用三次NURBS表示圆弧与整圆的算法研究

提出了一种实用的三次ＮＵＲＢＳ曲线表示圆弧及整圆的方法，并得出了各种情况下现成可用的结果。该方法用４个控制顶点所决定的一段ＮＵＲＢＳ曲线来表示一段小于１８０°的圆弧段；对于大于１８０°的圆弧则采用两段三次ＮＵＲＢＳ曲线来表示，当圆心角为３６０°时，则得到了整圆的表示。文中所述的方法统一、简单、符合对圆弧ＮＵＲＢＳ表示的要求。     

NURBS曲在G^1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的衫和造系统得到发展，对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一光滑程度的合成曲面，满足局部设计和修改的目的，本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ＾１光滑拼接条件，得到相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一人ＮＵＲＢ     

NURBS曲面G~1光滑拼接算法

非均匀有理Ｂ样条（ＮＵＲＢＳ）曲线、曲面造型方法，是当前ＣＡＤ／ＣＡＭ领域研究热点之一，大量的基于ＮＵＲＢＳ的实用造型系统得到发展。对ＮＵＲＢＳ而言，虽然具有参数连续性，但为了实用需要，仍需构造具有一定光滑程度的合成曲面，满足局部设计和修改的目的。本文给出了实用的具有二次公共边界曲线的ＮＵＲＢＳ曲面片Ｇ１光滑拼接条件，得到了相应控制顶点、权系数的具体算法；对于一个已知ＮＵＲＢＳ曲面，构造另一个ＮＵＲＢＳ曲面，使其达到Ｇ１拼接是简单易行的。     

NURBS造型方法的研究与实现

本文通过ＮＵＲＢＳ造型方法关键技术的研究，在我校自主开发的超人（ＳｕｐｅｒＭａｎ）ＣＡＤ／ＣＡＭ集成系统中成功地实现了基于复合曲线ＮＵＲＢＳ蒙皮曲面的造型功能，双向蒙皮曲面的造型功能和ＮＵＲＢＳ曲线曲面的可控修形功能，建立了窒完全基于ＮＵＲＢＳ方法的ＣＡＤ／ＣＡＭ系统，大量的工程应用和实例验证，表明本研究的ＮＵＲＢＳ方法是一种适用于复杂外形产品设计与制造的，灵活有效的造型方法。     

NURBS表示圆弧曲线的实用方法

根据推导得到二次和三次ＮＵＲＢＳ表示圆弧曲线的实用方法,给出计算控制顶点及其权值的表式,同时对该方法的适用范围进行了详细探讨,澄清了以往应用三次ＮＵＲＢＳ表示圆弧曲线的一点错误认识。     

NURBS双向蒙皮造型方法的研究与实现

本文通过对ＮＵＲＢＳ造型方法技术的研究，在自主开发的ＳｕｐｅｒＭａｎＣＡＤ／ＣＡＭ集成系统中成功地实现了基于复杂曲线的ＮＵＲＢＳ双向蒙皮曲面的造型功能，从而建立了完全基于ＮＵＲＢＳ方法的ＣＡＤ／ＣＡＭ系统．通过大量的工程应用和实例验证表明所研究的ＮＵＲＢＳ方法是一种适用于复杂外形产品设计与制造的、灵活有效的造型方法．     

COMPUTING AREAS BOUNDED BYRATIONAL BEZIER CURVES

ＣＯＭＰＵＴＩＮＧＡＲＥＡＳＢＯＵＮＤＥＤＢＹＲＡＴＩＯＮＡＬＢＥＺＩＥＲＣＵＲＶＥＳＧｕｏｊｉｎＷａｎｇ；ＴｈｏｍａｓＷ．ＳｅｄｅｒｂｅｒｇＣＯＭＰＵＴＩＮＧＡＲＥＡＳＢＯＵＮＤＥＤＢＹＲＡＴＩＯＮＡＬＢＥＺＩＥＲＣＵＲＶＥＳ￥ＧｕｏｊｉｎＷａｎｇ...     

CURVE AND SURFACE INTERPOLATIONBY SUBDIVISION ALGORITHMS

ＣＵＲＶＥＡＮＤＳＵＲＦＡＣＥＩＮＴＥＲＰＯＬＡＴＩＯＮＢＹＳＵＢＤＩＶＩＳＩＯＮＡＬＧＯＲＩＴＨＭＳＲｕｉｂｉｎＱｕＣＵＲＶＥＡＮＤＳＵＲＦＡＣＥＩＮＴＥＲＰＯＬＡＴＩＯＮＢＹＳＵＢＤＩＶＩＳＩＯＮＡＬＧＯＲＩＴＨＭＳ￥ＲｕｉｂｉｎＱｕＡｂｓｔｒａ...     

矩形板动力响应计算的迁移子结构法

矩形板动力响应计算的迁移子结构法杜国君，陈继荣（东北重型机械学院，齐齐哈尔）ＴＨＥＴＲＡＮＳＦＥＲＳＵＢＳＴＲＵＣＴＵＲＥＭＥＴＨＯＤＡＢＯＵＴＤＹＮＡＭＩＣＲＥＳＰＯＮＳＥＣＡＬＣＵＬＡＴＩＯＮＯＦＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＳ￥ＤｕＧｕｏ－ｊ...     

CONSTRUCTING RECTANGULAR NURBS SURFACES IN TERMS OF THREE OF ITS FOUR BOUNDARY CURVES

ＣＯＮＳＴＲＵＣＴＩＮＧＲＥＣＴＡＮＧＵＬＡＲＮＵＲＢＳＳＵＲＦＡＣＥＳＩＮＴＥＲＭＳＯＦＴＨＲＥＥＯＦＩＴＳＦＯＵＲＢＯＵＮＤＡＲＹＣＵＲＶＥＳＷａｎｇＬａｚｈｕ；ＺｈｕＸｉｎｘｉｏｎｇＣＯＮＳＴＲＵＣＴＩＮＧＲＥＣＴＡＮＧＵＬＡＲＮＵＲＢＳＳＵＲ...     

Representing Quadric Surfaces Using NURBS Surfaces

A method for representing quadrc surfaces using NURBS is presented.By means of the necessary and sufficient conditons for NURBS curves to precisely represent circular arcs and other conics,quadric surfaces can be represented by NURBS surfaces with fewer control vertices.The method can be used not only for NURBS surface representation of quadric surfaces,but also for rounding polyhedrons.Many examples are given in the paper.     

Accurate parametrization of conics by NURBS

One argument often given to explain the popularity of NURBS (nonuniform rational B-spline) is that it permits the definition of free-form curves and surfaces (as do most spline models). It also provides an exact representation of conic sections and thus of a large set of curves and surfaces used intensively in CAD: circular arcs, circles, cylinders, cones, spheres, surfaces of revolution and so forth. Nevertheless, few published works discuss the mathematical properties behind the representation of conics by NURBS except for two monographs by Piegl and Tiller (1995) and Farin (1995). The article does not pretend to fill this theoretical lack but rather deals with the following problems: all known NURBS representations of curves and surfaces based on conics have only a C^l continuity. Moreover, no technique exists that would eventually allow us to find a parametrization with a higher level of continuity. The parametrization resulting from the NURBS representation of conics can deviate significantly from the ideal are length (that is, uniform) parametrization. The only known solution to reduce this deviation is to increase the number of control points of the spline by using refinement algorithms, for instance, but such a process converges only slowly to the uniform parametrization. The solution proposed uses an original reparametrization process called zigzag reparametrization, based on a specific family of rational polynomials     

圆弧标准型有理三次Bézier表示的内在性质研究

给出圆弧带参数的标准型有理三次Bézier表示的一种实用形式,讨论参数对内控制点、两内权因子及肩点的影响。详细分析该参数与圆弧非标准型二次Bézier表示下的权因子之间的内在关系,参数值的变化对应了一个有理线性参数变换。最后讨论了圆弧标准型有理三次Bézier表示的反算问题。     

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.     

NURBS细分曲线算法

从基于差商算子定义B样条的角度,在对B样条基函数进行细分基础上提出了一种NURBS细分曲线算法,应用在自由型曲线生成和形状控制上具有良好的实际效果,完全具备了参数NURBS曲线的重要性质。最后给出了细分曲线生成圆及圆弧的实例。     

闭合圆锥曲线的2次周期性NURBS表示

本文探讨了用2次周期性NURBS表示闭合圆锥曲线的方法,导出了精确表示椭圆、抛物线和双曲线的条件,并给出了精确表示圆弧的充要条件;提出了用2次NURBS精确表示圆的新方法.和其它基于矩形控制多边形的同类方法相比,其控制点数最少.用2次非周期性NURBS(2次有理Bezier样条)和2次均匀有理B样条描述圆锥曲线的方法均是本文的一个特例.     

NURBS曲线曲面的显式矩阵表示及其算法

从 B样条的差商定义出发 ,提出差商展开系数的概念 ,通过差商展开系数显式解析表示式的导出 ,得到任意次 NU RBS曲线曲面系数矩阵的显式解析表示式 ,并给出了求差商展开系数和 NURBS曲线曲面系数矩阵的数值算法 .文中给出的方法适用于一切 NU RBS曲线曲面 ,包括有理和非有理的 Bézier、均匀和非均匀的 B样条曲线曲面 .相应的数值算法计算简单 ,易于实现 .差商展开系数解析表示式为 NU RBS曲线曲面的表示、转换和节点插入、升阶等基本运算以及与差商相关的问题的研究提供了一个统一的构造性工具和应用方法 .     

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.     

Data Approximation Using Biarcs

. An algorithm for data approximation with biarcs is presented. The method uses a specific formulation of biarcs appropriate for parametric curves in Bézier or NURBS formulation. A base curve is applied to obtain tangents and anchor points for the individual arcs joining in G 1 continuity. Data sampled from circular arcs or straight line segments is represented precisely by one biarc. The method is most useful in numerical control to drive the cutter along straight line or circular paths.