Minkowski isoperimetric-hodograph curves

General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case of rational general offsets, namely, Minkowski isoperimetric-hodograph curves. Differential geometric topics in the Minkowski plane, including the notion of normality, Frenet frame, Serret–Frenet equations, involutes and evolutes are introduced. These lead to an elegant process from which an explicit parametric representation of the general offset curves is derived. Using the duality between indicatrix and isoperimetrix and between involutes and evolutes, rational curves with rational general offsets are characterized. The dual Bézier notion is invoked to characterize the control structure of Minkowski isoperimetric-hodograph curves. This characterization empowers the constructive process of freeform curve design involving offsetting techniques.     

基于正交完备U-系统的参数曲线图组表达

为了探索参数曲线图组的频谱性质,引进一类属于L2[0,1]的正交完备分片k次多项式系统（简称U-系统）.该系统下的U级数展开式具有良好的平方逼近及一致逼近性质,而且能用有限项U级数实现对分段k次多项式参数曲线图组的精确表达;基于U-系统理论,给出了用以计算给定几何图组U-谱的信息转换算法,该算法具有直观、简便、快速的特点.构建的数据处理平台可用于几何信息的分析与综合,并且能在信息安全（信息隐藏、数字水印）及模式识别等方面有实用价值;提供了用U-系统表达参数曲线图组的实验图例.     

Parameterization in Finite Precision

Certain classes of algebraic curves and surfaces admit both parametric and implicit representations. Such dual forms are highly useful in geometric modeling since they combine the strengths of the two representations. We consider the problem of computing the rational parameterization of an implicit curve or surface in a finite precision domain. Known algorithms for this problem are based on classical algebraic geometry, and assume exact arithmetic involving algebraic numbers. In this work we investigate the behavior of published parameterization algorithms in a finite precision domain and derive succinct algebraic and geometric error characterizations. We then indicate numerically robust methods for parameterizing curves and surfaces which yield no error in extended finite precision arithmetic and, alternatively, minimize the output error under fixed finite precision calculations. Received January 8, 1997; revised August 27, 1998.     

Geometric Hermite interpolation for space curves

This paper considers the geometric Hermite interpolation for spacial curves by parametric quartic Bézier curve. In additon to position and tangent direction, the curvature vector is prescribed at each knot. We prove that under appropriate assumptions the interpolant exists locally with one degree of freedom. Moreover, we prove the interpolant is 6th order accurate.     

Geometric properties estimation from discrete curves using discrete derivatives

Accurate geometric properties estimation from discrete curves is an important problem in many application domains, such as computer vision, pattern recognition, image processing, and geometric modeling. In this paper, we propose a novel method for estimating the geometric properties from discrete curves based on derivative estimation. We develop derivative estimation by defining the derivative of a discrete function at a point, which will be called the discrete derivative. Similarly, the second and higher order discrete derivatives at that point are also defined, and their convergence is demonstrated by theory analysis. These definitions of the different order discrete derivatives provide a simple and reliable way to estimate the derivatives from discrete curves. Based on the discrete derivatives, classical differential geometry can be discretized, and the geometric properties are estimated from discrete curves by using differential geometry theory. The proposed method is independent of any analytic curve and estimates the geometric properties directly from discrete data points, which makes it robust to the geometric shapes of discrete curves. Another advantage of the proposed method is the robustness to noise because of the calculation characteristics of the discrete derivatives. The proposed method is evaluated and compared with other existing methods in the experiments with both synthetic and real discrete curves. The test results show that the proposed method has good performance, and is robust to noise and suitable for different curve shapes.     

CAD-based sampling for CMM inspection of models with sculptured features

This paper addresses sampling models for trimmed sculptured surfaces, and multiple patches with boundary conditions. A CAD-based sampling system is developed and implemented in this work. The sculptured features are sampled along their isoparametric curves. These curves are then used to re-construct the model geometry using the skinning of cross section curves. We refer to the re-constructed model as the substitute geometry. The problem is to determine the sample curve locations such that the substitute geometry satisfies certain geometric conditions. These are, the form error, and the continuity of the substitute geometry across the boundaries of adjacent surfaces. Three criteria are integrated to determine the sample locations: the surface curvature change, the substitute geometry deviation from the CAD model, and the significance of trimmed portions of the surface. A boundary representation-based methodology for the sampling of trimmed surfaces is developed and implemented. This methodology is extended to handle n-sided surfaces obtained through filling n-sided regions with quadrilateral surface patches, and models that may include multiple surface patches. Furthermore, a tool to assess the sampling plans based on the continuity of the substitute geometry across boundaries of adjacent surface patches is developed. The developed algorithms, their implementations, and case studies are presented in this paper.     

一类三角Bézier曲线的形状特征

在几何造型的许多应用中,良好的曲线形状应该消除不必要的奇点和拐点,因此 往往需要预知与分析参数曲线的各种形状特征,以避免出现奇异形状的设计风险。为了快速确 定参数曲线的形状特征,利用锥面的齐次性简化了参数曲线的形状条件,得出了一类带 2 个形 状参数的二次三角 Bézier 曲线的尖点条件锥和 2 张重结点边界条件锥;3 张特征锥面及其切平 面将特征空间划分为不同的特征区域。曲线的形状特征完全由特征点在特征空间的分布区域决 定。用垂直于坐标轴的平面切割特征空间,可得到基于包络与拓扑映射方法的所有形状条件分 布图。进而讨论了形状参数变化对各特征区域的影响,相关结果可使设计者明确如何配置控制 顶点或者调节形状参数,使得生成曲线为全局凸或局部凸曲线,或具有所需要的奇点与拐点, 或将当前曲线形状调节为另一种所需的形状。     

Survey of Continuities of Curves and Surfaces

This survey presents an overview to various types of continuity of curves and surfaces, in particular parametric (Cⁿ), visual or geometric (Cⁿ, Gⁿ), Frenet frame (Fⁿ), and tangent surface continuity (Tⁿ), and discusses the relation with curve and surface modeling, visibility of (dis) continuities, and graphics rendering algorithms. It is the purpose of this paper to provide an overview of types of continuity, and to put many terms and definitions on a common footing in order to give an understanding of the subject. 1991 Computing Reviews Classification: I.3.5 [Computer Graphics] Computational geometry and object modeling. I.3.7 [Computer Graphics] Three-Dimensional Graphics and Realism.     

Construction of several second- and fourth-order geometric partial differential equations for space curves

Geometric partial differential equations for curves and surfaces are used in many fields, such as computational geometry, image processing and computer graphics. In this paper, a few differential operators defined on space curves are introduced. Based on these operators, several second-order and fourth-order geometric flows for evolving space curves are constructed. Some properties of the changing rates of the arc-length of the evolved curves and areas swept by the curves are discussed. Short-term and long-term behaviors of the evolved curves are illustrated.     

Computer modelling and finite element analysis of spiral triangular strands

In this paper a new mathematical geometric model of spiral triangular wire strands with a construction of (3 + 9) and (3 + 9 + 15) wires is proposed and an accurate computational two-layered triangular strand 3D solid modelling, which is used for a finite element analysis, is presented. The present geometric model fully considers the spatial configuration of individual wires in the strand. The three dimensional curve geometry of wires axes in the individual layers of the triangular strand consists of straight linear and helical segments. The derived mathematical representation of this curve is in the form of parametric equations with variable input parameters which facilitate the determination of the centreline of an arbitrary circular wire of the right and left hand lay triangular one and two-layered strands. Derived geometric equations were used for the generation of accurate 3D geometric and computational strand models. The correctness of the derived parametric equations and performance of the generated strand model are controlled by visualizations. The 3D computational model was used for a finite element behaviour analysis of the two-layered triangular strand subjected to tension loadings. Illustrative examples are presented to highlight the benefits of the proposed geometric parametric equations and computational modelling procedures by using the finite element method.     

A shape design system using volumetric implicit PDEs

Solid modeling based on partial differential equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDE-based methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of elliptic PDEs for boundary value problems to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves and iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space. The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary and additional constraints.     

Equivalence between Closed Connected <Emphasis Type="Italic">n</Emphasis>-<Emphasis Type="Italic">G</Emphasis>-Maps without Multi-Incidence and <Emphasis Type="Italic">n</Emphasis>-Surfaces

Many combinatorial structures have been designed to represent the topology of space subdivisions and images. We focus here on two particular models, namely the n-G-maps used in geometric modeling and computational geometry and the n-surfaces used in discrete imagery. We show that a subclass of n-G-maps is equivalent to n-surfaces. To achieve this, we provide several characterizations of n-surfaces. Finally, the proofs being constructive, we show how to switch from one representation to another effectively.

Implicit curves and surfaces in CAGD

The role of implicit curves and surfaces in computer-aided geometric design (CAGD) are described. The ways in which the study of implicit algebraic curves and surfaces draws on algebraic geometry are reviewed. The implicitization of parametric curves and surfaces, parameterization of implicits, and techniques used to circumvent conversions between implicit and parametric representations are discussed     

基于Bayes-MeTiS网格划分的3D几何重构

为提升 3D模型几何重构过程的压缩效率,提出一种基于MeTiS网格划分的贝叶斯3D模型几何重构算法。首先,在编码端 采用MeTiS方法 对原始3D网格进行子网划分,采用随机线性矩阵对子网几何形状进行编码,并对边界节点的邻居节点使用伪随机数生成器进行数据序列构建;然后,利用贝叶斯算法进行几何模型重构算法的设计,在理论上给出了均值、方差矩阵以及模型参数学习规则,实现了3D模型的几何重构;最后,将其与图傅里叶光谱压缩(GFT)、最小二乘压缩(LMS)和基于压缩感知的图傅里叶光谱压缩(CSGFT)等算法进行仿真对比。结果表明,所提方法具有较高的比特率压缩指标以及较低的重构误差,计算效率明显提高。     

Simultaneous aligning and smoothing of surface triangulations

In this work we develop a procedure to deform a given surface triangulation to obtain its alignment with interior curves. These curves are defined by splines in a parametric space and, subsequently, mapped to the surface triangulation. We have restricted our study to orthogonal mapping, so we require the curves to be included in a patch of the surface that can be orthogonally projected onto a plane (our parametric space). For example, the curves can represent interfaces between different materials or boundary conditions, internal boundaries or feature lines. Another setting in which this procedure can be used is the adaption of a reference mesh to changing curves in the course of an evolutionary process. Specifically, we propose a new method that moves the nodes of the mesh, maintaining its topology, in order to achieve two objectives simultaneously: the piecewise approximation of the curves by edges of the surface triangulation and the optimization of the resulting mesh. We will designate this procedure as projecting/smoothing method and it is based on the smoothing technique that we have introduced for surface triangulations in previous works. The mesh quality improvement is obtained by an iterative process where each free node is moved to a new position that minimizes a certain objective function. The minimization process is done on the parametric plane attending to the surface piece-wise approximation and to an algebraic quality measure (mean ratio) of the set of triangles that are connected to the free node. So, the 3-D local projecting/smoothing problem is reduced to a 2-D optimization problem. Several applications of this method are presented.     

Projectively invariant classes of geometric continuity for CAGD

A new classification of geometric continuity is presented. It is based on the simultaneous application of the concept of ‘contact of order r” to curves (or surfaces) and to their tangent surfaces and generalized tangent surfaces; this implies projective invariance. The developments are illustrated by algorithms for geometrically continuous Bézier curves. Moreover, we obtain new properties of certain splines derived by variational formulations.     

From on-line sketching to 2D and 3D geometry: a system based on fuzzy knowledge

The paper describes the development of a fuzzy knowledge-based prototype system for conceptual design. This real time system is designed to infer user's sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles; arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tool, but also accepts user interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry.     

Designing parametric blends: surface model and geometric correspondence

A model based on the curve-modulation technique is proposed for designing a parametric blend between two parametrically defined surfaces. The modulants are the cross section and the spine curves. The end position and tangency conditions of the cross section are discussed. Its intermediate shape can be left to the designer's choice, and generally depends on the application at hand. Various alternative choices, that fulfill the end conditions are suggested; this provides the flexibility demanded by various applications. The spine is designed as the intersection of two derived surfaces. Two alternative surface derivations are proposed, and both are discussed in some detail with their merits and demerits. The first derivation generates the contact curves automatically and relieves the designer from specifying them; the second one accepts contact curves specified by the designer, and generates the spine accordingly. Both of them are equally important in CAD/CAM and solid modelling applications.     

Adaptive H∞ Control of Piecewise‐Linear Systems with Parametric Uncertainties and External Disturbances

In this paper, we study the adaptive H_∞ control problem for piecewise linear systems with parametric uncertainties and external disturbances. Motivated by the vector projection technique and dissipation theory, we propose a piecewise H_∞ controller and its associated projection‐type parameter adaptation law, which are capable of guaranteeing the system stability and minimizing the effect of disturbances. Moreover, the common quadratic Lyapunov function method is used such that all of the synthesis conditions are formulated as linear matrix inequities and thus can be solved efficiently. Finally, a numerical example is provided to illustrate the results.     

Mapping from parametric characteristics to generalized frequency response functions of non-linear systems

Based on the parametric characteristic of the nth-order generalized frequency response function (GFRF) for non-linear systems described by a non-linear differential equation (NDE) model, a mapping function from the parametric characteristics to the GFRFs is established, by which the nth-order GFRF can be directly written into a more straightforward and meaningful form in terms of the first order GFRF, i.e., an n-degree polynomial function of the first order GFRF. The new expression has no recursive relationship between different order GFRFs, and demonstrates some new properties of the GFRFs which can explicitly unveil the linear and non-linear factors included in the GFRFs, and reveal clearly the relationship between the nth-order GFRF and its parametric characteristic, as well as the relationship between the nth-order GFRF and the first order GFRF. The new results provide a novel and useful insight into the frequency domain analysis and design of non-linear systems based on the GFRFs. Several examples are given to illustrate the theoretical results.