Blind digital watermarking of rational Bézier and B-spline curves and surfaces with robustness against affine transformations and Möbius reparameterization

We present a blind watermarking scheme for rational Bézier and B-spline curves and surfaces which is shape-preserving and robust against the affine transformations and Möbius reparameterization that are commonly used in geometric modeling operations in CAD systems. We construct a watermark polynomial with real coefficients of degree four which has the watermark as the cross-ratio of its complex roots. We then multiply the numerator and denominator of the original curve or surface by this polynomial, increasing its degree by four but preserving its shape. Subsequent affine transformations and Möbius reparameterization leave the cross-ratio of these roots unchanged. The watermark can be extracted by finding all the roots of the numerator and denominator of the curve or surface: the cross-ratio of the four common roots will be the watermark. Experimental results confirm both the shape-preserving property and its robustness against attacks by affine transformations and Möbius reparameterization.     

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

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Feature sensitive re-sampling of point set surfaces with Gaussian spheres

Feature sensitive simplification and re-sampling of point set surfaces is an important and challenging issue for many computer graphics and geometric modeling applications.Based on the regular sampling of the Gaussian sphere and the surface normals mapping onto the Gaussian sphere,an adaptive re-sampling framework for point set surfaces is presented in this paper,which includes a naive sampling step by index propagation and a novel cluster optimization step by normalized rectification.Our proposed re-sampling scheme can generate non-uniformly distributed discrete sample points for the underlying point sets in a feature sensitive manner.The intrinsic geometric features of the underlying point set surfaces can be preserved efficiently due to our adaptive re-sampling scheme.A novel splat rendering technique is adopted to illustrate the efficiency of our re-sampling scheme.Moreover,a numerical error statistics and surface reconstruction for simplified models are also given to demonstrate the effectiveness of our algorithm in term of the simplified quality of the point set surfaces.     

Parametric G n blending of curves and surfaces

We introduce a simple blending method for parametric curves and surfaces that produces families of parametrically defined, G ⁿ –continuous blending curves and surfaces. The method depends essentially on the parameterizations of the curves/surfaces to be blended. Hence, the flexibility of the method relies on the existence of suitable parameter transformations of the given curves/surfaces. The feasibility of the blending method is shown by several examples. The shape of the blend curve/surface can be changed in a predictable way with the aid of two design parameters (thumb weight and balance).     

Local and global analysis of parametric solid sweeps

In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a “local analysis”, which covers parametrizations and singularities, as well as a “global theory” which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the “funnel” which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface–surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and non-decomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.     

Angular interpolation of bi-parameter curves

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders.     

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.     

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.     

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.     

Computation of optimal composite re-parameterizations

Rational re-parameterizations of a polynomial curve that preserve the curve degree and [0,1] parameter domain are characterized by a single degree of freedom. The “optimal” re-parameterization in this family (that comes closest under the L₂ norm to arc-length parameterization) can be identified by solving a quadratic equation, but may exhibit too much residual parametric speed variation for motion control and other applications. Closer approximations to arc-length parameterizations require more flexible re-parameterization functions, such as piecewise-polynomial/rational forms. We show that, for fixed nodes, the optimal piecewise-rational parameterization of the same degree is defined by a simple recursion relation, and we analyze its convergence to the arc-length parameterization. With respect to the new curve parameter, this representation is only of C⁰ continuity, although the smoothness and geometry of the curve are unchanged. A C¹ parameterization can be obtained by using continuity conditions, rather than optimization, to fix certain free parameters, but the objective function is then highly non-linear and does not admit a closed-form optimization. Empirical results from implementations of these methods are presented.     

起重机参数化技术应用研究综述

随着社会的进步,人们对起重机的安全提出了更高、更严格的要求,传统的设计、仿真方法已难以满足起重机设计的现实需求。从参数化技术在起重机领域的应用现状展开研究,并结合国家当前对起重机提出安全节能的要求,指出应大力发展参数化技术的应用研究,提升参数化仿真在起重机优化设计、事故反演、安全评估等领域的应用能力,重点研究参数化设计,通过不断吸收新理念和新技术,提升参数化设计水平,以不断提高起重机质量,保证起重机的安全。     

基于NURBS方法的气动外形优化设计

采用NURBS曲线曲面,对钝锥弹头和钝双锥弹体建立参数化曲面模型,取NURBS曲线控制点作为设计参数,应用高超声速面元法求解气动力特性,在给定设计约束下,采用遗传算法进行气动外形优化设计,并对优化结果进行了比较分析。结果表明,采用NURBS方法构造参数化外形,并结合优化技术可方便快速地获得所需最优外形;与应用二次曲线构造参数化外形相比,该方法对弹体形状控制更加灵活,并可局部修改弹头曲线形状。因此,基于NURBS方法发展整套的系统优化设计算法很有现实意义和应用价值。     

Canal surfaces with rational contour curves and blends bypassing the obstacles

In this paper, we will present an algebraic condition, see (20), which guarantees that a canal surface, given by its rational medial axis transform (MAT), possesses rational generalized contours (i.e., contour curves with respect to a given viewpoint). The remaining computational problem of this approach is how to find the right viewpoint. The canal surfaces fulfilling this distinguished property are suitable for being taken as modeling primitives when some rational approximations of canal surfaces are required. Mainly, we will focus on the low-degree cases such as quadratic and cubic MATs that are especially useful for applications. To document a practical usefulness of the presented approach, we designed and implemented two simple algorithms for computing rational offset blends between two canal surfaces based on the contour method which do not need any further advanced formalism (as e.g. interpolations with MPH curves). A main advantage of the designed blending technique is its simplicity and also an adaptivity to choose a suitable blend satisfying certain constrains (avoiding obstacles, bypassing other objects, etc.). Compared to other similar methods, our approach requires only one SOS decomposition for the whole family of rational canal surfaces sharing the same silhouette, which significantly simplifies the computational complexity.     

基于约束的装配图参数化设计

介绍在参数化零部件库的基础上，通过定义参数表达式形式的装配关系或约束，来实现装配图设计的参数化设计，在设计者对零部件或装配尺寸的参数作出修改后，可自动进行装配图调整，既能充分体现设计者的设计意图，又能快速地按照设计者意图自动生成装配图，达到零部件参数化和装配设计参数化的统一。     

Degenerate parametric curves

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.     

Curve implicitization using moving lines

It is a classical result that two corresponding pencils of lines intersect in a conic section, and likewise any conic section can be expressed as the intersection of two pencils of lines. We here extend the idea of pencils to higher degree families lines, and show that any planar rational curve can be expressed as the intersection of two families of lines. This extension leads to a more efficient implicitization algorithm for curves, in which, for example, the implicit equation of a degree four rational curve can generally be expressed as the determinant of a 2 × 2 matrix (Bezout's resultant produces a 4 × 4 matrix and Sylvester's resultant an 8 × 8 matrix).     

Homogeneous bounding boxes as tools for intersection algorithms of rational bézier curves and surfaces

In the divide-and-conquer algorithm for detecting intersections of parametric rational Bézier curves (surfaces), we use bounding boxes in recursive rough checks. In this paper, we replace the conventional bounding box with a homogeneous bounding box, which is projectively defined. We propose a new rough check algorithm based on it. One characteristic of the homogeneous bounding box is that it contains a rational Bézier curve (surface) with weights of mixed signs. This replacement of the conventional bounding box by the homogeneous one does not increase the computation time.     

Photographic composite detection using circles

In this work, we propose several new methods for detecting photographic composites using circles. In particular, we focus on three kinds of scenes: (1) two coplanar circles with the same radius; (2) a single circle with its discriminable center; (3) a single circle with geometric constraints for camera calibration. For two circles’ situation, we first estimate the focal length based on the equality of the sizes of two coplanar circles, and then estimate the normal vector of the world circle plane. Inconsistencies in the angles among the normal vectors (Each circle determines a normal vector) are used as evidence of tampering. On the other hand, for the single circle case, we warp the circle to make metric measurement. To demonstrate the effectiveness of the approach, we present results for synthetic and visually plausible composite images.