Intersecting a freeform surface with a general swept surface

We present efficient and robust algorithms for intersecting a rational parametric freeform surface with a general swept surface. A swept surface is given as a one-parameter family of cross-sectional curves. By computing the intersection between a freeform surface and each cross-sectional curve in the family, we can solve the intersection problem. We propose two approaches, which are closely related to each other. The first approach detects certain critical points on the intersection curve, and then connects them in a correct topology. The second approach converts the intersection problem to that of finding the zero-set of polynomial equations in the parameter space. We first present these algorithms for the special case of intersecting a freeform surface with a ruled surface or a ringed surface. We then consider the intersection with a general swept surface, where each cross-sectional curve may be defined as a rational parametric curve or as an implicit algebraic curve.     

Proper real reparametrization of rational ruled surfaces

Let K⊆R be a computable field. We present an algorithm to decide whether a proper rational parametrization of a ruled surface, with coefficients in K(i), can be properly reparametrized over a real (i.e. embedded in R) finite field extension of K. Moreover, in the affirmative case, the algorithm provides a proper parametrization with coefficients in a real extension of K of degree at most 2.     

三次可展Bezier曲面的构造

可展曲面在很多的工程领域里,尤其在机械工程设计中有着重要的作用,例如飞机机翼、汽车车身、船体、鞋和服装等的设计与制造等。在空间的一平面上分别生成2条3次Bezier曲线,该平面绕一固定轴旋转不同角度,生成两个相交的平面,这2条3次Bezier曲线跟随旋转,分别位于两相交平面上,并由这两条曲线生成直纹面。根据直纹面可展的充要条件,求解出未知的设计曲线和伴随曲线的控制顶点,最终生成3次可展Bezier曲面。     

Implicit modeling from polygon soup using convolution

We present a novel method for creating implicit surfaces from polygonal models. The implicit function is defined by convolving a kernel with the triangles in the polygonal model. By adopting a piecewise quartic polynomial kernel function with a finite support, we derive a convolution model that has a closed-form solution, and thus can be efficiently evaluated. The user only needs to specify an effective radius of influence to generate an implicit surface of desired closeness to the polygonal model. The resulting implicit surface is fast to evaluate, not requiring accumulating evaluation results using any hierarchical data structure, and can be efficiently ray-traced to reveal the detailed features.     

A computerised minimum distance algorithm for machining of sculptured surfaces

This paper presents a unique minimum distance algorithm for machining of sculptured surfaces on computer numerical control (CNC) machines. The algorithm has been used to develop a system that graphically simulates the cutter path of the defined surfaces and generates the required CNC programs. The paper presents the experimental results of the surface roughness, tool size, and computed numerical control (NC) points for various tolerance values used on several sculptured surfaces designed by the system and actually machined on a CNC machining centre using the generated NC program from the system.     

基于隐式曲面的3维树木建模

为了利用隐式曲面构造出光滑拼接的3维树木枝条模型,同时义能避免3维树木建模中枝条曲面的单一性,以增强树木模型局部细节的真实感描述.提出了一种基于隐式曲面的3维树木建模方法,该方法首先将基于骨架的隐式曲面与具有多项式密度分布的直线骨架卷积曲面结合起来进行3维树木建模;然后采用BlobTree结构来组合隐式曲面原型,并用优化融合来消除树木枝条的融合突起;最后用PCM等隐式曲面建模技术来模拟3维树木表面的局部细节,并采用实际树木图片中的颜色概率分布进行纹理填充.实验结果表明,该方法不仅可以重构出具有多样性的光滑3维树木枝条,并能够逼真地生成树权脊梁、树木突起等局部细节特征.     

Creating and Rendering Convolution Surfaces

Implicit surfaces obtained by convolution of multi-dimensional primitives with some potential function, are a generalisation of popular implicit surface models: blobs, metaballs and soft objects. These models differ in their choice of potential function but agree upon the use of underlying modelling primitives, namely, points. In this paper a method is described for modelling and rendering implicit surfaces built upon an expanded set of skeletal primitives: points, line segments, polygons, arcs and planes. An analytical solution to the convolution is described. This solution offers a more accurate and robust representation of the resultant implicit surface than previous methods. An algorithm for ray-tracing the surfaces formed through convolution of any combination of these primitives is also outlined.     

Ratioquadrics: an alternative model for superquadrics

This paper presents a new family of 2D curves and its extension to 3D surfaces, respectively, calledrationconics andratioquadrics that have been designed as alternatives to the well-known superconics and superquadrics. This new model is intended as an improvement to the original one on three main points: first, it involves lower computation cost and provides better numerical robustness; second, it offers higher order continuities (C ¹/G ² orC ²/G ² instead ofC ⁰/G ⁰); and third, it provides a greater variety of shapes for the resulting curves and surfaces. All these improvements are obtained by replacing the signed power function involved in the formulation of superconics and superquadrics by linear or quadratic rational polynomials.     

Graphics processing unit‐based triangulations of Blinn molecular surfaces

Computing the surface of a molecule (e.g., a protein) plays an important role in the analysis of its geometric structure as needed in the study of interactions between proteins, protein folding, protein docking, and so forth. There are a number of algorithms for the computation of molecular surfaces and their triangulations, but only a few take advantage of graphics processing unit computing. This paper describes a graphics processing unit‐based marching cubes algorithm to triangulate molecular surfaces. In the end of the paper, a performance analysis of three implementations (i.e., serial CPU, CUDA, and OpenCL) of the marching cubes‐based triangulation algorithm takes place as a way to realize beforehand how molecular surfaces can be rendered in real‐time in the future. Copyright © 2011 John Wiley & Sons, Ltd.     

Equidistant Smoothing of Polyhedra with Arbitrary Topologies

Smoothing of polyhedron with arbitrary topology is an important issue in CAGD and CAD/CAM, but so far it is deemed to be difficult to smooth the complex corners of a polyhedron. In this paper, the concept of distance surfaces of a surface and a solid is introduced, and the incisive properties of such surfaces are addressed which provide a theoretical foundation for modifying a general corner. The method is based on making constricted volume and the maximum distance the volume can be constricted is given too. It is shown that by the proposed method in this paper any polyhedron can be G¹ smoothed with quadraic and, sometimes toroidal surfaces. The new approach is suitable for engineering design and NC machining. The associated algorithm based on the classification theorem of corners is simple, fast and robust.     

A new curve-based approach to polyhedral machining

This paper presents a new approach to three-axis NC tool path generation for sculptured surfaces. In the proposed curve-based approach, the gouge-free tool paths are generated from a polyhedral model of the STL (stereolithography) format. The polyhedral model is offset by a local-offsetting scheme. Then, the offset elements such as triangular facets, trimmed cylinders, and trimmed spheres are sliced by a series of drive planes. The curve segments on a drive plane are sorted, trimmed and linked, while the concave gouge is removed during the trimming process. The method is implemented on a PC, and some illustrative examples are provided in this paper. The main advantage of the proposed method is that the tool path can be generated from a polyhedral model without any concave and convex gouge, especially on an NC machine that supports NURBS interpolation. Other advantages and disadvantages of the proposed model are also discussed.     

Catmull-Clark细分曲面的形状调整

提出一种调整细分曲面形状的算法．该算法用cosα(Ck)取代C-B样条的形状因子α，并将Ck的定义区间从[-1，1]扩大到[-1，∞)；然后用这种扩展了的GB样条来构造catmull—clark细分曲面；使得生成细分曲面的形状不仅能够在C-B样条的范围内可调，而且还能在标准的catmull-clark细分曲面和初始的控制网格之间任意调整．该算法保留了C-B样条和catmull-clark细分曲面的主要特点，如精确表示圆柱体、处理任意拓扑结构的控制网格等。     

Cutting cubesvisualizing implicit surfaces by adaptive polygonization

A method for visualizing implicit surfaces is discussed. This method makes as few assumptions as possible concerning the surface and representation of its defining function. The surface may contain singularities, for instance, because it has self-intersections or it is reducible. A userdefined part of space is filled by a set of cubes, cutting pieces (called facets) off the surface. The set of cubes is controlled by an octree converging to the surface. The set of resulting facets can be taken as a piecewise linear approximation, which is sufficiently close to the given surface with respect to criteria specified by the user. Finally, some examples obtained with this method are presented.     

参数曲面导矢界估计及其在曲面绘制中的应用

目的是改进参数曲面的绘制效率。采用的方法是在参数曲线导矢界的基础上，进一步提出了3次参数曲面导矢界的公式。基于这些公式，能够较好地解决了参数曲面绘制算法过程中出现的重复绘制的问题和不连续的问题，进一步提高了曲面绘制的效果和效率。     

Qualitative and quantitative comparisons of B-spline offset surface approximation methods

Surface offset is one of the most useful operations in Computer Aided Geometric Design (CAGD). However, an implementation of this operator is not trivial primarily because the offset surface, in general, does not have the same representation as the original surface. Hence, it is difficult or impossible to represent an exact offset in a system with limited surface forms. For this reason, some CAGD surface offset operators produce results that are, at times, unsatisfactory. In this article, we discuss surface offset approximation methods in B-spline environment: both the original surfaces and their offsets are B-spline surfaces. This article summarizes research contained in the first named author's PhD thesis.     

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.     

三角网格上五次齐次代数曲面的重构

提出三角网格上重建代数曲面的一种方法,利用三次控制曲面来构造五次具有"齐次"形式的GC<'1>光滑曲面,所构造的代数曲面具有2次精度、局部性好、计算量低、自由参数几何意义明确的优点;而且这个五次代数曲面在与一簇特殊的平面相交时,交线为一个四次代数曲线和一条直线,从而化简了这类曲面参数化的计算量.     

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.     

Two algorithms for volume-preserving approximations of surfaces of revolution

The paper deals with volume-preserving approximations of surfaces of revolution. The approximating surfaces are generated only by line segments and circular arcs of a constant radius r. Further, for r > 0, the approximating surfaces are visually C¹ surfaces. For r = 0, developable C⁰ surfaces are obtained (consisting of either congruent cylinders or frustums of cones of revolution). Two algorithms are discussed. The first algorithm preserves the volume enclosed by a surface of revolution and the planes of every two latitude circles; the approximating surface is, however, no longer a surface of revolution. The second algorithm applies an approximating surface of revolution; however, the volume preservation no longer holds globally.     

G2 quasi-developable Bezier surface interpolation of two space curves

Surface development is used in many manufacturing planning operations, e.g., for garments, ships and automobiles. However, most freeform surfaces used in design are not developable, and therefore the developed patterns are not isometric to the original design surface. In some domains, the CAD model is created by interpolating two given space curves. In this paper, we propose a method to obtain a G² quasi-developable Bezier surface interpolating two arbitrary space curves. The given curves are first split into a number of piecewise Bezier curves and elemental Bezier patches each of which passes through four splitting points are constructed. All neighboring elemental patches are G² connected and they are assembled optimally in terms of the degree of developability (the integral Gaussian curvature). Experiments show that the final composite Bezier surface is superior to a lofted one which is defined regardless of the final surface developability.