三次可展Bezier曲面的构造

可展曲面在很多的工程领域里,尤其在机械工程设计中有着重要的作用,例如飞机机翼、汽车车身、船体、鞋和服装等的设计与制造等。在空间的一平面上分别生成2条3次Bezier曲线,该平面绕一固定轴旋转不同角度,生成两个相交的平面,这2条3次Bezier曲线跟随旋转,分别位于两相交平面上,并由这两条曲线生成直纹面。根据直纹面可展的充要条件,求解出未知的设计曲线和伴随曲线的控制顶点,最终生成3次可展Bezier曲面。     

参数曲面导矢界估计及其在曲面绘制中的应用

目的是改进参数曲面的绘制效率。采用的方法是在参数曲线导矢界的基础上，进一步提出了3次参数曲面导矢界的公式。基于这些公式，能够较好地解决了参数曲面绘制算法过程中出现的重复绘制的问题和不连续的问题，进一步提高了曲面绘制的效果和效率。     

Normal orientation methods for 3D offset curves, sweep surfaces and skinning

Computing an offset curve to a 3D curve involves the basic problem of normal orientation in space. The same thing is encountered when interpolating surfaces with 3D sweep or skinning methods. Different methods for normal orientation are discussed an this article. A new method is presented for normal orientation on general parametric curves.     

Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces

This paper studies geometric design of developable composite Bézier surfaces from two boundary curves. The number of degrees of freedom (DOF) is characterized for the surface design by deriving and counting the developability constraints imposed on the surface control points. With a first boundary curve freely chosen, (2m+3), (m+4), and five DOFs are available for a second boundary curve of a developable composite Bézier surface that is G⁰, G¹, and G², respectively, and consists of m consecutive patches, regardless of the surface degree. There remain five and (7-2m) DOFs for the surface with C¹ and C² continuity. Allowing the end control points to superimpose produces Degenerated triangular patches with four and three DOFs left, when the end ruling vanishes on one and both sides, respectively. Examples are illustrated to demonstrate various design methods for each continuity condition. The construction of a yacht hull with a patterned sheet of paper unrolled from 3D developable surfaces validates practicality of these methods in complex shape design. This work serves as a theoretical foundation for applications of developable composite Bézier surfaces in product design and manufacturing.     

Subdivision models for varying-resolution and generalized perturbations

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. In this paper, we are interested in both smooth and non-smooth subdivision schemes. We propose two models that generalize the subdivision operation and can yield both smooth and non-smooth schemes in a controllable way:

• (1) The ‘varying-resolution’ model allows a structured access to the various resolutions of the refined data, yielding certain patterns. This model generalizes the standard subdivision iterative operation and has interesting interpretations in the geometrical space and also in creativity-oriented domains, such as music. As an infrastructure for this model, we propose representing a subdivision scheme by two dual rules trees. The dual tree is a permuted rules tree that gives a new operator-oriented view on the subdivision process, from which we derive an ‘adjoint scheme’.

• (2) The ‘generalized perturbed schemes’ model can be viewed as a special multi-resolution representation that allows a more flexible control on adding the details. For this model, we define the terms ‘template mask’ and ‘tension vector parameter’.

The non-smooth schemes are created by the permutations of the ‘varying-resolution’ model or by certain choices of the ‘generalized perturbed schemes’ model. We then present procedures that integrate and demonstrate these models and some enhancements that bear a special meaning in creative contexts, such as music, imaging and texture. We describe two new applications for our models: (a) data and music analysis and synthesis, which also manifests the usefulness of the non-smooth schemes and the approximations proposed, and (b) the acceleration of convergence and smoothness analysis, using the ‘dual rules tree’.     

细分曲面求交裁剪算法研究

基于细分曲面的参数化表示,研究了细分曲面的精确求交、裁剪算法。首先对控制网格建立局部坐标系,将细分曲面表示为一系列小的面片,并对每个控制顶点赋予参数值。然后用改进的轮廓删除法细分控制网格,在关联曲面间进行相交性检测,得到近似交点及其参数值,再用迭代法求得精确解。根据用户指定的裁剪区域确定交线的走向,将被裁剪曲面的控制网格面分为保留面、裁剪面和删除面,设置每个裁剪面的裁剪域,从而实现细分曲面的精确裁剪。算例表明,该文的方法简单、有效。     

Spline functions in chemistry: approximation of surfaces over triangle domains

Chemists often come across triangle domains – usually with the basic simplex in ?². A smooth surface is needed for approximating the chemical properties between the measured data for solving some model (differential) equations numerically.

Our research group has been working on approximating ternary chemical surfaces of two special fields by smooth functions (vapour – Liquid equilibrium data and explosion-limit surfaces of ternary gas systems).

A mathematical solution was given in both fields by special spline surfaces, and for visualization, our own software (TRIGON) was used. In this paper the summarized chemical background of each problem is provided, the mathematical solutions, the newest theoretical developments and their results are discussed.     

Biconic subdivision of surfaces of revolution and its applications in intersection problems

This paper presents a novel method for the subdivision of surfaces of revolution. We develop a new technique for approximating the genertrix by a series of pairs of conic sections. By using an error estimate based on convex combination, an efficient least-squares approach is proposed that yields near-optimal fitting. The resulting surface approximation is shown to be more efficient than other tessellation methods in terms of the number of fitting segments. This in turn allows us to implement efficient and robust algorithms for such surfaces. In particular, novel intersection techniques based on the proposed subdivision method are introduced for the two most fundamental types of intersections – line/surface and surface/surface intersections. The experimental results show that our method outperforms conventional methods significantly in both computing time and memory cost.     

Feasibility study of robotic fibre placement on intersecting multi-axial revolution surfaces

In this paper, the first steps towards using a robotic workcell for the automated fibre placement (AFP) manufacturing of Y-shaped tubes are proposed. The proposed workcell is constituted of a standard serial manipulator holding the fibre placement toolhead combined to a rotary table on which the part where the fibres must be laid out is attached. The investigations carried out in this work explore the feasibility of this setup and more precisely the path planning aspect. To this aim two novel path planning algorithms are presented generalizing the techniques proposed in the literature for open-contoured and cylindrical surfaces. In the first, the maximal geodesic curvature typically allowed in AFP is disregarded to generate continuous paths with a constant placement angle on the branches without any gaps or overlaps. Subsequently, a second algorithm, taking into account this curvature constraint, is presented. These algorithms were implemented in a software using the MATLAB™ suite. Finally, an algorithm to optimize the motion of the robotic system is presented and simulations are discussed.     

具有变动边界的过渡面的热传导方程构造法

提出过渡面的热传导方程构造方法。使用热传导方程、初值条件和变动的边界条件,得到过渡面的偏微分方程模型,再利用分离变量法或者数值方法得到模型的解,则方程的像空间就是满足连续性要求的过渡面。数值实验也表明这种方法是可行的、有效的。     

几何网络中静态数据管理问题的近似策略

针对静态数据管理问题,设计了link cost在不满足三角不等式的情况下几何网络中此问题的近似算法。通过引入两个受限的数据安置作对比,经过类似于均态分析的算法分析,在给定相关的参数的情况下,所给的近似算法具有常数的近似度。不过,网络中link cost的最大值与最小值之比是已知的。     

Two algorithms for three-layer channel routing

A channel router is an important design aid in the design automation of VLSI circuit layout. Many algorithms have been developed based on various wiring models with routing done on two layers. With the recent advances in VLSI process technology, it is possible to have three independent layers for interconnection. In this paper two algorithms are presented for three-layer channel routing. The first assumes a very simple wiring model. This enables the routing problem to be solved optimally in a time of $\text{O}\text{(}\text{n}\text{log}\text{n}\text{)}$. The second algorithm is for a different wiring model and has an upper bound of $\text{O(}\text{n}{}^2\text{)}$ for its execution time. It uses fewer horizontal tracks than the first algorithm. For the second model the channel width is not bounded by the channel density.     

A particular class of spline in reconstruction of revolution surfaces from 3-D data measured by CMM

In this paper, we tackle the problem of revolution surface reconstruction starting from a cloud of points measured by a coordinate-measuring machine. The proposed method is a mix of two mathematical techniques: approximation by λ integral quasi-interpolating spline and linear algebraic transformation. It takes into account that a surface of revolution can be described by rotating a single cross-section around a symmetric axis and the measured points are affected by manufacturing inaccuracy and measurement uncertainty. An experimental example shows that this approach gives more accurate results with respect to B-spline approximation.     

Automating extrusion design: a case study in geometric and topological reasoning for mechanical design

A model for topological and geometric reasoning about mechanical designs is described and demonstrated by a computer program using the 2D domain of extrusion cross-sections as a case study. The design system uses a feature-based representation and has two stages: first, parametric design, in which a numerical, iterative technique is used to vary existing parameters and second, topological design (the central focus of this research), in which the extrusion cross-section is modified by adding new parameters and/or deleting old ones. This paper summarizes recent research related to topological design, then describes representation and reasoning as modelled in the computer program, presents some test cases, and outlines future directions for geometric and topological reasoning in mechanical design domains.     

Dynamic programming interpretation of construction-type plant layout algorithms and some results

This paper presents a dynamic programming interpretation of some existing construction-type algorithms for the CAD of plant layouts. The two sub-problems of constructing a layout by these algorithms are first, the selection of a department for next placement, and second, locating it in the available space in an optimum manner. The first problem is tackled by adopting a heuristic. Successive choices of this nature lend themselves to a dynamic programming interpretation. The second problem is solved by allocating those blocks of area in the vicinity of existing departments which minimize the cost of materials handling between the new and the existing departments. It is found that application of dynamic programming significantly improves the efficiency of existing construction-type procedures. Some new heuristics for the first sub-problem are also examined. Computational results on test problems indicate that more complex policies might yield better results when the number of departments becomes larger.     

带形状参数样条曲线的研究

如何通过调整形状参数修改曲线形状是计算机辅助几何设计中一个有意义的研究课题.为了有效地利用形状参数来调整曲线的形状,增强修改曲线的灵活性,研究了5种带形状参数B样条曲线的表示方法及性质,这些曲线模型都可以通过改变形状参数的取值,调整曲线接近控制多边形的程度,从而得到不同位置的连续曲线.分析了每种造型方法的形状参数对曲线形状的影响,给出了形状参数的适用范围,比较了5种造型方法的特点.通过大量的公式推导和实验,提出了利用形状参数不同取值来表示一些自由曲线的新方法,并用实例进行了说明.实验证明,C-B样条曲线、带形状参数的均匀B样条曲线、带形状参数双曲多项式的均匀B样条曲线、带形状参数三角多项式的均匀B样条曲线都可利用形状参数的特定取值表示一些工业领域常用的自由曲线,这比起用控制顶点表示同样的自由曲线更为简单.     

边界满足C0或C1连续条件的光顺过渡面的构造

针对联结两个曲面片的过渡曲面构造问题,使用偏微分方程中的热传导方程,详细研究了如何构造边界满足C⁰或C¹连续条件的光顺过渡面的偏微分方程模型,讨论了初始曲面及方程参数对光顺过渡面的影响。