The Applications of Wavelets in Hierarchical Representations and Smoothing of Curves and Surfaces

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一种构造任意类三次三角曲线的方法

在自由曲线曲面造型中,一般多以多项式为基函数构造参数曲线曲面,而在三角函数空间中也能构造参数曲线曲面.给出了一种构造任意类三次三角参数曲线的方法,该法以三次多项式曲线的基本性质为基础,从而构造出的曲线与对应的三次多项式曲线具有几乎完全相似的性质,而且所构造的曲线能精确表示圆弧、椭圆弧、抛物线弧等二次曲线,为曲线曲面造型提供了一种新方法.     

构造任意参数曲线曲面的简捷方法

本文探讨了用控制参量形式表示各种参数三次曲线、曲面和二次Ｂｅｚｉｅｒ、二次Ｂ－ｓｐｌｉｎｅ曲线、曲面的基函数统一表达式。采用改变控制参数取值的方法构造所需的各种曲线、曲面，为曲线，曲面造型提供了一种简捷的数学方法，还讨论了参量的不同取值所对应的不同种类曲线、曲面的几何特性。     

Optimal approximate conversion of spline surfaces

The paper introduces an effective method for approximate conversion of spline surfaces. The method uses mainly geometric continuity conditions, parameter transformations and nonlinear optimization techniques. Degree reduction to bicubic and biquintic surfaces is introduced, curvature oriented segmentation is developed. The method can be extended to merging spline surfaces and to spline approximation of offset surfaces.     

C1三次插值样条曲线曲面

利用Bézier曲线的端点插值性质,得到了构造三次插值样条曲线曲面的一种改进的基函数——BB基函数。由BB基函数构造了C¹保形三次插值样条曲线;构造了C¹双三次插值样条曲面。     

Canopy surface technique for parametric blending

A method based on canopy surfaces is presented for blending parametric surfaces. The blend designed using this method gives freedom to the designer in selecting i) the primitive surfaces to be blended, ii) the contact curves lying on them (in which the blend meets the primitives), iii) the endtangent directions along the contact curves (which are used to ensure tangentplane continuity), and iv) the shape of the cross-sectional curve. An important feature of this method, which is not seen in earlier methods, is the use of end tangents to ensure theC ¹ continuity of iso-parametric curves across the junctions between the blend and the primitives in addition to the tangent-plane continuity.     

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C¹ bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.     

形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足,提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数;然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线,并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题;最后定义相应的组合样条曲面,并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质,而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型,因此方程结构相对较为简单,符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足,是一种实用的自由曲线曲面造型方法。     

Canal surfaces with rational contour curves and blends bypassing the obstacles

In this paper, we will present an algebraic condition, see (20), which guarantees that a canal surface, given by its rational medial axis transform (MAT), possesses rational generalized contours (i.e., contour curves with respect to a given viewpoint). The remaining computational problem of this approach is how to find the right viewpoint. The canal surfaces fulfilling this distinguished property are suitable for being taken as modeling primitives when some rational approximations of canal surfaces are required. Mainly, we will focus on the low-degree cases such as quadratic and cubic MATs that are especially useful for applications. To document a practical usefulness of the presented approach, we designed and implemented two simple algorithms for computing rational offset blends between two canal surfaces based on the contour method which do not need any further advanced formalism (as e.g. interpolations with MPH curves). A main advantage of the designed blending technique is its simplicity and also an adaptivity to choose a suitable blend satisfying certain constrains (avoiding obstacles, bypassing other objects, etc.). Compared to other similar methods, our approach requires only one SOS decomposition for the whole family of rational canal surfaces sharing the same silhouette, which significantly simplifies the computational complexity.     

Parameterdarstellung kubischer und bikubischer Splines

The known representation of curves and surfaces by P. Bezier is applied to cubic and bicubic splines. In this form spline-conditions are particularly vivid and along easily manageable. A parametric representation naturally connected with spline-conditions allows moreover a most simple and clear correction of splines and bisplines.     

C-Bézier曲线曲面的扩展

运用积分定义的方式,构造了带多形状参数的C-Bézier曲线曲面,改变形状参数的值,能整体或局部调控曲线曲面的形状。它们包含C-Bézier曲线曲面为其特例且具有C-Bézier曲线曲面的主要性质。还给出了带2个形状参数的二次C-Bézier曲线段G¹和C¹拼接条件以及带3个形状参数的三次C-Bézier曲线段G¹、C¹和G²拼接条件。     

Approximate G continuous interpolation of a rectangular network of rational cubic curves

The problem of spanning a rectangular network of rational cubic curves with a smooth surface is discussed in this paper. Provided the network is compatible with a smooth surface, then algorithms for patch construction, optimization and subdivision are developed to construct an ‘approximately smooth’ surface, that is, G¹ continuous to within some tolerance, composed of rational bicubic patches. The algorithms have been applied in the die and mould industry. The toolmaker constructs a wireframe model of an EDM (electro-discharge machining) electrode and the algorithms automatically construct the surface model. For toolmaking companies, this simplifies the surface modelling process making a highly-specialized and time-consuming task virtually automatic.     

拟合任意空间曲面的三角函数方法

在ＣＡＧＤ中，曲线曲面拟合一般采用多项式作为基函数，而本文提出一种基于三角函数的拟合方法，它具有Ｂ样条的主要优点，并达到了Ｃ＾３连续性，而三次Ｂ样条只能达到Ｃ＾２，它还解决了多项式不能描述椭圆的问题，因此，这一方法适用于自由曲线曲面的设计。本文还结合实例说明了用此方法进行造型设计的技术。     

Bézier curves and surfaces with shape parameters

Bézier curves with n shape parameters and triangular Bézier surfaces with 3n(n+1)/2 shape parameters are presented in this paper. The geometric significance of the shape parameters and the geometric properties of these curves and surfaces are discussed. The shapes of the curves and the surfaces can be modified intuitively, foreseeably and precisely by changing the values of the shape parameters.     

Multiresolution for Algebraic Curves and Surfaces using Wavelets

This paper describes a multiresolution method for implicit curves and surfaces. The method is based on wavelets, and is able to simplify the topology. The implicit curves and surfaces are defined as the zero-valued piece-wise algebraic isosurface of a tensor-product uniform cubic B-spline. A wavelet multiresolution method that deals with uniform cubic B-splines on bounded domains is proposed. In order to handle arbitrary domains the proposed algorithm dynamically adds appropriate control points and deletes them in the synthesis phase.     

曲线曲面小波分解偏差估计

针对曲线曲面小波变换后低分辨模型和原始高分辨模型之间的偏差问题,提出离散偏差模型方法来估计偏差.通过定义偏差曲线,并在此基础上提出离散偏差曲线法来估计高分辨曲线和低分辨曲线之间的偏差:给定弓高误差限,用等误差方法离散偏差曲线,利用离散点估计偏差.同时,将该思想推广到曲面小波分解偏差估计上,定义了偏差曲面,并提出离散偏差曲面法来估计高分辨曲面和低分辨曲面之间的偏差.实验结果表明,文中算法既可以实现分解n次之后的低分辨模型的偏差估计,又可以通过减小离散误差提高估计精度.     

椭圆曲线的高精确逼近方法

应用扩展的三次均匀 B样条曲线 ,给出了逼近椭圆曲线的高精确逼近方法 ,论证了该方法精度 ,扩展了多项式参数曲线的应用。文中给出了逼近曲线的图例。     

Rational adaptive blends among obstacles in 3D by contour method

In this paper we will continue in investigating ‘contour method’ and its using for the computation of rational parameterizations of canal surfaces without a need of sum of squares (SOS) decomposition. Further approaches for constructing flexible smooth transitions between canal surfaces will be presented. Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space. Using this type of curves significantly simplifies the previous methods discussed in Bizzarri (2015), and mainly new situations, which could not have been handled with the previous setup, are successfully solved, now. Especially a method for constructing rational adaptive blends which bypass a given obstacle (or more given obstacles when needed) is thoroughly discussed and its functionality is demonstrated on a number of examples. The designed approach works not only for simple obstacles represented by one-dimensional medial axis transforms but also for more general obstacles described by two-dimensional medial surface transforms.     

Designing parametric blends: surface model and geometric correspondence

A model based on the curve-modulation technique is proposed for designing a parametric blend between two parametrically defined surfaces. The modulants are the cross section and the spine curves. The end position and tangency conditions of the cross section are discussed. Its intermediate shape can be left to the designer's choice, and generally depends on the application at hand. Various alternative choices, that fulfill the end conditions are suggested; this provides the flexibility demanded by various applications. The spine is designed as the intersection of two derived surfaces. Two alternative surface derivations are proposed, and both are discussed in some detail with their merits and demerits. The first derivation generates the contact curves automatically and relieves the designer from specifying them; the second one accepts contact curves specified by the designer, and generates the spine accordingly. Both of them are equally important in CAD/CAM and solid modelling applications.     

一类代数空间逼近样条

给定空间不共面的四个有序数据点,可以形成一个四面体。在四面体内,Bernstein-Bézier（B-B）形式定义两类正则实多项式代数曲面片,一类是二次的,一类是三次的。此两类曲面片在四面体内的交集为一条正则曲线段。先固定二次曲面片,并得到其参数形式,然后约简三次曲面片所对应的Bernstein系数,使之为带有三个形状调整的形状因子,其中两个分别代表曲线段端点处的曲率,另外一个作为形状的调整。利用二次曲面的参数形式,由三次曲面片可得到曲线的隐参数约束形式,从而得到曲线的参数形式。对给定的空间点列,利用两个形状因子较容易的拼接出G²-连续的逼近曲线,突破了现行代数曲线生成方法,即空间连续曲线均是通过三角形仿射变换,由B-B形式生成的平面弧拼接而成。