Curvature continuous interpolation of curve meshes

This paper presents a method for local construction of a curvature continuous (GC²) piecewise polynomial surface which interpolates a given rectangular curvature continuous quintic curve mesh. First, we create a C² quintic basic curve mesh, which interpolates the same grid points, preserves the tangent slopes and curvatures but not derivative vectors at the grid points. After estimating twist and higher order mixed partial derivatives for each grid point, we generate locally the so-called C² biquintic basic patches, which serve to compute the first and second order cross-derivative functions of the final interpolation surface. In order to match the tangents and second order derivative vectors of the original boundary curves at the grid points, these basic patches are reparametrized by 5 × 3 and 3 × 5 functions, which lead to vector-valued first and second order cross-derivative functions of degrees 7 and 9 of the final surface patches, and eventually lead to a GC² piecewise polynomial surface of degree 9 × 9, which is then converted to a GC² Bézier composite surface. By arranging the surface patches in a chess-board fashion, the degrees of the final surface patches can be 9 × 5 and 5 × 9. An example for the construction of a GC² ship hull, together with its color-coded curvature maps, is given to illustrate the method. This method is attractive because the final surface has a much lower degree than other similar methods, it allows flexible local modification of the original curve mesh and local editing of the interpolation surface, and it is easily integrated into state-of-the-art geometric modeling systems.     

C2 Local convexity-preserving interpolation through the control point form method

C ² continuous convexity-preserving parametric curves in ℝ² is presented. The control point form method, which was developed in the field of numerical grid generation, is used here to construct interpolating curves. Good control over the shape of the curves is obtained thanks to the capability of the proposed approach to interpolate not only the data points, but also some directions suitably associated with them.     

Curve Interpolation with Arbitrary End Derivatives

An algorithm for interpolating data points with end derivative constraints is presented. The interpolating curve passes through the given points and at the same time assumes the derivatives specified. For degree p interpolation, up to p−1 derivatives may be specified, resulting in C ^p−1 continuous curve interpolants. The method is useful to piece individual curve segments together or to create closed curves with various degrees of smoothness.     

Reducing control points in lofted B-spline surface interpolation using common knot vector determination

A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method.     

分形插值曲面

研究一类迭代函数系统和分形插值曲面的构造，得到分形插值曲面面连续的充分必要条件，由此给出了简单的分形超限插值方法和新的网格点插值方法；从而利用网格曲线或网格点上的特征生成连续的分形插值曲面，这种简单有效的方法可广泛地用于分形造型和数据拟合。     

Controllable Locality in C2 Interpolating Curves by B2-splines / S-splines

This paper presents a systematic scheme for controlling the local behaviour of C² interpolating curves, based on the cubic B2-splines and the quartic S-splines. Both splines have an additional control point obtained by knot- insertion or degree-elevation in each span of the conventional uniform cubic interpolating B-splines. The shape designer can choose the desired range of locality for each span and get the corresponding additional control point as a barycentric combination of interpolation points within the range, without solving any variational problem and simultaneous equations. The scheme is consistent over the entire curve subject to some typical end conditions. The class of the curves derived includes the conventional cubic interpolating B-splines. Examples demonstrate the behaviour of the new interpolating curves and the capability of the scheme.     

Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing

This paper addresses the problem of constructing a structured quadrilateral grid inside a given four-sided 2D region by a particular boundary-conforming mapping scheme—Hermite Interpolation Mapping (HIM). When the four given boundary curves are concave and convoluted, all boundary-conform mapping methods suffer from potential self-overlapping problem. Under HIM, the geometry of the grid depends on both the four boundary curves and the tangent vector functions associated with the curves. While the four boundary curves are fixed, the tangent functions in HIM can be varied to suit the need of controlling the characteristics of the mesh inside the given region to prevent self-overlapping. Besides tangent functions, the four twist vectors at the corners of the region can also be adjusted to influence the distribution of the inner grid elements. In our approach, a constrained functional optimization scheme is adopted to adjust the tangent functions and the twist vectors, adaptive to the geometry of the boundary curves, so that the resulting HIM will be free of self-overlapping. The optimization is carried out on the shape control energy that measures the overall mesh quality of the underlying HIM while the self-overlapping is strongly prevented in the form of constraints to the optimization. Experimental results show the promise of the proposed method as a practical and effective solution for structured grid generation.     

Darstellung und Korrektur symmetrischer Kurven und Flächen auf EDV-Anlagen

In this paper a simple representation of piecewise quadratic curves and biquadratic surfaces will be suggested which is useful for interpolating a given surface at equidistant and symmetric points. Corrections of the surface can easily be done.     

平面隐式曲线的Hermite 插值逼近

隐式曲线在医学图像处理、地理信息系统、数值场可视化等领域中有着重要应用。 在分析点采样和曲线逼近理论的基础上,提出一种运用Hermite 插值方法逼近平面隐式曲线的 算法。首先将曲线绘制区域网格化,在网格单元各边中通过线性插值计算曲线采样点;其次通 过计算采样点精简前后构成的曲线段之间产生的误差优化采样点;最后通过Hermite 插值法逼 近隐函数曲线。实验表明,通过该算法绘制出的曲线在采样点数量较少的情况下,其光滑度和 准确度仍较高。     

Nonuniform B-spline Mesh Fairing Method

A method for fairing a surface composed of a set of discrete data points distributed in anonrectangular topological mesh is presented.All curves are expressed by nonuniform cubic B-splinecurves.The fairing method is minimizing the elastic strain energy of mesh curves and of springs at-tached to the data points.The fairing surface can be generated by interpolating through the meshcurves.The generation and fairing of a ship hull surface is given as an example.     

Similarity based interpolation using Catmull–Clark subdivision surfaces

A new method for constructing a Catmull–Clark subdivision surface (CCSS) that interpolates the vertices of a given mesh with arbitrary topology is presented. The new method handles both open and closed meshes. Normals or derivatives specified at any vertices of the mesh (which can actually be anywhere) can also be interpolated. The construction process is based on the assumption that, in addition to interpolating the vertices of the given mesh, the interpolating surface is also similar to the limit surface of the given mesh. Therefore, construction of the interpolating surface can use information from the given mesh as well as its limit surface. This approach, called similarity based interpolation, gives us more control on the smoothness of the interpolating surface and, consequently, avoids the need of shape fairing in the construction of the interpolating surface. The computation of the interpolating surface’s control mesh follows a new approach, which does not require the resulting global linear system to be solvable. An approximate solution provided by any fast iterative linear system solver is sufficient. Nevertheless, interpolation of the given mesh is guaranteed. This is an important improvement over previous methods because with these features, the new method can handle meshes with large number of vertices efficiently. Although the new method is presented for CCSSs, the concept of similarity based interpolation can be used for other subdivision surfaces as well.     

Objective analysis of a two dimensional scalar field by successive corrections using a personal computer

The use of a successive correction method for interpolating data from irregularity spaced points to a regularly spaced grid has long been used in meteorological applications. This procedure was automated using a personal computer and a demonstration program was written based on the Cressman method. The program also produce isopleth analysis for standard meteorological scalar fields such as temperature and height at constant pressure levels.     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

G2 quasi-developable Bezier surface interpolation of two space curves

Surface development is used in many manufacturing planning operations, e.g., for garments, ships and automobiles. However, most freeform surfaces used in design are not developable, and therefore the developed patterns are not isometric to the original design surface. In some domains, the CAD model is created by interpolating two given space curves. In this paper, we propose a method to obtain a G² quasi-developable Bezier surface interpolating two arbitrary space curves. The given curves are first split into a number of piecewise Bezier curves and elemental Bezier patches each of which passes through four splitting points are constructed. All neighboring elemental patches are G² connected and they are assembled optimally in terms of the degree of developability (the integral Gaussian curvature). Experiments show that the final composite Bezier surface is superior to a lofted one which is defined regardless of the final surface developability.     

基于等值线的海区网格插值算法研究

为了将人类长期海洋观测所获取的矢量等值线数据与遥感反演等所获取的栅格数据进行比对运算,存在由等值线到海区网格的插值需求.以海洋表层水温矢量等值线为例,运用具有层次嵌套关系的三级分辨率海区网格,对等值线进行了离散化,通过建立离散点与海区网格之间的空问索引关系,并运用顾及方向的单点移面平均加权法,进行了海区网格点插值运算过程的研究.     

Curve interpolation in recursively generated B-spline surfaces over arbitrary topology

Recursive subdivision is receiving a great deal of attention in the definition of B-spline surfaces over arbitrary topology. The technique has recently been extended to generate interpolating surfaces with given normal vectors at the interpolated vertices. This paper describes an algorithm to generate recursive subdivision surfaces that interpolate B-spline curves. The control polygon of each curve is defined by a path of vertices of the polyhedral network describing the surface. The method consists of applying a one-step subdivision of the initial network and modifying the topology in the neighborhood of the vertices generated from the control polygons. Subsequent subdivisions of the modified network generate sequences of polygons each of which converges to a curve interpolated by the limit surface. In the case of regular networks, the method can be reduced to a knot insertion process.     

Triangular Curved Surface Construction of A Series of Arbitrary Disordered Points in Space

Constructing Bernstein-Bezier triangular interpolating curve surface interpolating a series of arbitrary disordered data points is of considerable importance for the design and modeling of surfaces with a variety of continuity information. In this article. a kind of simple and reliable algorithm that can process complex field triangular grid generating is presented, and a group of formulae for determining triangular curved surface with wholly C1 continuity are given. It can process arbitrary non-convex boundary and can be used to construct surfaces inner holes.     

Modeling,analysis and multi-objective optimization of twist extrusion process using predictive models and meta-heuristic approaches,based on finite element results

Recently, twist extrusion has found extensive applications as a novel method of severe plastic deformation for grain refining of materials. In this paper, two prominent predictive models, response surface method and artificial neural network (ANN) are employed together with results of finite element simulation to model twist extrusion process. Twist angle, friction factor and ram speed are selected as input variables and imposed effective plastic strain, strain homogeneity and maximum punch force are considered as output parameters. Comparison between results shows that ANN outperforms response surface method in modeling twist extrusion process. In addition, statistical analysis of response surface shows that twist extrusion and friction factor have the most and ram speed has the least effect on output parameters at room temperature. Also, optimization of twist extrusion process was carried out by a combination of neural network model and multi-objective meta-heuristic optimization algorithms. For this reason, three prominent multi-objective algorithms, non-dominated sorting genetic algorithm, strength pareto evolutionary algorithm and multi-objective particle swarm optimization (MOPSO) were utilized. Results showed that MOPSO algorithm has relative superiority over other algorithms to find the optimal points.     

Hermite interpolation with a pair of spirals

The Hermite interpolation problem in the plane considered here is to join two points and to match given unit tangent vectors and signed curvatures at the two points with various G² curves consisting of a pair of spirals. The rotation of the tangent vector of the interpolating curve from one point to the other is restricted to being less than π. The necessary and sufficient conditions for the existence of each of the various curves are given.     

带自由曲面的三视图重建算法研究

针对蕴含自由曲面的三视图,提出了通过恢复边界线和轮廓线进而根据这些恢复 的空间曲线重建自由曲面的算法。首先分析并证明了自由曲面在三视图中的投影性质,从而提 出边界投影和轮廓投影的匹配算法。针对视图中存在的被打断的样条曲线,提出了分段样条曲 线的爬坡算法来解决此类曲线不能匹配的问题。然后基于投影匹配序列重建出自由曲面的边界 线,再由边界投影上的点和轮廓投影端点的对应关系重建出空间轮廓线。由轮廓线等参采样构 造截面线并和边界线一起蒙皮生成最终自由曲面。本文提出的算法扩展了工程图的重建域。