Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .     

Reliable Collision Detection for Time-Dependent Parametric Surfaces

We improve on an algorithm by Von Herzen, Barr, and Zatz (HBZ) to detect geometric collisions between pairs of time-dependent parametric surfaces. The HBZ algorithm uses upper bounds on the parametric derivatives to guarantee detection of collisions and near misses, thus avoiding the defects of algorithms which sample the surfaces, possibly missing sharp spikes. Unfortunately, the user of the HBZ algorithm must supply not only routines computing the surface functions, but also routines bounding every component in the Jacobian of these surface functions over an arbitrary parametric range. Although they give helpful analyses for several types of surfaces, HBZ admit the need to provide Jacobian bounding routines as a disadvantage.We propose using interval arithmetic to bound functional values over a parametric input range, thus eliminating the need for the Jacobian entirely. Our interval version of the collision detection algorithm assumes neither bounded differentiability nor satisfaction of the Lipschitz criterion. Therefore, our code can detect geometric collisions for the much larger class of surface functions for which bounds on the function values can be computed using interval arithmetic. We contrast our code to that of HBZ and give timing comparisons.     

基于加权渐进插值的Loop 细分曲面等距逼近

等距曲面在CAD/CAM 领域有着重要的作用,由于细分曲面没有整体解 析表达式,使得计算细分曲面等距比参数曲面更加困难。针对目前已有的两种等距面逼近算 法进行了改进,利用加权渐进插值技术避免了传统细分等距逼近算法产生网格偏移的问题。 此外,提出了针对边界等距处理方案,使得等距后的细分曲面在内部和边界都均匀等距。该 方法无需求解线性方程组,具有全局和局部特性,能够处理闭网格和开网格,为Loop 细分 曲面数控加工奠定了良好的基础算法。最后给出的实例验证了算法的有效性。     

Geometric accuracy analysis for discrete surface approximation

In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density. This paper gives explicit formulae to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formulae can be directly applied to design sampling density for data acquisitions and surface reconstructions. Furthermore, we prove that the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace–Beltrami operators on the meshes are also convergent to those on the smooth surfaces. These theoretical results lay down the foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing.Practical algorithms for approximating surface Delaunay triangulations are introduced based on global conformal surface parameterizations and planar Delaunay triangulations. Thorough experiments are conducted to support the theoretical results.     

Surface approximation with rational B-splines

Many modern geometric modelers use nonuniform rational B-spline curves and surfaces as their canonical representations. Rational B-splines are a versatile representation, encompassing integral B-splines and the basic classical primitives such as conics, quadrics, and torii. However, rational B-splines representations other than these classical primitives have found little application in surface modeling. In this paper we develop approximation algorithms based on the general rational B-spline formulation. Numerical experiments indicate that rational B-splines allow a significantly more compact approximation of two classes of parametric surfaces in comparison to integral B-splines. The two classes of surfaces studied are generalized cylinders and offsets of a rational B-spline surface patch progenitor.     

有理三角曲面的分片线性逼近

有理三角曲面的分片线性逼近在参数曲面的求交、绘制等方面有着重要应用.已有研究主要采用曲面的二阶导矢界来估计逼近误差,而有理曲面的导矢界估计是一项困难的工作.为解决上述问题,利用齐次坐标,给出了一种定义域为任意三角形的有理三角曲面的分片线性逼近算法.该算法有效地避免了有理三角曲面的导矢界估计,并且离散段数可先验地给出.此外,通过重新参数化技术来缩小有理三角Bézier曲面的权因子之间的比值,进一步提高了算法的效率.     

Robust uniform triangulation algorithm for computer aided design

This paper presents a new robust uniform triangulation algorithm that can be used in CAD/CAM systems to generate and visualize geometry of 3D models. Typically, in CAD/CAM systems 3D geometry consists of 3D surfaces presented by the parametric equations (e.g. surface of revolution, NURBS surfaces) which are defined on a two dimensional domain. Conventional triangulation algorithms (e.g. ear clipping, Voronoi-Delaunay triangulation) do not provide desired quality and high level of accuracy (challenging tasks) for 3D geometry. The approach developed in this paper combines lattice tessellation and conventional triangulation techniques and allows CAD/CAM systems to obtain the required surface quality and accuracy. The algorithm uses a Cartesian lattice to divide the parametric domain into adjacent rectangular cells. These cells are used to generate polygons that are further triangulated to obtain accurate surface representation. The algorithm allows users to control the triangle distribution intensity by adjusting the lattice density. Once triangulated, the 3D model can be used not only for rendering but also in various manufacturing and design applications. The approach presented in this paper can be used to triangulate any parametric surface given in S(u,v) form, e.g. NURBS surfaces, surfaces of revolution, and produces good quality triangulation which can be used in CAD/CAM and computer graphics applications.     

Stripification of Free-Form Surfaces With Global Error Bounds for Developable Approximation

Developable surfaces have many desired properties in the manufacturing process. Since most existing CAD systems utilize tensor-product parametric surfaces including B-splines as design primitives, there is a great demand in industry to convert a general free-form parametric surface within a prescribed global error bound into developable patches. In this paper, we propose a practical and efficient solution to approximate a rectangular parametric surface with a small set of C ⁰ -joint developable strips. The key contribution of the proposed algorithm is that, several optimization problems are elegantly solved in a sequence that offers a controllable global error bound on the developable surface approximation. Experimental results are presented to demonstrate the effectiveness and stability of the proposed algorithm.     

一种参数多项式曲面片的逐点生成算法

在计算机绘图中，一般来说，曲线实际上是由折线代替，而曲面实为小平面拼接而成，在使计算量降到最低的情况下画出真正的曲线方面，已有许多文章研究了曲线的逐点生成方法，并取得了一定的进展，但是尚无有效的快速逐点生成曲面的方法，为了快速逐点生成曲面，在建立多项式函数递推计算公式和算法的基础上，给出了一种逐点生成参数多项式曲面片的算法，由于此算法中只用到整数加法运算，且点数的适当选取可使计算量达到极小，因此是一种很有效的算法，该方法还可以加以改进，而用于有理函数，这无疑对有理曲线曲面（如NURBS曲线曲面）的快速生成以及对计算机图形学的其他一些领域都是有意义的。     

A theoretical development for the computer generation and display of piecewise polynomial surfaces

Two algorithms for parametric piecewise polynomial evaluation and generation are described. The mathematical development of these algorithms is shown to generalize to new algorithms for obtaining curve and surface intersections and for the computer display of parametric curves and surfaces.     

Computing isophotos of surface of revolution and canal surface

Isophote of a surface consists of a loci of surface points whose normal vectors form a constant angle with a given fixed vector. It also serves as a silhouette curve when the constant angle is given as π/2. We present efficient and robust algorithms to compute isophotes of a surface of revolution and a canal surface. For the two kinds of surfaces, each point on the isophote is derived by a closed-form solution. To find each connected component in the isophote, we utilize the feature of surface normals. Both surfaces are decomposed into a set of circles, where the surface normal vectors at points on each circle construct a cone. The vectors which form a constant angle with given fixed vector construct another cone. We compute the parametric range of the connected component of the isophote by computing the parametric values of the surface which derive the tangential intersection of these two cones.     

On geometric interpolation of parametric surfaces

By exploiting the freedom in the choice of parametrization of a parametric surface, we show that there exist quadratic parametric surfaces that approximate a given parametric surface with the same approximation order as cubics, namely four, in the neighbourhood of a point where the Gaussian curvature of the surface is nonzero. This provides a first generalization to surfaces of earlier work for curves.     

Isotopic meshing of implicit surfaces

Implicit surfaces are given as the zero set of a function F:ℝ³→ℝ. Although several algorithms exist for generating piecewise linear approximations, most of these are based on a user-defined stepsize or bounds to indicate the precision, and therefore cannot guarantee topological correctness. Interval arithmetic provides a mechanism to determine global properties of the implicit function. In this paper we present an algorithm that uses these properties to generate a piecewise linear approximation of implicit curves and surfaces, that is isotopic to the curve or surface itself. The algorithm is simple and fast, and is among the first to guarantee isotopy for implicit surface meshing.     

Triangulation of 3D surfaces

A simple generator of graded triangular meshes on spatial surfaces is introduced in this paper. The algorithm is based on the approximation of the surface by tensor product polynomial patches which are uniquely mappable on a planar parametric space. Each of the patches is triangulated separately in its parametric space, using modified advancing front technique allowing for generation of pre-stretched elements, and the obtained triangulation is mapped back onto the original surface. Large effort has been devoted to the treatment of singularities arising on surfaces approximated by degenerated patches.     

Implicitization and parametrization of quadratic and cubic surfaces by μ-bases

Parametric and implicit forms are two common representations of geometric objects. It is important to be able to pass back and forth between the two representations, two processes called parameterization and implicitization, respectively. In this paper, we study the parametrization and implicitization of quadrics (quadratic parametric surfaces with two base points) and cubic surfaces (cubic parametric surfaces with six base points) with the help of μ-bases – a newly developed tool which connects the parametric form and the implicit form of a surface. For both cases, we show that the minimal μ-bases are all linear in the parametric variables, and based on this observation, very efficient algorithms are devised to compute the minimal μ-bases either from the parametric equation or the implicit equation. The conversion between the parametric equation and the implicit equation can be easily accomplished from the minimal μ-bases.     

Quart-parametric interpolations for intersecting paths

The intersecting path is an important tool path generation method. This paper proposes an approach for the quart-parametric interpolation of intersecting paths. The objective of our approach is that the intersecting paths for surface machining can be directly interpolated within the computer numerical control (CNC) system. This enables the CNC interpolator to process the intersecting paths without geometric approximation as in existing approaches and take into consideration any specific feedrate profiles and further machining dynamical issues along the path.The interpolation of the intersection of two general parametric surfaces is transferred into interpolation of its projection curves and the time trajectories of four parameters along the intersecting curves are obtained. Our strategy is to carry out the quart-parametric interpolation based on the projection interpolation. The feedrate control method is developed, and then the interpolation algorithms for two projection curves are proposed. An error reduction scheme is presented to alleviate point deviation from the drive parametric surface. Simulations of quart-parametric interpolation have been carried out to verify the effectiveness of the proposed algorithm.     

Accurate GPU-accelerated surface integrals for moment computation

We present algorithms for computing accurate moments of solid models that are represented using multiple trimmed NURBS surfaces and triangles. Our algorithms make use of programmable Graphics Processing Units (GPUs) to accelerate the moment computations. For NURBS surfaces, we evaluate the surface coordinates and normals accurately, with theoretical bounds, using our GPU NURBS evaluator. We have developed a framework that makes use of this data to evaluate surface integrals of trimmed NURBS surfaces in real time. Since typical solid models also contain flat planes that are usually tessellated into triangles, our framework also includes GPU acceleration of the moment contributions of such flat faces. Using our framework, we can compute volume and moments of solid models with theoretical guarantees. Our algorithms support local geometry changes, which is useful for providing interactive feedback to the designer while the solid model is being designed. We can compute the center of mass and check for stability of the solid model interactively. Other applications of such real-time moment computation include deformation modeling, animation, and physically based simulations.     

插值边界曲线的NURBS 近似极小曲面设计

探索性地设计了一个插值给定边界曲线的NURBS 近似极小曲面算法,弥补了当前NURBS 系统无法有效地设计工程所急需的一般NURBS 极小曲面的缺陷.运用NURBS 曲面的节点插入、Hybrid 多项式逼近等多种技术,将NURBS 曲面转化为相对简单的分片Bézier 曲面求解,并运用各子曲面片的控制顶点优化、整体曲面不断更新的迭代方法,成功地得到高精度的近似分片Bézier 极小曲面.最后,可以按用户的各种要求选择运用相应不同的迭代逼近算法,求取插值给定边界曲线的近似NURBS 极小曲面.     

Interactive display of large NURBS models

We present algorithms for interactive rendering of large-scale NURBS models. The algorithms convert the NURBS surfaces to Bezier surfaces, tessellate each Bezier surface into triangles, and render them using the triangle-rendering capabilities common to current graphics systems. We present algorithms for computing tight bounds on surface properties in order to generate high quality tessellation of Bezier surfaces. We introduce enhanced visibility determination techniques and present methods to make efficient use of coherence between successive frames. In addition, we also discuss issues in parallelization of these techniques. The algorithm also avoids polygonization anomalies like cracks. Our algorithms work well in practice and, on high-end graphics systems, are able to display models described using thousands of Bezier surfaces at interactive frame rates     

Trimming implicit surfaces

Algorithms for trimming implicit surfaces yielding surface sheets and stripes are presented. These two-dimensional manifolds with boundaries result from set-theoretic operations on an implicit surface and a solid or another implicit surface. The algorithms generate adaptive polygonal approximation of the trimmed surfaces by extending our original implicit surface polygonization algorithm. The presented applications include modeling several spiral shaped surface sheets and stripes (based on M. Eschers artworks) and extraction of ridges on implicit surfaces. Another promising application of the presented algorithms is modeling heterogeneous objects as implicit complexes.