Curves with rational Frenet-Serret motion

This paper examines a special type of rational curves called rational Frenet-Serret (RF) curves distinguished by the property that the motion of their Frenet-Serret frame is rational. It is shown that a rational curve is an RF curve if and only if it has rational speed and rational curvature. The paper derives a general representation formula for RF curves suitable for geometric design and provides a geometric survey of special RF curves. The special case of a cubic helix is examined thoroughly. Additionally the paper discusses several applications including examples for the design of rational sweeping surfaces, rational pipe surfaces and rational transition surfaces joining sweeps with G¹ continuity.     

On bending invariant signatures for surfaces

Isometric surfaces share the same geometric structure, also known as the "first fundamental form." For example, all possible bendings of a given surface that includes all length preserving deformations without tearing or stretching the surface are considered to be isometric. We present a method to construct a bending invariant signature for such surfaces. This invariant representation is an embedding of the geometric structure of the surface in a small dimensional Euclidean space in which geodesic distances are approximated by Euclidean ones. The bending invariant representation is constructed by first measuring the intergeodesic distances between uniformly distributed points on the surface. Next, a multidimensional scaling technique is applied to extract coordinates in a finite dimensional Euclidean space in which geodesic distances are replaced by Euclidean ones. Applying this transform to various surfaces with similar geodesic structures (first fundamental form) maps them into similar signature surfaces. We thereby translate the problem of matching nonrigid objects in various postures into a simpler problem of matching rigid objects. As an example, we show a simple surface classification method that uses our bending invariant signatures.     

Design and recovery of 2-D and 3-D shapes using rational Gaussian curves and surfaces

A new representation for parametric curves and surfaces is introduced here. It is in rational form and uses rational Gaussian bases. This representation allows design of 2-D and 3-D shapes, and makes recovery of shapes from noisy image data possible. The standard deviations of Gaussians in a curve or surface control the smoothness of a recovered shape. The control points of a surface in this representation are not required to form a regular grid and a scattered set of control points is sufficient to reconstruct a surface. Examples of shape design, shape recovery, and image segmentation using the proposed representation are given.     

A weighted mean approach to smooth parametric representation of polygon meshes

A method for representing genus-zero polygon meshes by smooth parametric surfaces is described. A surface is defined by a weighted sum of linear functions, each describing a polygon face in parametric form. Rational Gaussian blending functions that adapt to the size and shape of mesh faces are used as the weights. The proposed representation has a very high degree of continuity everywhere and provides a smoothness parameter that can be varied to produce surfaces at varying resolutions. It is shown that the representation facilitates geometry processing of meshes. The use of locally supported weight functions as an alternative to rational Gaussian weights is also discussed.     

有理Beta样条曲面造型系统设计与实现

通过对有理Ｂｅｔａ样条曲面性质有其仅因子几何意义的讨论，描述了此造型系统的结构和功能以及其于窗口设计，面向对象的事件驱动环境。同时给出了有理Ｂｅｔａ样条曲面的算法及实步骤。最后通过实例对系统的实现加以说明。     

Representation of Dupin cyclides using quaternions

Dupin cyclides are surfaces characterized by the property that all their curvature lines are circles or lines. Spheres, circular cylinders, cones and tori are particular examples. We introduce a bilinear rational Bézier-like formula with quaternion weights for parametrizing principal patches of Dupin cyclides. The proposed construction is not affine invariant but it is Möbius invariant, has lower degrees compared with the standard representation, and it is convenient for offsetting. Several important properties of Dupin cyclides can be recovered in terms of closed quaternion formulas: implicit equation, principal curvatures, representation as canal surfaces. Advantages of this approach are demonstrated by deriving a new formula for the Willmore energy of a principal patch.     

Cyclides in surface and solid modeling

It is shown that Dupin cyclides (C.P. Dupin, 1822), as surfaces in computer-aided geometric design (CAGD), have attractive properties such as low algebraic degree, rational parametric forms, and an easily comprehensible geometric representation using simple and intuitive geometric parameters. Their alternative representations permit the transition between forms when one or the other is more convenient for a specific purpose. Cyclides provide is useful extension of geometric coverage in solid modeling, primarily as blending surfaces for many commonly occurring situations. The geometry, properties, and uses of the Dupin cyclide in free-form surface modeling and blending are discussed     

Geometric Invariant Shape Representations Using Morphological Multiscale Analysis

In this paper, we present a new geometric invariant shape representation using morphological multiscale analysis. The geometric invariant is based on the area and perimeter evolution of the shape under the action of a morphological multiscale analysis. First, we present some theoretical results on the perimeter and area evolution across the scales of a shape. In the case of similarity transformations, the proposed geometric invariant is based on a scale-normalized evolution of the isoperimetric ratio of the shape. In the case of general affine geometric transformations the proposed geometric invariant is based on a scale-normalized evolution of the area. We present some numerical experiments to evaluate the performance of the proposed models. We present an application of this technique to the problem of shape classification on a real shape database and we study the well-posedness of the proposed models in the framework of viscosity solution theory.     

Tunnel-free voxelisation of rational Bézier surfaces

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bézier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bézier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bézier surface and then voxelising each of the rational Bézier surfaces. We indicate the advantages carrying out this procedure.     

Approximating a helix segment with a rational Bézier curve

From a differential geometric point of view a helix segment can be considered as a spatial generalization of a circular arc. Thus for problems of shape preservation and geometric modelling an approximate rational representation of a helix segment is of interest. In this paper rational Bézier curves of degree 4, 5 and 6 are presented that approximate a helix segment. The approximants fulfill certain geometric constraints. A generalized degree elevation for rational polynomials in Bézier representation is discussed and used for the construction.     

Planar shape representation and matching under projective transformation

This paper introduces a new representation for planar objects which is invariant to projective transformation. Proposed representation relies on a new shape basis which we refer to as the conic basis. The conic basis takes conic-section coefficients as its dimensions and represents the object as a convex combination of conic-sections. Pairs of conic-sections in this new basis and their projective invariants provides the proposed view invariant representation. We hypothesize that two projectively transformed versions of an object result in the same representation. We show that our hypothesis provides promising recognition performance when we use the nearest neighbor rule to match projectively deformed objects.     

三次有理双圆弧曲线与双圆弧直纹面

本文提出了由共面四点确定双圆弧曲线的方法，分析了它的几何性质，并建立了双圆弧曲线的三次有理参数形式的方程，它可在计算机上表示。作为应用，本文构造一类以双圆弧为横向截线的直纹曲面，包括其特殊情形双圆弧锥面与双圆弧柱面，这些算法对于计算机辅助形设计与数控技术是有益的。     

Lupaşq-Bézier曲线的几何求值算法及其应用

Lupa?q-Bézier曲线是一种以q-整数作为形状参数的广义Bézier曲线.本文构造了Lupa?q-Bézier曲线的一种新型几何求值算法,该算法倒数第二层2个节点的仿射组合与曲线相切.利用算法的相切性质得到Lupa?q-Bézier曲线导矢的一种新表示,并实现了Lupa?q-Bézier曲线的细分.特别地,二次...     

一种起始点无关的小波系数形状匹配

小波变换的多分辨率特征使其在计算机视觉中得到广泛的应用,在形状匹配中,小波变换对起始点的依赖制约了小波变换的应用。为了克服小波变换对起始点的依赖,引入Zernike矩,提出一种起始点无关的小波系数形状匹配算法。对输入图像进行预处理后提取目标轮廓,生成具有平移、尺度不变的形状链状表达,并通过小波变换进行多尺度分析。最后计算各个尺度下的各阶Zernike矩,来解决小波变换的起始点问题,实现形状表达的旋转不变性。实验结果表明该算法适用于轮廓较明显的目标,同时具有速度快、精度高、鲁棒性强的优点。     

CE-Bézier 曲面光滑拼接的研究与实现

针对CE-Bézier 曲面造型中复杂曲面难以用单一曲面来表示的问题,通过 分析CE-Bézier 曲线的唯一性,提出了一种新的CE-Bézier 曲面的光滑拼接技术。首先,在 分析第1 类CE-Bézier 曲线基函数及其端点性质的基础上,对第1 类CE-Bézier 曲线的唯一 性进行了研究,得出了对于同一条第1 类CE-Bézier 曲线可以有很多组不相同的控制顶点和 形状参数与之对应的结论;其次,利用该结论进一步给出了两相邻第1 类CE-Bézier 曲面片 间G1 光滑拼接的一般几何条件,并通过合理地选取形状参数,进一步简化了该曲面的G1 拼接条件;最后,给出了第1 类CE-Bézier 曲面光滑拼接的几何造型实例。实例结果表明, 该方法简单、直观、易实现,有效地增强了CE-Bézier 方法表达复杂曲线曲面的能力,可广 泛地应用于工程复杂曲面的造型系统中。     

A simple technique for NURBS shape modification

Non-uniform rational B-splines (NURBS) have become a de-facto industry standard primarily because they offer a unified mathematical form for representing both freeform shapes and analytical curves or surfaces. Despite these advantages, designers need software that lets them work with NURBS in a natural way. The paper presents a unified approach to NURBS shape modification that builds on a perspective functional transformation of arbitrary origin and provides a homogeneous interface based on simple, easily understood geometric concepts     

三角域上带形状参数的三次Bézier曲面

张量积Bézier曲面被成功地应用于商业CAD系统中,然而实际工程中的某些外形却无法依靠张量积形式实现.因此在CAGD中,三角Bézier曲面成为外部形状设计的主要工具之一.为了更加灵活地控制三角曲面的形状,构造了一组带形状参数的三次多项式基函数,它们是三角域上三次Bernstein基的扩展.利用该组基函数定义了三角域上带形状参数的多项式曲面.基函数和曲面分别具有Bernstein基和Bézier曲面的性质.在形状参数的取值范围内,三次Bézier三角曲面是它的特例.由于含有可调的形状参数,该曲面在形状修改与变形中具有更大的灵活性.形状参数具有明确的几何意义,参数越大曲面越逼近控制网格.实例表明,通过改变形状参数的取值可以调整曲面的形状,在CAGD中该方法是有效的.     

共形几何代数--几何代数的新理论和计算框架

共形几何代数是一个新的几何表示和计算工具.作为几何的高级不变量和协变量系统的结合,它为经典几何提供了统一和简洁的齐性代数框架,以及高效的展开、消元和化简算法,从而可以进行极其复杂的符号几何计算,在几何建模与计算方面表现出很大的优势.主要讲述共形几何代数的产生背景和意义,共形几何代数的数学理论和它最有特色的几个部分,包括Grassmann结构、统一几何表示和旋量作用、基本不变量系统和高级不变量系统、新的计算思想、展开和化简技术等.     

Development of a multi-resolution framework for NUBS

The piecewise polynomial B-spline representation is a flexible tool in Computer Aided Geometric Design (CAGD) for representing and designing the geometric objects. In the field of Computer Graphics (CG), Computer Aided Design (CAD), or Computer Aided Engineering (CAE), a very useful property for a given spline model is to have locally supported basis functions. This allows localized modification of the shape. Unfortunately this property can also become a serious disadvantage when the user wishes to edit the global shape of a complex object. A multi-resolution representation is proposed as a solution to alleviate this problem.In this work, we propose a multi-resolution representation for Non-uniform B-splines (NUBS). The proposed multi-resolution model has three features that it uses control point decimation strategy for decomposing NUBS curves and it is efficient in both time and space utilization. A comparative study of the proposed work is also made with an alternate approach in the literature, which is based upon knot decimation.