不规则三维曲面拼接的研究

介绍了双参数控制的双Ｂｅｔａ样条曲面的算法原理１边界拼接的处理方法，并用Ｃ语言编程完成对若干个双Ｂｅｔａ样条曲面的拼接．     

不规则三维曲面消隐问题的研究

本文介绍了双参数控制的双Ｂｅｔａ样条曲面的算法原理、消隐算法及实现方法，并用Ｃ语言编程实现了对双Ｂａｔｅ样条曲面的快速消隐     

一个求非均匀有理B—样条曲线曲面交点（线）的算法

本文论述了非均匀有理Ｂ－样条曲线曲面的杂交方法，在４维空间里，由非均匀有理Ｂ－样条曲线（面）分割成有理Ｂｅｚｉｅｒ样条曲线（面），进而用离散求效法求出其交点（线）。     

一个求非均匀有理B－样条曲线曲面交点（线）的算法

本文论述了非均匀有理Ｂ－样条曲线曲面的求交方法，在４维空间里，由非均匀有理Ｂ－样条曲线（面）分割成有理Ｂｅｚｉｅｒ样条曲线（面），进而用离散求交法求出其交点（线）．     

NURBS曲面在工程曲面设计中的应用

本文探讨了工程曲面设计中应用ＮＵＲＢＳ（非均匀有理Ｂ样条）曲面的方法，包括所设计曲面的预处理，控制顶点与节点的设计，以及通过调整权重修改曲面的方法等。     

参数样条曲面OFFSET的BP算法

提出了用ＢＰ神经网络计算参数样条曲面ＯＦＦＳＥＴ曲面的新方法。该方法能保证对于每一组给定的参数、参数样条曲面及其ＯＦＦＳＥＴ曲面上的点一一与之对应。文章以非均匀有理Ｂ样条曲面为例，给出计算实例。结果表明，该方法可行、稳定、具有实际应用价值。     

双Beta样条曲面造型系统的研究

该文介绍了双参数控制的双Ｂｅｔａ样条曲面的算法原理、边界条件处理、投影变换，以此为基础设计了自由曲面造型系统（ＦＳＭＳ），并用ＦＳＭＳ开发的实例做了验证分析。     

Beta样条插值的保凸方法

本文采用Ｂｅｔａ样条函数拟合曲线，提出了通过控制Ｂｅｔａ样条的控制矢量以保证拟合的保凸性，建立了以节点多边形与控制多边形之间关系为条件的保凸性判别准则。     

双Beta样条曲面在屏幕上自动调整和显示

介绍了双参数控制的双Ｂｅｔａ样条曲面的算法，投影变换，消隐处理，在屏幕上自动调整和显示的原理和实现方法，并给出应用实例。     

基于NURBS的船体曲面造型

介绍基于非均匀有理Ｂ样条的船体曲面造型的设计过程和方法，实现了三向光顺的船体曲面造型。     

旋转曲面CAD的有理B样条方法

本文给出了旋转曲面CAD的一种有理B样条新方法。只要轮廓曲线用非均匀有理二次B样条曲线来表示,就可相当方便地把旋转曲面表示成非均匀有理双二次B样条曲面。这样做,能把圆柱面、圆锥面、圆环面、球面乃至一 般旋转曲面的程序软件和自由曲面的程序软件都用有理B样条的格式统一处理,对CAD工作是有益的。文中附有旋转曲面和球面的生成实例。     

The -basis and implicitization of a rational parametric surface

The concept of a μ-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The μ-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a μ-basis to an arbitrary rational parametric surface. We show that: (1) the μ-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the μ-basis is in fact a basis of the moving plane module of the rational surface; and (3) the μ-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the μ-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.     

Surface approximation with rational B-splines

Many modern geometric modelers use nonuniform rational B-spline curves and surfaces as their canonical representations. Rational B-splines are a versatile representation, encompassing integral B-splines and the basic classical primitives such as conics, quadrics, and torii. However, rational B-splines representations other than these classical primitives have found little application in surface modeling. In this paper we develop approximation algorithms based on the general rational B-spline formulation. Numerical experiments indicate that rational B-splines allow a significantly more compact approximation of two classes of parametric surfaces in comparison to integral B-splines. The two classes of surfaces studied are generalized cylinders and offsets of a rational B-spline surface patch progenitor.     

Kernel modeling for molecular surfaces using a uniform solution

In this paper, a rational Bézier surface is proposed as a uniform approach to modeling all three types of molecular surfaces (MS): the van der Waals surface (vdWS), solvent accessible surface (SAS) and solvent excluded surface (SES). Each molecular surface can be divided into molecular patches, which can be defined by their boundary arcs. The solution consists of three steps: topology modeling, boundary modeling and surface modeling. Firstly, using a weighted α-shape, topology modeling creates two networks to describe the neighboring relationship of the molecular atoms. Secondly, boundary modeling derives all boundary arcs from the networks. Thirdly, surface modeling constructs all three types of molecular surfaces patch-by-patch, based on the networks and the boundary arcs. For an SES, the singularity is specially treated to avoid self-intersections. Instead of approximation, this proposed solution can produce precise shapes of molecular surfaces. Since rational Bézier representation is much simpler than a trimmed non-uniform rational B-spline surface (NURBS), computational load can be significantly saved when dealing with molecular surfaces. It is also possible to utilize the hardware acceleration for tessellation and rendering of a rational Bézier surface. CAGD kernel modelers typically use NURBSs as a uniform representation to handle different types of free-form surface. This research indicates that rational Bézier representation, more specifically, a bi-cubic or 2×4 rational Bézier surface, is sufficient for kernel modeling of molecular surfaces and related applications.     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

有理B样条曲面的区间隐式化

提出有理B样条曲面的区间隐式化方法,即对一个有理B样条曲面,寻求包含给定的曲面的区间隐式B样条曲面,使得区间隐式B样条曲面的"厚度"尽量小,同时尽量避免出现多余分支.该问题等价于求区间隐式B样条曲面的2个边界曲面.针对该问题建立一个最优化模型并求解.     

Surfaces with Pythagorean normals along rational curves

A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals (or shortly a PSN curve). These curves represent rational paths on the surface along which the surface possesses rational offset curves. Our aim is to study rational surfaces containing enough PSN curves. The relation with PN surfaces will be also investigated and thoroughly discussed. The algebraic and geometric properties of PSN curves will be described using the theory of double planes. The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical-control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it. A special attention will be devoted to rational surfaces with pencils of PSN curves.     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

有理曲面的区间隐式化

利用一个低阶多项式区间隐式曲面来包围所给的参数式有理曲面,并构造了一些关于区间隐式曲面厚度和微分张量的目标函数.在最小化这些目标函数的条件下,该区间隐式曲面的中心曲面可以近似地逼近有理曲面,其逼近的误差可以利用区间隐式曲面的区间宽度进行估计.最后提供了具体的算法和一些实例.     

Computing rational bisectors

Bisector construction plays an important role in many geometric computations. This article explains how to compute rational bisectors of point-surface and sphere-surface pairs. This article shows that the bisector of a point and a rational surface in R³ (3D space) is also a rational surface. This result implies that the bisector of a sphere and a surface with a rational offset is also a rational surface. Even a simple rational bisector between two spheres and that between a point and a sphere have many important applications in practice. The bisector between a cube and a sphere consists of various surface patches, some of them are the bisectors between portions of the sphere and the corners of the cube. An application that uses the bisector of two spheres (of different radii) occurs in computing an optimal path for an airplane trying to avoid radar detection. Assuming each radar has different intensity, we can model the influence regions with spheres of different radii. The optimal path must lie on the bisector surface of the spheres