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.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

Cubic Algebraic Spline Curves Design

In this paper we propose a construction method of the planar cubic algebraic splinecurve with endpoint interpolation conditions and a specific analysis of its properties. Thepiecewise cubic algebraic curve has G2 continuous contact with the control polygon at twoendpoints and is G2 continuous between each segments of itself. The process of this method issimple and clear, and provides a new way of thinking to design implicit curves.

CURDIS:A template for incremental curve discretization algorithms and its application to conics

We introduce CURDIS,a template for algorithms to discretize arcs of regular curves by incrementally producing a list of support pixels covering the arc.In this template,algorithms proceed by finding the tangent quadrant at each point of the arc and determining which side the curve exits the pixel according to a tailored criterion.These two elements can be adapted for any type of curve,leading to algorithms dedicated to the shape of specific curves.While the calculation of the tangent quadrant for various curves,such as lines,conics,or cubics,is simple,it is more complex to analyze how pixels are traversed by the curve.In the case of conic arcs,we found a criterion for determining the pixel exit side.This leads us to present a new algorithm,called CURDIS-C,specific to the discretization of conics,for which we provide all the details.Surprisingly,the criterion for conics requires between one and three sign tests and four additions per pixel,making the algorithm efficient for resource-constrained systems and feasible for fixed-point or integer arithmetic implementations.Our algorithm also perfectly handles the pathological cases in which the conic intersects a pixel twice or changes quadrants multiple times within this pixel,achieving this generality at the cost of potentially computing up to two square roots per arc.We illustrate the use of CURDIS for the discretization of different curves,such as ellipses,hyperbolas,and parabolas,even when they degenerate into lines or corners.

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.

C^n-CONTINUOUS B-TYPE SPLINE CURVES WITH ITSLOCALIZATION INTERPOLATION AND APPROXIMATION

This paper presents a class of Cn- continuous B- type spline curves with some paramet- ric factors.The length of their local support is equal to4.Taking the different values of the parametric factors,the curves can become free- type curves or interpolate a set of given points even mix the both cases.When the parametric factors satisfy the certain conditions,the degrees of the curves can be decreased as low as possible.Besides,when all the parametric factors tend to zero,the curves globally approximate to the control polygon.

Planar Cardinal Spline Curves with Minimum Strain Energy

In this paper, we focus on how to use strain energy minimization to obtain the optimal value of the fl＇ee parameter of the planar Cardinal spline curves. The unique solution can be easily obtained by minimizing an appropriate approximation of the strain energy. An example is presented to illustrate the effectiveness of our method.

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves,the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed.The quartic Hermite spline curves not only have the same interpolation and continuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C;continuity by the shape parameters when the interpolation conditions are fixed.

One Fairing Method of Cubic B-spline Curves Based on Weighted Progressive Iterative Approximation

A new method to the problem of fairing planar cubic B-spline curves is introduced in this paper. The method is based on weighted progressive iterative approximation （WPIA for short） and consists of following steps： finding the bad point which needs to fair, deleting the bad point, re-inserting a new data point to keep the structm-e of the curve and applying WPIA method with the new set of the data points to obtain the faired curve. The new set of the data points is formed by the rest of the original data points and the new inserted point. The method can be used for shape design and data processing. Numerical examples are provided to demonstrate the effectiveness of the method.

PH-spline approximation for Bézier curve and rendering offset

In this paper, a G1, C1, C2 PH-spline is employed as an approximation for a given Bzier curve within error bound and further renders offset which can be regarded as an approximate offset to the Bzier curve. The errors between PH-spline and the Bzier curve, the offset to PH-spline and the offset to the given Bzier curve are also estimated. A new algorithm for constructing offset to the Bzier curve is proposed.

Cylindrical Helix Spline Approximation of Spatial Curves

In this paper, we present a new method for approximating spatial curves with a G^1 cylindrical helix spline within a prescribed tolerance. We deduce the general formulation of a cylindrical helix, which has 11 freedoms. This means that it needs 11 restrictions to determine a cylindrical helix. Given a spatial parametric curve segment, including the start point and the end point of this segment, the tangent and the principal normal of the start point, we can always find a cylindrical segment to interpolate the given direction and position vectors. In order to approximate the known parametric curve within the prescribed tolerance, we adopt the trial method step by step. First, we must ensure the helix segment to interpolate the given two end points and match the principal normal and tangent of the start point, and then, we can keep the deviation between the cylindrical helix segment and the known curve segment within the prescribed tolerance everywhere. After the first segment had been formed, we can construct the next segment. Circularly, we can construct the G^1 cylindrical helix spline to approximate the whole spatial parametric curve within the prescribed tolerance. Several examples are also given to show the efficiency of this method.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Many-Knot Spline Interpolating Curves and Their Applications in Font Design

Many-knot spline interpolating is a class of curves and surfaces fitting method presentedin 1974. Many-knot spline interpolating curves are suitable to computer aided geometric design anddata points interpolation. In this paped, the properties of many-knot spline interpolating curves arediscussed and their applications in font design are considered. The differences between many-knotspline interpolating curves and the curves genoaed by exceeding-lacking adjuStment algorithm aregiven.

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

SHAPE ANALYSIS FOR A KIND OF RATIONAL PARAMETRIC CUBIC CURVES

A rational parametric planar cubic H spline curve is defined by a set of control vertices in a plane and percentage factors of line segments between every two control vertices. Movement of any control vertex affects three curve segments. This paper is the succession and development of reference of Tang Yuehong. We analyze the geometric features like cusps and inflection points in the curve and calculate the cusps and inflection points, then give a necessary and sufficient condition to the inflection points in the curve when it is non degenerative, and finally show that the curves have no cusps in the interval (0,1). In many applications, it is desirable to analyze the parametric curves for undesirable features like cusps and inflection points

B-Spline曲线聚风装置直线翼垂直轴风力机启动性能数值模拟

为了提升直线翼垂直轴风力机的启动性能,基于B-Spline曲线生成方法,提出一种具有流线型轮廓的聚风装置,将其分别安装在风轮顶端和底端,用以提升风轮附近的来流风速,使风轮汲取更多风能,达到更易启动的特点.选取聚风装置的5个结构参数进行设计:聚风装置与风轮间隙距离ΔL、底圆半径R_(1)、顶圆半径R_(2)、入口角度α_(1)和出口角度α_(2).设计方法采用二次旋转正交组合筛选最优模型,通过三维数值模拟研究聚风装置参数对直线翼垂直轴风力机启动性能的影响,获得了最佳外形参数组合.此外,对有无加装聚风装置的直线翼垂直轴风力机进行了静态三维数值模拟.结果表明,通过添加具有外凸流线型轮廓的聚风装置,直线翼垂直轴风力机的启动性能有显著提升.在风速较低的情况下,性能改善更加明显.聚风装置可使直线翼垂直轴风力机的平均启动力矩系数最大增加38.8%,峰值平均启动力矩系数最大可增加31.2%.

Robust Corner Detection Based on Multi-scale Curvature Product in B-spline Scale Space

这份报纸在 B 花键弯曲规模空间的框架论述一种多尺度的弯曲产品角落察觉技术。规模产品功能在不同规模从轮廓的弯曲产品被导出。角落被 thresholding 作为本地最大值构造越过几规模的弯曲产品结果。通过规模产品，本地化精确性和察觉表演能显著地以 CNN 标准被改进。实验也证明那个建议方法显示出坚韧性到高频率细节并且提供有希望的察觉结果。

Paths of algebraic hyperbolic curves

Cubic algebraic hyperbolic （AH） Bezier curves and AH spline curves are defined with a positive parameter a in the space spanned by {1, t, sinht, cosht}. Modifying the value of a yields a family ofAH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called ＂path of AH curve＂ （AH Bezier and AH spline curves） when a changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.

On the Parametric Modeling of Turbine Blade Section Curve

Geometric parameters of the turbine blade are classified according to their destined functions, and the mathematical definition of those parameters in the section curve is introduced in detail. Some parts of the section curve shape can be adjusted freely, offering more flexibility to designers.