case, we defend the idea that each point of the sampling should move with the surface using the information given by the motion to allow fast reconstruction. In this article, we tackle the problem of producing a good evolving sampling of a deforming surface S, and maintaining its DT along the motion. The surface is known only through a projection operator (O ₁):ℝ³→S, and a normal operator (O ₂) that returns the oriented normal at a point on the surface. On that basis, we offer some perspectives on how reconstruction algorithms can be extended to the tracking of deforming surfaces.     

Aircraft identification using a bilinear surface representation of radar data

Low frequency radar scattering data is used for the identification of aircraft. It is shown that such radar data lies on two-dimensional surfaces in n-space. A bilinear approximation for these surfaces is described. Surface intersections using this approximation can be found simply and directly without solving a system of n simultaneous nonlinear equations. This intersection information can be used to show separability and effect feature reduction. The approximation is utilized to construct a modified nearest neighbor algorithm, which is evaluated by computer simulation experiments. These experiments showed a phenomenon of “bias”, where one aircraft data surface is more susceptible to misclassification in the presence of noise than the surface corresponding to another aircraft. This “bias” observed is shown to be related to the surface characteristics of the data surfaces involved, specifically proximity and relative curvature of corresponding points on the two surfaces.     

Improved hydrogen-sensing performance of a Pd/GaN Schottky diode with a surface plasma treatment approach

The improved hydrogen-sensing performance of a Pd/GaN Schottky diode with a simple surface treatment is demonstrated. The studied device with an inductively coupled-plasma (ICP)-treatment shows both the good sensitivity and fast response. A high hydrogen detection sensing response of 2.05 × 10⁵, under exposing to a 10,000 ppm H₂/air gas at room temperature, is obtained. It is found that, due to the increased surface roughness, more hydrogen atoms are adsorbed on the active layer which leads to the substantial increase of current change. In addition, the studied device shows a stable and widespread reverse voltage operating regime (−0.3 to −3 V) and a fast response about of 2.9 s. Therefore, this simple surface treatment approach gives the promise for hydrogen sensing applications.     

The spreading of a viscous microdrop on a solid surface

The spreading of a liquid microdrop across a solid surface is examined using the interface formation model. This model allows for variable surface tension at constant temperature and a flow induced Maragoni effect, by incorporating irreversible thermodynamics into the continuum model. The model is solved for small Capillary number and small Reynolds number. This problem has been considered before for much larger drops in Shikhmurzaev (Phys Fluids 9:266, 1997a), which examined the spreading of a drop for ε = τ U _CL/R ≪ 1, where U _CL is the speed of the moving contact line across the solid surface, τ is the surface tension relaxation time of the viscous liquid, and R is a typical length scale for the size of the drop. This paper extends that work by examining ε = O(1), which will be shown to be the appropriate scaling for very small liquid drops, on the scale of micrometres or less.     

Detection of protein aggregation with a Bloch surface wave based sensor

We present the innovative application of a Bloch surface wave based sensor to the detection of protein aggregation. In Hen Egg White Lysozyme (HEWL) solutions, aggregates are discriminated from the monomeric forms in a label-free detection scheme.     

Computing point-set surfaces with controlled spatial variation of residuals

This paper presents an accurate method for computing point-set surfaces from input data that can suppress the noise effect in the resulting point-set surface. This is accomplished by controlling spatial variation of residual errors between the input data and the resulting point-set surface and offsetting any systematic bias. More specifically, this method first reduces random noise of input data based on spatial autocorrelation statistics: the statistics Z via Moran’s I. The bandwidth of the surface is adjusted until the surface reaches desired value of the statistics Z corresponding to a given significance level. The method then compensates for potential systematic bias of the resultant surface by offsetting along computed normal vectors. Computational experiments on various point sets demonstrate that the method leads to an accurate surface with controlled spatial variation of residuals and reduced systematic bias.