Computing offsets of trimmed NURBS surfaces

Offsetting of trimmed NURBS surfaces is one of the widely used functionalities in the design and manufacture simulations of composite laminates. This paper presents an approach for the offsetting of a trimmed NURBS surface. The approach has been developed mainly to meet the stringent accuracy requirements in the simulation of composite laminate design and manufacturing processes. However, the approach is applicable for the offset of a general trimmed NURBS surface. Though the method is based on known techniques in literature, the practical approach and the treatment of the subject presented is unique and has not been reported earlier. The basic approach consists of offsetting the underlying surface, offsetting of all the trimming loops and the creation of offset trimmed surface using the offset surface and the offset trimming loops. This is a unified offset approach for trimmed surfaces where in the offset of underlying surface and the offset of trimming loops are obtained using the same approach. The approach has better error control and results in less number of control points. Further the approach can be extended to obtain offsets of a general B-Rep. The approach has been used in the creation of offset surfaces of various aircraft components.     

Computing non-self-intersecting offsets of NURBS surfaces

A new approach for the computation of non-self-intersecting offset surface of a single G₁ continuous NURBS surface has been presented. The approach recognizes special surfaces, i.e. planes, spheres, cones and cylinders, and offsets them precisely. An approximate offset surface within the specified tolerance is computed for a general free form surface. The method for a general free form surface consists of (1) sample offset surface based on second derivatives; (2) eliminate sample points which can give self-intersections; (3) surface fitting through the remaining sample points; and (4) removal of all the removable knots of the surface. The approach checks for self-intersections in the offset surface and removes the same automatically, if any. The non-self-intersecting offsets for surface of extrusion and surface of revolution are obtained by removing the self-intersections in the offset generator and profile curves respectively using point sampling, cleaning of sampled points, curve fitting and knot removal. The approach has better control on error. It generates offset surface with less number of control points and degree. The methodology works only for a class of problems where in the offset of a single G₁ surface is still a single connected surface without having any holes. The offset methodology has been demonstrated through three types of surfaces namely surface of revolution, surface of extrusion and a general free form surface. This approach has been extensively used in creation of offset surfaces of composite laminate components. The presented approach can also be used to check for self-intersections in any general surface and to remove the same, if any, with little modifications, as long as the cleaned surface is a single connected surface.     

Dynamic highlight line generation for locally deforming NURBS surfaces

The highlight line model is a powerful tool in assessing the quality of a surface. Efficient highlight line generation is especially important for an interactive design environment. In this paper, a method for dynamic generation of highlight lines on a locally deforming NURBS surface is presented. The method generates frames of the deforming surface and the corresponding highlight lines by directly modifying the current highlight lines using a Taylor expansion technique, instead of going through a tracing process. The highlight lines computation process adopted here enables a unified distance surface to generate all highlight lines in the highlight line family. The computation process is facilitated by looking up pre-calculated information of the tessellation mesh and an indexing technique for the distance surface. The indexing technique is presented to determine when the highlight line model should be re-generated and, to facilitate the highlight line re-generation process. The new technique is suitable for interactive design environments and animation applications as the updating process takes only one subtraction and one vector inner product to get the new parameters for each new node.     

NURBS曲面形状修改的一种改进算法

提出一种修改NURBS曲面形状的新算法。该算法运用约束优化方法,通过改变控制顶点和权因子修改NURBS曲面的形状。实验证明,该算法可以获得令人满意的效果。     

Optimizing conformality of NURBS surfaces by general bilinear transformations

The conformality of NURBS surfaces greatly affects the results of rendering and tessellation applications. To improve the conformality of NURBS surfaces, an optimization algorithm using general bilinear transformations is presented in this paper. The conformality energy is first formulated and its numerical approximation is then constructed using the composite simpson’s rule. The initial general bilinear transformation is obtained by approximating the conformal mapping of its 3D discretized mesh using a least square method, which is further optimized by the Levenberg–Marquardt method. Examples are given to show the performance of our algorithm for rendering and tessellation applications.     

Modifying free-formed NURBS curves and surfaces for offsetting without local self-intersection

This paper presents an algorithm of modifying free-formed NURBS curve/surface for offsetting without local self-intersecting. The method consists of (1) sampling a number of points from a progenitor curve/surface based on second derivatives; (2) checking the curvature or maximum curvature of the progenitor curve/surface at the sampled points; (3) inserting corresponding knots of sampled points; (4) repositioning control points till the curvature/maximum curvature of the curve/surface everywhere are less than the reciprocal of offset distance. The method is efficient and is able to obtain better offsetting results.     

基于能量优化的NURBS曲面几何特征修改

NURBS作为曲面造型技术中的重要方法之一,在计算机辅助几何设计和计算机图形学领域中有着广泛的应用。针对NURBS曲面的几何特征修改,提出了基于能量优化的修改方法。通过对NURBS曲面的控制顶点进行扰动,以曲面的应力能改变为目标函数并使之最小化,实现了NURBS曲面上给定多个点处的位置、一阶偏导矢、二阶偏导矢和法矢等几何特征的修改。数值实例表明该方法用于微调时,可实现对曲面局部形状的多种修改效果,便于交互设计。     

Representing Quadric Surfaces Using NURBS Surfaces

A method for representing quadrc surfaces using NURBS is presented.By means of the necessary and sufficient conditons for NURBS curves to precisely represent circular arcs and other conics,quadric surfaces can be represented by NURBS surfaces with fewer control vertices.The method can be used not only for NURBS surface representation of quadric surfaces,but also for rounding polyhedrons.Many examples are given in the paper.     

Interactive rendering of NURBS surfaces

NURBS (Non-uniform rational B-splines) surfaces are one of the most useful primitives employed for high quality modeling in CAD/CAM tools and graphics software. Since direct evaluation of NURBS surfaces on the GPU is a highly complex task, the usual approach for rendering NURBS is to perform the conversion into Bézier surfaces on the CPU, and then evaluate and tessellate them on the GPU. In this paper we present a new proposal for rendering NURBS surfaces directly on the GPU in order to achieve interactive and real-time rendering. Our proposal, Rendering Pipeline for NURBS Surfaces (RPNS), is based on a new primitive KSQuad that uses a regular and flexible processing of NURBS surfaces, while maintaining their main geometric properties to achieve real-time rendering. RPNS performs an efficient adaptive discretization to fine tune the density of primitives needed to avoid cracks and holes in the final image, applying an efficient non-recursive evaluation of the basis function on the GPU. An implementation of RPNS using current GPUs is presented, achieving real-time rendering rates of complex parametric models. Our experimental tests show a performance several orders of magnitude higher than traditional approximations based on NURBS to Bézier conversion.     

Efficient registration of NURBS geometry

The paper presents an implementation of free-form surface registration in relation to inspection of engineering components, defined as NURBS. Registration is principally performed through the Iterative Closest Point (ICP) method. The time-critical step in ICP was found to be the determination of the closest points on NURBS to a given point in space. Significant speed improvements were achieved through the adoption of a dual surface representation, involving approximation of NURBS entities by a polyhedral mesh. A criterion for sufficient polyhedral approximation was derived and implemented, producing encouraging results. Original solutions are suggested in order to further improve the computational speed. Extensive testing has been carried out, showing that the proposed registration method handles a full six degrees of freedom and achieves global convergence. Performance of the implemented algorithms is discussed with reference to registration of a turbine blade airfoil.     

Computing constant offsets of a NURBS B-Rep

An approach for the automatic offset of a NURBS B-Rep has been presented which can be used for a class of manifold B-Reps. The approach offsets each of the trimmed surfaces (faces) of the B-Rep and then removes the gaps and intersections between offset faces automatically, if any. The offset B-Rep is then created by sewing all the updated offset faces. The practical approach and the treatment of the subject presented are unique and have not been reported earlier. The approach works under the assumption that the number of faces in both base and offset B-Reps is the same. It further assumes that the faces are at least G₁ continuous and non-self intersecting after constant offset. The present approach has given accurate and robust offsets of many lay-up surfaces (B-Reps) in the composite laminate modeling. The approach has better error control. Some of the offsets of lay-up surfaces that are generated using this approach have been presented.     

Fast Ray Tracing NURBS Surfaces

In this paper,a new algorithm wit extrapolation process for computing the ray/surface intersection is presented.Also,a ray is defined to be the intersection of two planes,which are non-orthogonal in general,in such a way that the number of multiplication operations is reduced.In the preprocessing step,NURBS surfaces are subdivded adaptively into rational Bezier patches.Parallelepipeds are used to enclose the respective patches as tightly as possible Therefore,for each ray that hits the enclosure(i.e.,parallelepiped)of a patch the intersection points with the parallelepiped‘s faces can be used to yield a good starting point for the following iteration.The improved Newton iteration with extrapolation process saves CPU time by reducing the number of iteration steps.The intersection scheme is facter than previous methods for which published performance data allow reliable comparison.The method may also be used to speed up tracing the intersection of two parametric surfaces and oter operations that need Newton iteration.     

NURBS扫描体的逼近算法研究

提供了一种NURBS扫描体的逼近方法。该方法主要步骤：(1)通过系列平面切割,把NURBS曲面(实体)进行降维处理,变成平面曲线;(2)为曲线设置局部标架;(3)在局部标架下求出每一曲线在每一时刻的极值点后将其转换成原曲线的奇异点;(4)使用marching cubes算法剔除扫描体内部点,保留扫描体边界上的奇异点;(5)由所有保留点拟合成奇异曲面。本算法能较好地逼近NURBS扫描体,其逼近精度可通过控制切割精度和扫描过程中时间间隔的选取而得到有效控制。     

Fast Dynamic Tessellation of Trimmed NURBS Surfaced1

Trimmed NURBS (non-uniform rational B-splines) surfaces are being increasingly used and standardized in geometric modeling applications. Fast graphical processing of trimmed NURBS at interactive speeds is absolutely essential to enable these applications. which poses some unique challenges in software, hardware, and algorithm design. This paper presents a technique that uses graphical compilation to enable fast dynamic tessellation of trimmed NURBS surfaces under highly varying transforms. We use the concept of graphical data compilation. through which we preprocess the NURBS surface into a compact, view-independent form amenable for fast per-frame extraction of triangles. Much of the complexity of processing is absorbed during compilation. Arbitrarily complex trimming regions are broken down into simple regions that are specially designed to facilitate tessellation before rendering. Potentially troublesome cases of degeneracies in the surface are detected and dealt with during compilation. Compilation enables a clean separation of algorithm-intensive and compute-intensive operations, and provides for parallel implementations of the latter. Also, we exercise a classification technique while processing trimming loops. which robustly takes care of geometric ambiguities and deals with special cases while keeping the compilation code simple and concise.     

GPU-accelerated Hausdorff distance computation between dynamic deformable NURBS surfaces

We present a parallel GPU-accelerated algorithm for computing the directed Hausdorff distance from one NURBS surface to another, within a bound. We make use of axis-aligned bounding-box hierarchies that bound the NURBS surfaces to accelerate the computations. We dynamically construct as well as traverse the bounding-box hierarchies for the NURBS surfaces using operations that are optimized for the GPU. To compute the Hausdorff distance, we traverse this hierarchy after culling bounding-box pairs that do not contribute to the Hausdorff distance. Our contribution includes two-sided culling tests that can be performed in parallel using the GPU. The culling, based on the minimum and maximum distance ranges between the bounding boxes, eliminates bounding-box pairs from both surfaces that do not contribute to the Hausdorff distance simultaneously. We calculate accuracy bounds for our computed Hausdorff distance based on the curvature of the surfaces. Our algorithm runs in real-time with very small guaranteed error bounds for complex NURBS surfaces. Since we dynamically construct our bounding-box hierarchy, our algorithm can be used to interactively compute the Hausdorff distance for models made of dynamic deformable surfaces.     

Number-theoretic functions which are equivalent to number of divisors

Let d(n) denote the number of positive integral divisors of n. In this paper we show that the Möbius function, μ(N), can be computed by a single call to an oracle for d(n). We also show that any function that depends solely on the exponents in the prime factorization of N can be computed by at most log₂ N calls to an oracle for d(N).     

An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches

Designers require a means of designing complex free-form surfaces easily and intuitively. One general approach to designing such surfaces is to first define a curve mesh consisting of characteristic lines, such as cross sections and boundary curves, then to interpolate the curve mesh using free-form surfaces. NURBS surfaces are widely used but make the interpolation of an irregular curve mesh difficult. This has been a major limiting constraint on designers. In this paper, we propose a new surface representation that enables the smooth interpolation of an irregular curve mesh with NURBS curves and surfaces.     

A robust conforming NURBS tessellation for industrial applications based on a mesh generation approach

A NURBS tessellation technique is presented with the goal to robustly approximate CAD surfaces that define the boundary of complicated three dimensional geometric shapes with a minimum number of triangles. The minimization is achieved by generating anisotropic triangles in the three dimensional space. New procedures are presented to handle numerical stability issues due to the anisotropy. The tessellation is generated using a mesh generation viewpoint, as opposed to the more classical viewpoint of graphical visualization of surfaces in CAD. This ensures topological conformity of the resulting mesh. A tiered approximation approach is used for speed and robustness. Degeneracies associated with NURBS curves and surfaces are given special attention as they occur frequently in naval and aerospace conceptual-to-early design process. Analogies with a classical mesh generation process are discussed and several numerical examples illustrate the method.     

Trimmed NURBS曲面的展开研究

本文分析复杂曲面中的TrimmedNURBS(裁剪面)曲面表达,讨论曲面曲线的展开算法,在此基础上详细描述TrimmedNURBS曲面展开算法的实现步骤,并通过实例介绍展开算法在IGES格式的TrimmedNURBS曲面展开的具体应用。本研究主要用于解决三维柱面设计中经常会遇到螺纹的设计、齿轮的设计、螺旋槽的等设计问题。