Matrix-valued 4-point spline and 3-point non-spline interpolatory curve subdivision schemes

The objective of this paper is to study and construct matrix-valued templates for interpolatory curve subdivision. Since our investigation of this problem was motivated by the need of such subdivision stencils as boundary templates for interpolatory surface subdivision, we provide both spline and non-spline templates that are necessarily symmetric, due to the lack of direction-orientation in carrying out surface subdivision in general. For example, the minimum-supported Hermite interpolatory C¹ cubic spline curve subdivision scheme, with the skew-symmetric basis function for interpolating first derivatives, does not meet the symmetry specification. Non-spline C² interpolatory templates constructed in this paper are particularly important, due to their smaller support needed to minimize undesirable surface oscillations, when adopted as boundary templates for interpolatory C² surface subdivision. The curve subdivision templates introduced in this paper are adopted as boundary stencils for interpolatory surface subdivision with matrix-valued templates.     

五点二重逼近细分法

提出了一种新的构造曲线的算法——五点二重逼近细分法。利用细分格式 的生成多项式讨论了该细分格式的一致收敛性及Ck 连续性。该细分格式带有一个张力参数 μ, 通过选取不同的μ值,可以分别生成C1~C5 连续的极限曲线。特别是当μ=9/256 时, 细 分格式生成的极限曲线可以达到C7 连续。最后给出了五点二重逼近曲线细分的实例,表明 了这种细分格式是有效的。     

曲线插值的一种具有还圆性的细分方法

传统的线性四点插值细分方法不能表示圆等非多项式曲线,为了解决这种 问题,基于几何特性提出了一种带有一个参数的四点插值型曲线细分方法。细分过程中,过 相邻三插值点作圆,过相邻二插值点的圆弧有两个中点,将其加权平均得到新插值点,文中 给出了插值公式和算法描述。所给方法具有还圆性,可以实现保凸性。实例分析对比了本方 法与多种细分方法的差异,说明本方法是有效的,当参数取值较小时,曲线靠近控制多边形。     

对Nira的四点插入生成曲线方法的进一步改进

本文对一种基于四点插入的曲线生成方法进行了进一步讨论,分析了其更一般的表达式,并在对曲线控制局部性讨论的基础上进一步分析了其C~1连续性质,给出了控制点处切向表达式的一般形式,最后给出了一种方便地控制曲线形状(控制点“硬度”)的方法,使这种曲线生成法成为一种实用的生成法。     

Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision

Many engineering design applications require geometric modeling and mechanical simulation of thin flexible structures, such as those found in the automotive and aerospace industries. Traditionally, geometric modeling, mechanical simulation, and engineering design are treated as separate modules requiring different methods and representations. Due to the incompatibility of the involved representations the transition from geometric modeling to mechanical simulation, as well as in the opposite direction, requires substantial effort. However, for engineering design purposes efficient transition between geometric modeling and mechanical simulation is essential. We propose the use of subdivision surfaces as a common foundation for modeling, simulation, and design in a unified framework. Subdivision surfaces provide a flexible and efficient tool for arbitrary topology free-form surface modeling, avoiding many of the problems inherent in traditional spline patch based approaches. The underlying basis functions are also ideally suited for a finite-element treatment of the so-called thin-shell equations, which describe the mechanical behavior of the modeled structures. The resulting solvers are highly scalable, providing an efficient computational foundation for design exploration and optimization. We demonstrate our claims with several design examples, showing the versatility and high accuracy of the proposed method.     

曲线插值约束的LOOP细分曲面

提出基于Loop细分方法的曲线插值方法,不需要修改细分规则,只需以插值曲线的控制多边形为中心多边形,向其两侧构造对称三角网格带,该对称三角网格带将收敛于插值曲线。因此,包含有该三角网格带的多面体网格的极限曲面将经过插值曲线。若要插值多条相交曲线只需在交点处构造全对称三角网格。运用该方法可在三角网格生成的细分曲面中插值多达六条的相交曲线。     

Analysis and design of Hermite subdivision schemes

Published online: 14 May 2002     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.