Canopy surface technique for parametric blending

A method based on canopy surfaces is presented for blending parametric surfaces. The blend designed using this method gives freedom to the designer in selecting i) the primitive surfaces to be blended, ii) the contact curves lying on them (in which the blend meets the primitives), iii) the endtangent directions along the contact curves (which are used to ensure tangentplane continuity), and iv) the shape of the cross-sectional curve. An important feature of this method, which is not seen in earlier methods, is the use of end tangents to ensure theC ¹ continuity of iso-parametric curves across the junctions between the blend and the primitives in addition to the tangent-plane continuity.     

Theory of contact for geometric continuity of parametric curves

This paper develops a theory of contact for piecewise parametric curves based on the differential geometry of evolutes, polar curves, and binormal indicatrices. This theory is completely geometric, independent of parametrization and generalizes to any order. Two sets of dimensionless, characteristic numbers describing the local geometry of a curve up tonth order are defined. These characteristic numbers can be used to describe conditions for higher order contacts in an algebraic fashion. The same characteristic numbers can also be used to interpret contact conditions of up tonth order in terms of the geometry of higher evolutes and binormal indicatrices. The resulting geometric contact conditions are used to design piecewise parametric curves for Computer Aided Geometric Design (CAGD) applications.     

Interactive 3D garment design with constrained contour curves and style curves

This paper presents interactive techniques to design 3D garments conveniently and precisely with constrained contour curves and style curves. Contour curves, including silhouette curves and cross section curves, are used for garment surface modeling. Style curves, including seam lines, dart lines, notch lines and grain lines, are introduced for designing patterns on the triangular garment surfaces. Contour curves are extracted automatically from the boundaries of garment sub-meshes. The definitions and resolving rules of various constraints are introduced for editing the contour curves conveniently. Style curves are generated by projecting control points onto 3D garment triangular surfaces. In order to draw the style curves validly, some constraints are also introduced according to the craft requirements of pattern design. Furthermore, the effects of style curves in pattern flattening are analyzed, which can guide the designer to draw style curves more reasonably and enhance the pattern design quality. Finally, some examples are given to show that our methods can make the 3D garment design more flexible and friendly, and style curves can be applied into design patterns on 3D triangular surface for shoes, toys and so on.     

Drawing curves onto a cloud of points for point-based modelling

Point-based geometric models are gaining popularity in both the computer graphics and CAD fields. A related design/modelling problem is the focus of the reported research: drawing curves onto digital surfaces represented by clouds of points. The problem is analyzed and solved, and a set of ‘design tools’ are proposed which allow the user/designer to efficiently perform ‘product development’ (alternative name: ‘detail design’) tasks which require efficient processing of a ‘digital surface’. The primary tool is a robust and efficient point projection algorithm combined with a smoothing technique for producing smooth ‘digital curves’ lying onto the cloud surface. The new design tools are tested on a real-life industrial example with very satisfactory results, which are thoroughly presented in the paper.     

基于曲率优化的机器人砂带磨削轨迹规划

针对复杂曲面零件砂带磨削编程效率低、精度差的问题,基于B样条曲线曲面重构和机器人离线编程技术,提出了一种根据关键接触点曲率值生成工业机器人磨削轨迹的方法.首先,利用零件表面上需要进行砂带磨削的关键接触点和积累弦长参数化法构造节点矢量,从而计算出磨削轨迹的B样条基函数;其次,根据控制顶点反求矩阵得到全部未知控制点和3次B样条加工曲线;然后,分析关键接触点之间的曲率变化率和弧长,对关键点细化生成符合磨削工艺要求的目标点;最后,通过求解双3次B样条插值曲面方程获得目标点的加工姿态.以水龙头磨削为例进行试验,结果表明曲率优化算法磨削的零件表面轮廓形状明显优于截面法,且其粗糙度值能稳定在0.082 μm左右,可以有效提高工件表面加工质量.     

构造任意参数曲线曲面的简捷方法

本文探讨了用控制参量形式表示各种参数三次曲线、曲面和二次Ｂｅｚｉｅｒ、二次Ｂ－ｓｐｌｉｎｅ曲线、曲面的基函数统一表达式。采用改变控制参数取值的方法构造所需的各种曲线、曲面，为曲线，曲面造型提供了一种简捷的数学方法，还讨论了参量的不同取值所对应的不同种类曲线、曲面的几何特性。     

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.     

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.     

有理多结点样条插值曲线及曲面

鉴于多结点样条曲线（MSIC）是一种点点通过的插值样条曲线,因此在多结点样条插值曲线研究的基础上,给出了有理多结点条插值曲线和有理多结点样条插值曲面的定义,并讨论了有理多结点样条的性质,对有理多结 样条曲线和有理多结点样条曲面的光滑拼接问题进行了讨论,此外,还对有理多结点样条在计算机辅助几何设计中的若干应用问题进行了说明。     

Representation of piecewise continuous algebraic surfaces in terms of B-splines

A method for representing shape using portions of algebraic surfaces bounded by rectangular boxes defined in terms of triple product Bernstein polynomials is described and some of its properties are outlined. The method is extended to handle piecewise continuous algebraic surfaces within rectangular boxes defined in terms of triple products of B-spline basis functions. Next, two techniques for sculptured shape creation are studied. The first is based on geometric manipulation of existing primitives and the second on approximation/interpolation of lower dimensional entities using least-squares techniques based on singular value decomposition. In addition, several interrogation techniques, such as contouring, ray tracing and curvature evaluation, used in the design and analysis of piecewise continuous algebraic surfaces are discussed.     

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.     

Detecting regions with undesirable curvature

The use of Bézier curves or Bézier surfaces as ‘faired’ interpolation (approximation) curves or surfaces occasionally yields inflection points or changes of sign of the Gaussian curvature which are undesirable in some applications, notably the fairing of ship lines (surfaces) or the fairing of car body surfaces. In the following investigation we use a polarity to detect such undesirable points, then these points can be removed by the use of the well known Bézier techniques.     

C-Bézier曲线曲面的扩展

运用积分定义的方式,构造了带多形状参数的C-Bézier曲线曲面,改变形状参数的值,能整体或局部调控曲线曲面的形状。它们包含C-Bézier曲线曲面为其特例且具有C-Bézier曲线曲面的主要性质。还给出了带2个形状参数的二次C-Bézier曲线段G¹和C¹拼接条件以及带3个形状参数的三次C-Bézier曲线段G¹、C¹和G²拼接条件。     

异度隐函数样条曲线曲面

隐式曲线曲面被广泛应用于曲线曲面插值、逼近与拼接. 通过添加辅助曲线曲面,提出异度隐函数样条曲线曲面方法,并对其插值性、凸性与正则性进行分析. 具体实例表明,异度隐函数样条提供了次数低、构造简单、灵活性好的曲线曲面插值与拼接方法.     

Fairing of Parametric Cubic B-spline Curves and Bicubic B-spline Surfaces

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Collision and intersection detection of two ruled surfaces using bracket method

Collision and intersection detection of surfaces is an important problem in computer graphics and robotic engineering. A key idea of our paper is to use the bracket method to derive the necessary and sufficient conditions for the collision of two ruled surfaces. Then the numerical intersection curve can be characterized. The cases for two bounded ruled surfaces are also discussed.     

一种构造任意类三次三角曲线的方法

在自由曲线曲面造型中,一般多以多项式为基函数构造参数曲线曲面,而在三角函数空间中也能构造参数曲线曲面.给出了一种构造任意类三次三角参数曲线的方法,该法以三次多项式曲线的基本性质为基础,从而构造出的曲线与对应的三次多项式曲线具有几乎完全相似的性质,而且所构造的曲线能精确表示圆弧、椭圆弧、抛物线弧等二次曲线,为曲线曲面造型提供了一种新方法.     

一类双参数类四次三角Bézier曲线及其扩展

给出了一类双参数的类四次三角Bézier曲线及其扩展曲线的定义,得到了该类曲线及其扩展曲线的性质,给出了两段双参数的类四次三角Bézier曲线[G1(C1),G2(C2)]及两段扩展曲线[G1(C1),G2(C2)]光滑拼接的充要条件,并讨论了这两类曲线的应用。算例表明,该类曲线及其扩展曲线在曲线造型,特别是在非对称图形的造型中,具有很强的描述能力。     

Rational adaptive blends among obstacles in 3D by contour method

In this paper we will continue in investigating ‘contour method’ and its using for the computation of rational parameterizations of canal surfaces without a need of sum of squares (SOS) decomposition. Further approaches for constructing flexible smooth transitions between canal surfaces will be presented. Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space. Using this type of curves significantly simplifies the previous methods discussed in Bizzarri (2015), and mainly new situations, which could not have been handled with the previous setup, are successfully solved, now. Especially a method for constructing rational adaptive blends which bypass a given obstacle (or more given obstacles when needed) is thoroughly discussed and its functionality is demonstrated on a number of examples. The designed approach works not only for simple obstacles represented by one-dimensional medial axis transforms but also for more general obstacles described by two-dimensional medial surface transforms.     

Geometric Approach to Measure-Based Metric in Image Segmentation

The Mumford-Shah functional and related algorithms for image segmentation involve a tradeoff between a two-dimensional image structure and one-dimensional parametric curves (contours) that surround objects or distinct regions in the image. We propose an alternative functional that is independent of parameterization; it is a geometric functional given in terms of the surfaces representing the data and image in the feature space. The Γ-convergence technique is combined with the minimal surfaces theory to yield a global generalization of the Mumford-Shah segmentation function.