非均匀B样条曲面顶点及法向插值

本文以非均匀Catmull-Clark细分模式下的轮廓删除法为基础,通过在细分网格中定义模板并调整细分网格的顶点位置,为非均匀B样条曲面顶点及法向插值给出了一个有效的方法．该细分网格由待插顶点形成的网格细分少数几次而获得．细分网格的顶点被分为模板内的顶点和自由顶点．各个模板内的顶点通过构造优化模型并求解进行调整,自由顶点用能量优化法确定．这一方法不仅避免了求解线性方程组得到控制顶点的过程,而且在调整顶点的同时也兼顾了曲面的光顺性．     

具有任意自由度的B样条非均匀细分*

为了便于工程实际应用,非均匀细分方法现在已经成为计算机图形学和几何建模中的热点问题。本文提出一个具有任意自由度的B样条非均匀细分算法,其实现与B样条均匀细分即Lane–Riesenfeld细分方法相似。该算法包含了非均匀d环结构生成的双重控制点,其中d环相似于d度均匀B样条曲线的Lane-Riesenfeld算法中均匀的d环结构。Lane-Riesenfeld算法是由B样条曲线基函数的连续卷积公式直接得出的,而本文的算法是blossoming方法的一个扩展。对于非均匀B样条曲线来说,本文的节点插入方法比之前的方法更简单更有效。     

一种对称非均匀细分曲面算法

为了得到能更好应用于CAD系统的细分曲面造型方法,提出一种基于B-样条的对称非均匀细分算法,其中的思想和均匀Lane-Riesenfeld节点插入算法相似。基于B-样条的节点插入算法,以Blossoming为工具,计算出细分后的新控制顶点。细分后得到的极限曲面由张量积样条曲面组成,在奇异点达到2C连续。与传统的细分曲面算法相比,该细分曲面算法具有良好的局部支撑性,大大降低了算法的复杂度,而且该算法是对称的,不用考虑定向问题。     

Doo-Sabin细分算法在动态模式下的推广

提出一种基于均匀三角多项式B样条的动态保凸细分算法，它可以看作Doo-Sabin细分算法在动态模式下的一个推广．其细分规则基于张量积曲面细分模式的几何意义，不仅可以生成旋转曲面等特殊曲面，而且可以根据参数来控制细分曲面的形状．最后运用传统的离散傅里叶技术和特征根方法证明了该细分算法的收敛性．     

一种任意次非均匀B 样条的细分算法

类似于经典的、应用于任意次均匀B 样条的Lane-Riesenfeld 细分算法, 提出了一种任意次非均匀B 样条的细分算法,算法包含加细和光滑两个步骤,可生成任意 次非均匀B 样条曲线。算法是基于于开花方法提出的,不同于以均匀B 样条基函数的卷积 公式为基础的Lane-Riesenfeld 细分算法。通过引入两个开花多项式,给出了算法正确性的 详细证明。算法的时间复杂度优于经典的任意次均匀B 样条细分算法,与已有的任意次非 均匀B 样条细分算法的计算量相当。     

Doo-Sabin细分模式的尖锐特征造型

通过推广准均匀二次B样条的节点插入算法,对边界面、折痕面、角点面等特征面给出新的细分规则,从而使Doo-Sabin细分模式可以表示边界、折痕、角点、刺点等尖锐特征,且特征处不受拓扑结构的限制.在特征附近进行了连续性分析,所得到的极限曲面具有分片G1连续性.该算法既可以设计有特征的、任意拓扑的复杂曲面,又可以精确地表示球面、柱面、锥面等工程技术中常用的二次曲面,在CAD/CAM领域具有广泛的应用前景.     

NURBS曲线曲面的多分辨率几何建模

针对均匀和准均匀B样条小波多分辨率建模表示能力和适应性的不足,基于离散内积和非均匀B样条节点插入算法建立了一种非均匀半正交B样条小波,并进一步论述了其在非均匀B样条曲线曲面中的多分辨率设计.最后通过实例对非均匀B样条曲线曲面中的多分辨率建模进行了说明和验证.     

电磁散射目标的NURBS建模

采用非均匀有理B样条(NURBS)对复杂目标进行建模,利用Cox-DeBoor算法将NURBS曲线曲面转化为更适合数值计算的有理Bézier曲线曲面,采用后向面判别法处理不同面元间的遮挡问题,并给出导弹雷达散射截面的计算结果。     

有理Bezier曲线的非均匀细分算法

ｄｅ Ｃａｓｔａｌｊａｕ算法很早就用于Ｂｅｚｉｅｒ曲线、曲面的细分。但对于有理Ｂｅｚｉｅｒ曲线，当某些点出现大权时，固定ｔ＝１／２的均匀细分算法失效。本文分析了失效的原因并提出了一种新的非均匀细分方法。通过分析和比较，证明了新方法非常有效，可以很好地应用于实践。     

基于非均匀Catmull-Clark细分方法的曲线插值

带有复杂型曲线插值约束的细分曲面的生成,是计算机图形学及几何造型技术等领域所关心的一个问题.鉴于此,提出了一种高效的可以插值三次NURBS曲线的细分曲面生成方法.只需在被插值曲线的控制多边形两侧构造具有对称性质的四边形,构成对称网格带;证明了对该对称网格带应用Sederberg等人提出的非均匀Catmull-Clark细分规则以后,它将收敛于这条被插值曲线.因此,含有这种对称网格带的多面体网格的细分极限曲面即为满足曲线插值约束的细分曲面.应用该方法,既可以插值单条NURBS曲线,也可以插值由多条NURBS曲线组成的曲线网格.因此,该方法广泛适用于产品外形和图形软件设计.     

Non-Uniform Subdivision Surfaces with Sharp Features

Sharp features are important characteristics in surface modelling. However, it is still a significantly difficult task to create complex sharp features for Non-Uniform Rational B-Splines compatible subdivision surfaces. Current non-uniform subdivision methods produce sharp features generally by setting zero knot intervals, and these sharp features may have unpleasant visual effects. In this paper, we construct a non-uniform subdivision scheme to create complex sharp features by extending the eigen-polyhedron technique. The new scheme allows arbitrarily specifying sharp edges in the initial mesh and generates non-uniform cubic B-spline curves to represent the sharp features. Experimental results demonstrate that the present method can generate visually more pleasant sharp features than other existing approaches.     

A subdivision scheme for Poisson curves and surfaces

The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.     

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points.The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.     

细分蒙皮曲面

对细分曲面在曲面造型中的应用进行了研究，并着重于蒙皮曲面造型技术．所提方法在传统的蒙皮曲面构造过程中引入细分方法，有效地避免了因截面曲线的相容性处理而产生的数据量激增的问题；最后生成的蒙皮曲面能够精确插值预先设计的截面曲线，并且可以在指定的截面曲线处产生折痕效果．     

A Blend of Subdivision Surfaces and NURBS

In present paper, the contour deletion method is developed both to blend surfaces and to fill N-sided holes, which is used for subdividing the NURBS surface. First, according to the non-uniform Catmull-Clark subdivision principle, surfaces are blended. The non-uniform Catmull-Clark subdivision method is constructed, which build the surface through interpolating corner vertices and boundary curves. Then the contour deletion method is adapted to remove the controlling mesh boundary contour in the process of segmentation iteration. Last, N sided-hole is filled to generate a integral smooth continuous surface. This method not only guarantee that the blending surface and base surface patches have C2 continuity at the boundary, but also greatly improve the smoothness of the N-side hole filling surface. The results show that, this method simplifies the specific computer-implemented process, broads the scope of application of subdivision surfaces, and solves the incompatible problem between the subdivision surface and classical spline. The resulting surface has both advantages of the subdivision surface and classical spline, and also has better filling effect.     

Single-knot wavelets for non-uniform B-splines

We propose a flexible and efficient wavelet construction for non-uniform B-spline curves and surfaces. The method allows to remove knots in arbitrary order minimizing the displacement of control points when a knot is re-inserted. Geometric detail subtracted from a shape by knot removal is represented by an associated wavelet coefficient replacing one of the control points at a coarser level of detail. From the hierarchy of wavelet coefficients, perfect reconstruction of the original shape is obtained. Both knot removal and insertion have local impact. Wavelet synthesis and analysis are both computed in linear time, based on the lifting scheme for biorthogonal wavelets. The method is perfectly suited for multiresolution surface editing, progressive transmission, and compression of spline curves and surfaces.     

Generalized Hermitian Radial Basis Functions Implicits from polygonal mesh constraints

In this work we investigate a generalized interpolation approach using radial basis functions to reconstruct implicit surfaces from polygonal meshes. With this method, the user can define with great flexibility three sets of constraint interpolants: points, normals, and tangents; allowing to balance computational complexity, precision, and feature modeling. Furthermore, this flexibility makes possible to avoid untrustworthy information, such as normals estimated on triangles with bad aspect ratio. We present results of the method for applications related to the problem of modeling 2D curves from polygons and 3D surfaces from polygonal meshes. We also apply the method to problems involving subdivision surfaces and front-tracking of moving boundaries. Finally, as our technique generalizes the recently proposed HRBF Implicits technique, comparisons with this approach are also conducted.     

A New Approach for Direct Manipulation of Free-Form Curve

There is an increasing demand for more intuitive methods for creating and modifying free-form curves and surfaces in CAD modeling systems. The methods should be based not only on the change of the mathematical parameters, such as control points, knots, and weights, but also on the user's specified constraints and shapes. This paper presents a new approach for directly manipulating the shape of a free-form curve, leading to a better control of the curve deformation and a more intuitive CAD modeling interface. The user's intended deformation of a curve is automatically converted into the modification of the corresponding NURBS control points and knot sequence of the curve. The algorithm for this approach includes curve elevation, knot refinement, control point repositioning, and knot removal. Several examples shown in this paper demonstrate that the proposed method can be used to deform a NURBS curve into the desired shape. Currently, the algorithm concentrates on the purely geometric consideration. Further work will include the effect of material properties.