Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.

Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions

This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.

Magnetic Field Curves and Magnetic Equipotential Surfaces around Crossing Electrical Wires Replacing Classical Magnetic Field Lines

This article is based on a recent model specifically defining magnetic field values around electrical wires. With this model, calculations of field around parallel wires were obtained. Now, this model is extended with the new concept of magnetic equipotential surface to magnetic field curves around crossing wires. Cases of single, double, and triple wires are described. Subsequent article will be conducted for more general scenarios where wires are neither infinite nor parallel.

A new method for specular curved surface defect inspection based on reflected pattern integrity

Defect inspection of specular curved surface is a challenging job. Taking steel balls for example, a new method based on reflected pattern integrity recognition is put forward. The specular steel ball surface will totally reflect the patterns when it is placed inside a dome-shaped light source, whose inner wall is modified by patterns with certain regular. Distortion or intermittence of reflected pattern will occur at the defective part, which indicates the pattern has lost its integrity. Based on the integrity analysis of reflected pattern images？ surface defects can be revealed. In this paper, a set of concentric circles are used as the pattern and an image processing algorithm is customized to extract the surface defects. Results show that the proposed method is effective for the specular curved surface defect inspection

STUDY ON THE METHOD OF CONSTRUCTING RATIONAL CUBIC CURVES AND CURVED SURFACE INCLUDING CONTROLLING PARAMETERS

By adopting the method of controlling parameters this paper describes the construction of various kinds of cubic curve segment and curved surface fragment with rational and non rational parameters, and discusses the relationship between controlling parameters, weighted factors and types, kinds and characteristics of curve segments and curved surface fragments. A mathematical method is provided for CAGD with abundant connotations, broad covering region, convenience, flexibility and direct simplicity.

RESEARCH ON NURBS SKINNING SURFACE VIA COMPLEX CURVES

In order to design the production with complex external shapes, a newmethod is put forward using non-uniform rational B-spline(NURBS)curves to unifythe description of complex curves composed of several segments with different degrees,and then these complex curves are used to construct NURBS skinning surface. Somekinds of skills are used to dispose the knot of NURBS curves and surfaces for practicalproblems. Finally, the method is verified by several complex examples.

Study on curved surface fitting model using GPS and leveling in local area

The height anomaly surface is fitt and the quasi-geoid can be obtained when the height anomaly is determined with the geometric analytic method. Therefore, some mathematical models to fit height anomaly surface using GPS, leveling and terrain data in a local area, including the polynomial fitting model, the multi-surface function fitting model, the motion surface fitting model and the fitting model of little flexibility deformation of thin board, are given. Then the digital characteristics are analyzed with the curved surface theory. The General curvature and the mean curvature of surface are concluded. The advantage, disadvantage and application of the above models are discussed. The effect of terrain undulation on height anomaly is considered in the surface fitting models. The practical case indicates that these models are of validity and practicability. It is concluded that the above models can give the good fitting results at the centimeter level. But the polynomial fitting model is worse than the other models.

A synthetic aperture radar sea surface distribution estimation by n-order Bézier curve and its application in ship detection

To dates,most ship detection approaches for single-pol synthetic aperture radar（SAR） imagery try to ensure a constant false-alarm rate（CFAR）.A high performance ship detector relies on two key components：an accurate estimation to a sea surface distribution and a fine designed CFAR algorithm.First,a novel nonparametric sea surface distribution estimation method is developed based on n-order Bézier curve.To estimate the sea surface distribution using n-order Bézier curve,an explicit analytical solution is derived based on a least square optimization,and the optimal selection also is presented to two essential parameters,the order n of Bézier curve and the number m of sample points.Next,to validate the ship detection performance of the estimated sea surface distribution,the estimated sea surface distribution by n-order Bézier curve is combined with a cell averaging CFAR（CA-CFAR）.To eliminate the possible interfering ship targets in background window,an improved automatic censoring method is applied.Comprehensive experiments prove that in terms of sea surface estimation performance,the proposed method is as good as a traditional nonparametric Parzen window kernel method,and in most cases,outperforms two widely used parametric methods,K and G0 models.In terms of computation speed,a major advantage of the proposed estimation method is the time consuming only depended on the number m of sample points while independent of imagery size,which makes it can achieve a significant speed improvement to the Parzen window kernel method,and in some cases,it is even faster than two parametric methods.In terms of ship detection performance,the experiments show that the ship detector which constructed by the proposed sea surface distribution model and the given CA-CFAR algorithm has wide adaptability to different SAR sensors,resolutions and sea surface homogeneities and obtains a leading performance on the test dataset.

Programmable robotized‘transfer-and-jet’printing for large,3D curved electronics on complex surfaces

Large,3D curved electronics are a trend of the microelectronic industry due to their unique ability to conformally coexist with complex surfaces while retaining the electronic functions of 2D planar integrated circuit technologies.However,these curved electronics present great challenges to the fabrication processes.Here,we propose a reconfigurable,mask-free,conformal fabrication strategy with a robot-like system,called robotized‘transfer-and-jet’printing,to assemble diverse electronic devices on complex surfaces.This novel method is a ground-breaking advance with the unique capability to integrate rigid chips,flexible electronics,and conformal circuits on complex surfaces.Critically,each process,including transfer printing,inkjet printing,and plasma treating,are mask-free,digitalized,and programmable.The robotization techniques,including measurement,surface reconstruction and localization,and path programming,break through the fundamental constraints of 2D planar microfabrication in the context of geometric shape and size.The transfer printing begins with the laser lift-off of rigid chips or flexible electronics from donor substrates,which are then transferred onto a curved surface via a dexterous robotic palm.Then the robotic electrohydrodynamic printing directly writes submicrometer structures on the curved surface.Their permutation and combination allow versatile conformal microfabrication.Finally,robotized hybrid printing is utilized to successfully fabricate a conformal heater and antenna on a spherical surface and a flexible smart sensing skin on a winged model,where the curved circuit,flexible capacitive and piezoelectric sensor arrays,and rigid digital–analog conversion chips are assembled.Robotized hybrid printing is an innovative printing technology,enabling additive,noncontact and digital microfabrication for 3D curved electronics.

Conformal manufacturing of soft deformable sensors on the curved surface

Health monitoring of structures and people requires the integration of sensors and devices on various 3D curvilinear,hierarchically structured,and even dynamically changing surfaces.Therefore,it is highly desirable to explore conformal manufacturing techniques to fabricate and integrate soft deformable devices on complex 3D curvilinear surfaces.Although planar fabrication methods are not directly suitable to manufacture conformal devices on 3D curvilinear surfaces,they can be combined with stretchable structures and the use of transfer printing or assembly methods to enable the device integration on 3D surfaces.Combined with functional nanomaterials,various direct printing and writing methods have also been developed to fabricate conformal electronics on curved surfaces with intimate contact even over a large area.After a brief summary of the recent advancement of the recent conformal manufacturing techniques,we also discuss the challenges and potential opportunities for future development in this burgeoning field of conformal electronics on complex 3D surfaces.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Young’s modulus surface and Poisson’s ratio curve for tetragonal crystals

This paper gives the general expressions for the compliance s′ijkl, Young＇s modulus E（hkl） and Poisson＇s ratio v（hkl, θ） along arbitrary loading direction [hkl] for tetragonal crystals. The representation surface for which the length of the radius vector in the [hkl] direction equals E（hkl） and representation curve for which the length of the radius vector with angle θ deviated from the reference directions [001^-], [100], [001^-], [101^-] and [112^-] equals v（100, θ）, v（001, θ）, v（110,θ）, v（101,θ） and v（111, θ） respectively, are constructed for nine tetragonal crystals （ammonium dihydrogen arsenate, ammonium dihydrogen phosphate, barium titanate, indium, nickel sulfate, potassium dihydrogen arsenate, potassium dihydrogen phosphate, tin and zircon）. The characteristics of them are analysed in detail.

Curved surface effect and emission on silicon nanostructures

The curved surface （CS） effect on nanosilicon plays a main role in the activation for emission and photonic manipulation. The CS effect breaks the symmetrical shape of nanosilicon on which some bonds can produce localized electron states in the band gap. The investigation in calculation and experiment demonstrates that the different curvatures can form the characteristic electron states for some special bonding on the nanosilicon surface, which are related to a series of peaks in photoluminecience （PL）, such as LN, LNO, Lo1, and Lo2 lines in PL spectra due to Si-N, Si-NO, Si=O, and Si-O-Si bonds on curved surface, respectively. Si-Yb bond on curved surface of Si nanostructures can provide the localized states in the band gap deeply and manipulate the emission wavelength into the window of optical communication by the CS effect, which is marked as the Lyb line of electroluminescence （EL） emission.

Residual strain evaluation of curved surface by grating-transferring technique and GPA

This paper investigates an advanced grating-transferring technique combined with geometric phase analysis (GPA) for residual strain evaluation of curved surface.A standard holographic grating is first transferred to a pre-produced epoxy resin film and then consolidated to a test region of curved surface.With a rubber mold and silicone rubber the deformed grating is replicated to a sheet metal after hole-drilling for release of residual stress.After that the grating is transferred from the sheet metal to the glass plate,which would be served as an analyzer grating (specimen grating).By GPA the local strain distributions related to the phase difference between the reference grating and analyzer grating for the released stress can be evaluated.A validation test has been conducted on the weld joint of a stainless steel tube and the obtained results demonstrate the ability of the method in measuring the residual strain of curved surface.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

Magnetorheological finishing of low-gradient curved surfaces based on four-axis linkage technique

Based on the distribution characteristic of magnetic field along the polish wheel,the four-axis linkage technique is advanced to replace a standard five-axis one to figure low-gradient optical surfaces with a raster tool-path in magnetorheological finishing(MRF).After introducing the fundaments of such simplification,the figuring reachability of a four-axis system for the low-gradient optics was theoretically analyzed.Further validation including magnetic field intensity and influence function characteristic was performed to establish its application.To demonstrate the correctness,feasibility and applicability of such technique,a K4 spherical part was figured by two iterations of MRF with surface form error improved to 0.219λPV and 0.027λRMS.Meanwhile,the surface roughness was also improved a lot in MRF process.These theoretical analyses and experimental results both indicate that high form accuracy and excellent surface quality can be obtained by using the four-axis linkage technique in the process of figuring low-gradient optical elements,and the four-axis linkage system undoubtedly is much more easy to control and much more economical.

Subdivision surface modeling system based on arbitrary topological curves network and combined subdivision

Arbitrary topological curve network has no restriction in topology structure,so it has more powerful representing ability in defining complex surfaces.A complex surface modeling system is presented based on arbitrary topological curve network and the improved combined subdivision method,its functions including creating and editing curve network,and generating and modifying curve network＇s interpolated surface.This modeling system can be used to the process of products＇concept design,and its applications is also significant to the development of subdivision method.

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000, Wu presented two new types of generalized Ball curves, one of which is called an NB1 curve located between the Wang-Ball curve and the Said-Ball curve. In this article, the authors aim to discuss properties of NB1 curves and surfaces, including the recursive algorithms, conversion algorithms between NB1 and Bezier curves and surfaces, etc. In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.

A fast direct point-by-point generating algorithm for B Spline curves and surfaces

Traditional generating algorithms for B Spline curves and surfaces require approximation methods where how to increment the parameter to get the best approximation is problematic; or they take the pixel-based method needing matrix trans- formation from B Spline representation to Bézier form. Here, a fast, direct point-by-point generating algorithm for B Spline curves and surfaces is presented. The algorithm does not need matrix transformation, can be used for uniform or nonuniform B Spline curves and surfaces of any degree, and has high generating speed and good rendering accuracy.

Numerical treatment for Darcy-Forchheimer flow of carbon nanotubes due to an exponentially stretching curved surface

This article gives a numerical report to two dimensional(2D)Darcy-Forchheimer flow of carbon-water nanofluid.Flow is instigated by exponential extending curved surface.Viscous liquid in permeable space is described by Darcy-Forchheimer.The subsequent arrangement of partial differential equations is changed into ordinary differential framework through proper transformations.Numerical arrangements of governing frameworks are set up by NDSolve procedure.Outcomes of different sundry parameters on temperature and velocity are examined.Skin friction and heat transfer rate are also shown and inspected.