基于一元对称幂基的等距曲面有理逼近算法

给出了基于一元对称幂基的等距曲面蒙面逼近新算法。利用一元对称幂基逼近张量积Bézier曲面u向曲线的等距曲线,得到一组等距逼近曲线,取固定的v值,得到一组数据点,用反算控制顶点的方法得到过这组数据点的v向曲线。对这两组曲线用蒙面算法得到逼近的有理等距曲面。该算法计算简单,将二元等距曲面有理逼近转化为一元曲线有理逼近,同时方便地解决了整体误差问题,随着对称幂基阶数的升高,可以得到较理想的逼近效果。     

NURBS边界曲面直接生成法

由于非均匀有理B样条(NURBS)曲面的复杂性,传统NURBS边界曲面的生成是先构造孔斯曲面,再由孔斯曲面向NURBS曲面转换得到,其操作过程比较烦琐。针对此问题,提出了NURBS边界曲面直接生成算法,该算法根据给定的四条NURBS边界曲线,结合孔斯曲面生成方法直接插值生成NURBS曲面,从而避免了通过孔斯曲面向NURBS曲面转换所带来的计算代价,因此同传统方法相比,具有较低的计算代价。实验结果表明:该算法简化了曲面生成步骤,减少了曲面转换过程的计算量,生成的曲面边界信息明确,且连续性好。     

基于截面数据的C1插值融合曲面重建

为了避免NURBS曲面重建需要进行节点矢量相容的问题，提出了一种双方向融合插值的[C1]参数曲面重建方法，该方法先后分段插值截面上连续的数据点、截面曲线以构造样条曲线和曲面片，并引入融合算法进行曲线、曲面拼接，从而得到光滑的待建曲面。该方法不会产生由节点插入所带来的大量的数据冗余以及复杂的计算过程，同时采用了融合的思想来处理曲线、曲面的拼接，改良了传统参数曲线、曲面拼接方法需要满足边界条件的缺陷。     

插值边界曲线的NURBS 近似极小曲面设计

探索性地设计了一个插值给定边界曲线的NURBS 近似极小曲面算法,弥补了当前NURBS 系统无法有效地设计工程所急需的一般NURBS 极小曲面的缺陷.运用NURBS 曲面的节点插入、Hybrid 多项式逼近等多种技术,将NURBS 曲面转化为相对简单的分片Bézier 曲面求解,并运用各子曲面片的控制顶点优化、整体曲面不断更新的迭代方法,成功地得到高精度的近似分片Bézier 极小曲面.最后,可以按用户的各种要求选择运用相应不同的迭代逼近算法,求取插值给定边界曲线的近似NURBS 极小曲面.     

双二次NURBS曲面间的最短距离

利用双二次Bezier曲面为非负的充要条件，给出了分别位于两张双二次NURBS曲面上的点是否为这两张曲面间距离最近的点的判别方法．曲面非负是指在此显式曲面上没有位于XOY平面下方的点．在这一基础上，得到了计算双二次NURBS曲面间最短距离的一个算法．最后，给出了实验结果并将文中算法与采用分层数据结构的常用算法进行了比较。     

四点法及保凸算法

四点插值细分法（简称四点法）是一种离散插值方法，在曲线和曲面造型中有较广泛的应用。本文给出了四点法的一个保凸算法。     

螺旋曲面的等距曲面

考察了螺旋曲面生成两种等距曲面的方法：由螺旋面直接作等距曲面,由螺旋遄的端面型线生成等距曲线,再由等距曲线生成螺旋面,提出了截向等距曲面的概念,指出了与通常意义下的等距曲面之间的关系,并给出了它们之间的关系表达式。最后的实例表明,两种等距曲面的互不蕴含关系。两种方法在具有自主版权的三维造型系统Ｇｅｍｓ５．０中得到实现。     

实用的可展曲面上的C~1曲线插值方法

曲面上的曲线插值是计算机辅助几何设计的重要课题之一.利用可展曲面可与平面贴合的性质,构造一个等距对应将可展曲面展成平面,从而将可展曲面上的曲线插值归结为通常的R2上插值曲线的构造,并证明所得的插值曲线为C1连续.最后以柱面、锥面以及切线曲面为例构造插值曲线,图例显示该算法具有满意的效果.     

可展曲面上G^2连续的曲线插值方法

根据微分几何理论,给出一种可展曲面上G2连续的曲线插值算法.构造一等距对应将可展曲面展成平面,从而将原问题转化为通常的平面上的曲线插值问题.在R2上利用二次三角B样条曲线插值型值点列,无需反算控制顶点,证明了所得的可展曲面上的插值曲线是G2连续的.理论推导和实例均表明,该算法具有推广应用的广阔前景.     

NURBS曲面生成算法及仿真研究

讨论了一种生成NURBS曲面的算法,用C语言实现了该算法,并利用MATLAB进行仿真对该算法进行验证。在算法中讨论了曲面及其等距面生成方法以及曲面生成技术中相关的一些技术,如曲线段间参数过渡、曲面生成模式、曲面生成的实时性、改变曲面的形状等。仿真结果证明了算法的有效性。     

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.     

平面NURBS曲线的导矢及其等距线

本文首先给出了计算NURBS曲线导矢的递推公式,在此基础上,给出了生成平面NURBS曲线等距线的算法。     

Computational methods for blending surface approximation

Blending surfaces form a smooth transition between two distinct, intersecting surfaces or smoothly join two or more disconnected surfaces and are normally procedural surfaces which are difficult to exchange and to interrogate in a reliable and efficient manner. In this paper, an approximation method for blending surfaces which are curvature continuous to the underlying surfaces with a non-uniform rational B-spline (NURBS) surface is presented. The use of NURBS is important since it facilitates the exchange of geometric information between various computer aided design and manufacturing systems. In the method, linkage curves on the underlying surfaces are approximated to within a specified tolerance and cross-link curves are created using the linkage curves, a directional curve and the parametric partial derivatives of the underlying surfaces. Cross-link curves are lofted to form the blending surface and an adaptive sampling procedure is used to test the blending surface against specified tolerances. Cross-link curves are added, where necessary, and the surface relofted until the continuity conditions are satisfied to within specified tolerances. Examples illustrate the applicability of the method.     

Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets

An isogeometric topological shape optimization method is developed, using a dual evolution of NURBS curves and level sets; the NURBS curves feature the exact representation of geometry and the level sets help to detect and guide the topological variation of NURBS curves. The implicit geometry by the level sets is transformed into the parametric NURBS curves by minimizing the difference of velocity fields in both representations. A gradient-based optimization problem is formulated, based on the evolution of the NURBS curves. The control points of NURBS curves are taken as design variables. The necessary response and design sensitivity are computed by an isogeometric boundary integral equation method (BIEM) using the NURBS curves. The design sensitivity is obtained on fixed grids and utilized as the velocity to update the Hamilton–Jacobi equation for the level sets. To obtain the whole velocity field on the fixed grids, an interpolation and velocity extension scheme are employed. The developed method provides accurate response and enhanced sensitivity using isogeometric BIEM. Also, additional post-processing is not required to communicate with CAD systems since the optimal design is represented as NURBS curves. Numerical examples demonstrate the accuracy of design sensitivity on fixed grids and the feasibility of shape and topological optimization.     

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation.When all th knot spans are identical,the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves.If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.     

NURBS曲面的等距曲面算法

本文将ＮＵＲＢＳ曲线的有理ｄｅＢｏｏｒ算法推广到ＮＵＲＢＳ曲面点的计算，由此可以得到ＮＵＲＢＳ曲面上点的单位法矢量，供其应用于ＮＵＲＢＳ曲面等曲面的生成。该算法几何意义明显，算法简洁，易于编程实现。     

Approximate T-spline surface skinning

This paper considers the problem of constructing a smooth surface to fit rows of data points. A special class of $T$-spline surfaces is examined, which is characterized to have a global knot vector in one parameter direction and individual knot vectors from row to row in the other parameter direction. These $T$-spline surfaces are suitable for lofted surface interpolation or approximation. A skinning algorithm using these $T$-spline surfaces is proposed, which does not require the knot compatibility of sectional curves. The algorithm consists of three main steps: generating sectional curves by interpolating data points of each row by a $B$-spline curve; computing the control curves of a skinning surface that interpolates the sectional curves; and approximating each control curve by a $B$-spline curve with fewer knots, which results in a $T$-spline surface. Compared with conventional $B$-spline surface skinning, the proposed $T$-spline surface skinning has two advantages. First, the sectional curves and the control curves of a $T$-spline surface can be constructed independently. Second, the generated $T$-spline skinning surface usually has much fewer control points than a lofted $B$-spline surface that fits the data points with the same error bound. Experimental examples have demonstrated the effectiveness of the proposed algorithm.     

平面NURBS曲线的等距线算法：圆弧法矢近似法

本文根据产生曲线的特征点与它的等距线的特征点的对应关系,给出了一种平面NURBS曲线的等距线表示方法——圆弧法矢近似法。这种方法的特点是:(1)等距线与产生曲线具有统一的NURBS表示;(2)计算简单、几何意义明确、近似精度高。     

The feedrate scheduling of NURBS interpolator for CNC machine tools

This paper proposes an off-line feedrate scheduling method of CNC machines constrained by chord tolerance, acceleration and jerk limitations. The off-line process for curve scanning and feedrate scheduling is realized as a pre-processor, which releases the computational burden in real-time task. The proposed method first scans a non-uniform rational B-spline (NURBS) curve and finds out the crucial points with large curvature (named as critical point) or G⁰ continuity (named as breakpoint). Then, the NURBS curve is divided into several NURBS sub-curves using curve splitting method which guarantees the convergence of predictor–corrector interpolation (PCI) algorithm. The suitable feedrate at critical point is adjusted according to the limits of chord error, centripetal acceleration and jerk, and at breakpoint is adjusted based on the formulation of velocity variation. The feedrate profile corresponding to each NURBS block is constructed according to the block length and the given limits of acceleration and jerk. In addition, feedrate compensation method for short NURBS blocks is performed to make the jerk-limited feedrate profile more continuous and precise. Because the feedrate profile is established in off-line, the calculation of NURBS interpolation is extremely efficient for CNC high-speed machining. Finally, simulations and experiments with two free-form NURBS curves are conducted to verify the feasibility and applicability of the proposed method.