Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

Parameterizing surfaces with certain special support functions,including offsets of quadrics and rationally supported surfaces

We discuss rational parameterizations of surfaces whose support functions are rational functions of the coordinates specifying the normal vector and of a given non-degenerate quadratic form. The class of these surfaces is closed under offsetting. It comprises surfaces with rational support functions and non-developable quadric surfaces, and it is a subset of the class of rational surfaces with rational offset surfaces. We show that a particular parameterization algorithm for del Pezzo surfaces can be used to construct rational parameterizations of these surfaces. If the quadratic form is diagonalized and has rational coefficients, then the resulting parameterizations are almost always described by rational functions with rational coefficients.     

Image‐based modeling of 3D objects with curved surfaces

This paper addresses an image‐based method for modeling 3D objects with curved surfaces based on the non‐uniform rational B‐splines (NURBS) representation. The user fits the feature curves on a few calibrated images with 2D NURBS curves using the interactive user interface. Then, 3D NURBS curves are constructed by stereo reconstruction of the corresponding feature curves. Using these as building blocks, NURBS surfaces are reconstructed by the known surface building methods including bilinear surfaces, ruled surfaces, generalized cylinders, and surfaces of revolution. In addition to them, we also employ various advanced techniques, including skinned surfaces, swept surfaces, and boundary patches. Based on these surface modeling techniques, it is possible to build various types of 3D shape models with textured curved surfaces without much effort. Copyright © 2007 John Wiley & Sons, Ltd.     

Ribbon-based transfinite surfaces

One major issue in CAGD is to model complex objects using free-form surfaces of general topology. A natural approach is curvenet-based design, where designers directly create and modify feature curves. These are interpolated by smoothly connected, multi-sided patches, which can be represented by transfinite surfaces, defined as a combination of side interpolants or ribbons. A ribbon embeds Hermite data, i.e., prescribed positional and cross-derivative functions along boundary curves.The paper focuses on two transfinite schemes: the first is an enhanced and extended variant of a multi-sided generalization of the classical Coons patch (Várady et al., 2011); the second one is based on a new concept of combining doubly curved composite ribbons, each one interpolating three adjacent sides. Main contributions include various ribbon parameterizations that surpass former methods in quality and computational efficiency. It is proven that these surfaces smoothly interpolate the prescribed ribbon data. Both formulations are based on non-regular convex polygonal domains and distance-based, rational blending functions. A few examples illustrate the results.     

插值两个平行平面曲线的可展B样条曲面

两条位于平行平面上的分别是[n]次和[n+1]次B-样条曲线,以这两条曲线作边界生成一个直纹面,即[(n,n+1)]次B样条曲面,得到了该直纹面为可展曲面的充要条件,构造了（2,3）次可展B－样条曲面。通过对以上可展B样条曲面进行凸性分析,得到了（n,n+1）次可展B样条曲面为凸曲面的充要条件。给出了几个（2,3）次B-样条可展曲面。     

插值平面曲线的可展B样条曲面及凸性分析

用两条位于两任意平面上的均匀B样条曲线作边界曲线,形成一个直纹面,得到了该直纹面为可展曲面的充要条件,解决了(n,n)次可展B样条曲面的设计问题,构造了(2,2)次B样条可展曲面。通过对以上可展B样条曲面进行凸性分析,得到了(n,n)次可展B样条曲面为凸曲面的充要条件。     

On modeling with rational ringed surfaces

A surface in Euclidean space is called ringed (or cyclic) if there exists a one-parameter family of planes that intersects this surface in circles. Well-known examples of ringed surfaces are the surfaces of revolution, (not only rotational) quadrics, canal surfaces, or Darboux cyclides. This paper focuses on modeling with rational ringed surfaces, mainly for blending purposes. We will deal with the question of rationality of ringed surfaces and discuss the usefulness of the so called P-curves for constructing rational ringed-surface-blends. The method of constructing blending surfaces that satisfy certain prescribed constraints, e.g. a necessity to avoid some obstacles, will be presented. The designed approach can be easily modified also for computing $n$-way blends. In addition, we will study the contour curves on ringed surfaces and use them for computing approximate parameterizations of implicitly given blends by ringed surfaces. The designed techniques and their implementations are verified on several examples.     

Analysis of improved positioning in five-axis ruled surface milling using envelope surface

This article covers side milling of ruled surfaces using a milling cutter. Flank milling is useful for machining objects such as impellers, turbine blades, fan vanes and all workpieces defined by non-developable, ruled surfaces. In the present article, we first introduce two types of positioning on ruled surfaces developed within the Toulouse Mechanical Engineering Laboratory. The positioning studied is taken from the geometric situation not taking the instantaneous speed of rotation of the milling cutter into account. The swept profile of the tool is then determined based on the tool motion. Having defined the envelope surface, we seek to analyse improved and standard positioning errors comparing envelope surfaces with the ruled surface. We then introduce an example to illustrate positioning developed through a first theoretical study before experimentation including machining and measurement of the test piece. Finally, we give our conclusions as to the validity of improved positioning without taking the instantaneous speed of rotation of the milling cutter into account.     

MESSAGE——具有自由曲面造型功能的立体造型系统

MESSAGE是一具有自由曲面造型功能的立体造型系统,它能够处理二次曲面、旋转曲面、 广义sweeping曲面、自由曲面以及由平面和上述曲面围成的三维物体,系统以平面及双三次 有理B样条曲面为基本曲面模型,提供了布尔运算、glue运算及多种局部修改运算.系统还 具有一个独立的几何数据库,可以支持一些重要的工程应用.     

Proper Reparametrization of Rational Ruled Surface

In this paper,we present a proper reparametrization algorithm for rational ruled surfaces.That is,for an improper rational parametrization of a ruled surface,we construct a proper rational parametrization for the same surface. The algorithm consists of three steps.We first reparametrize the improper rational parametrization caused by improper supports.Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally,the problem is reduced to the proper reparametrization of planar rational algebraic curves.     

基于卷积曲面的几何造型方法

卷积曲面是一种很有潜力的隐式曲面造型方法。由于卷积的可叠加性质,使得其生成的曲面很光滑,不会产生凸包和裂缝。卷积曲面造型是基于骨架的,其骨架可以是曲线或曲面,也可以以物体轮廓线为骨架原型来获取所需要的物体造型。在卷积造型的过程中,通过对骨架加权可以精细地控制结果曲面的形状。在介绍了卷积曲面造型的各类方法后,分析了其特点和发展方向。     

Minimal rational parametrizations of canal surfaces

It is well known that canal surfaces defined by a rational spine curve and a rational radius function are rational. The aim of the present paper is to construct a rational parameterization of low degree. The author uses the generalized stereographic projection in order to transform the problem to a parameterization problem for ruled surfaces. Two problems are discussed: parameterization with boundary conditions (design of canal surfaces with two curves on it, as is the case for rolling ball blends) and parameterization without boundary conditions.     

Kernel modeling for molecular surfaces using a uniform solution

In this paper, a rational Bézier surface is proposed as a uniform approach to modeling all three types of molecular surfaces (MS): the van der Waals surface (vdWS), solvent accessible surface (SAS) and solvent excluded surface (SES). Each molecular surface can be divided into molecular patches, which can be defined by their boundary arcs. The solution consists of three steps: topology modeling, boundary modeling and surface modeling. Firstly, using a weighted α-shape, topology modeling creates two networks to describe the neighboring relationship of the molecular atoms. Secondly, boundary modeling derives all boundary arcs from the networks. Thirdly, surface modeling constructs all three types of molecular surfaces patch-by-patch, based on the networks and the boundary arcs. For an SES, the singularity is specially treated to avoid self-intersections. Instead of approximation, this proposed solution can produce precise shapes of molecular surfaces. Since rational Bézier representation is much simpler than a trimmed non-uniform rational B-spline surface (NURBS), computational load can be significantly saved when dealing with molecular surfaces. It is also possible to utilize the hardware acceleration for tessellation and rendering of a rational Bézier surface. CAGD kernel modelers typically use NURBSs as a uniform representation to handle different types of free-form surface. This research indicates that rational Bézier representation, more specifically, a bi-cubic or 2×4 rational Bézier surface, is sufficient for kernel modeling of molecular surfaces and related applications.     

Quasi-Developable B-Spline Surface Design with Control Rulings

We propose a method for generating a ruled B-spline surface fitting to a sequence of pre-defined ruling lines and the generated surface is required to be as-developable-as-possible. Specifically, the terminal ruling lines are treated as hard constraints. Different from existing methods that compute a quasi-developable surface from two boundary curves and cannot achieve explicit ruling control, our method controls ruling lines in an intuitive way and serves as an effective tool for computing quasi-developable surfaces from freely-designed rulings. We treat this problem from the point of view of numerical optimization and solve for surfaces meeting the distance error tolerance allowed in applications. The performance and the efficacy of the proposed method are demonstrated by the experiments on a variety of models including an application of the method for path planning in 5-axis computer numerical control (CNC) flank milling.     

Automatic parameretization of rational curves and surfaces 1: conics and conicoids

Given the implicit equation for degree two curves (conics) and degree two surfaces (conicoids), algorithms are described here, which obtain their corresponding rational parametric equations (a polynomial divided by another). These rational parameterizations are considered over the fields of rationals, reals and complex numbers. In doing so, solutions are given to important subproblems of finding rational and real points on the given conic curve or conicoid surface. Further polynomial parameterizations are obtained whenever they exist for the conics or conicoids. These algorithms have been implemented on a VAX-780 using VAXIMA.     

Smooth blending of subdivision surfaces

A technique for merging freeform objects modeled with subdivision surfaces is proposed. The method takes into consideration the possible distortion of the surfaces while the smoothness of the blending surface connecting the objects is maintained. A blend region between the given object surfaces is located. The topology of the surfaces in the vicinity of the blend region is refined and the boundary curves of the blend region are smoothed to reduce possible distortion in the blending surface. A blend curve between the two surfaces is used for connecting the given surfaces. Vertices on the blend curve are regular whereas extraordinary vertices are restricted to locate along the boundary of the blend region. Locations of the blend vertices are adjusted such that the curvature of the surfaces along the boundaries of the blend region is minimized. This ensures extraordinary vertices on the boundaries to lie on a relatively flat region, and hence, minimizes the distortion of the surfaces in the merging process.     

一种应用参数曲面的动态自由变形方法

本文提出了一种动态自由变形方法，被变形物体首先附在两张参数曲面上，当参数曲面的形状发生变化时，物体会随之变形。这两张参数曲面分别被称为形状曲面和高度曲面，其中形状曲面用于控制变形物体的形状，高度曲面用于控制物体沿着形状曲面法向的高度。     

Cone spline approximation via fat conic spline fitting

Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fitting. An efficient fitting algorithm is also proposed for fat conic spline fitting with controllable tolerances. Several examples about approximation of general developable surfaces or other types of ruled surfaces by cone spline surfaces are presented.     

Topological correction of hypertextured implicit surfaces for ray casting

Hypertextures are a useful modelling tool in that they can add three-dimensional detail to the surface of otherwise smooth objects. Hypertextures can be rendered as implicit surfaces, resulting in objects with a complex but well defined boundary. However, representing a hypertexture as an implicit surface often results in many small parts being detached from the main surface, turning an object into a disconnected set. Depending on the context, this can detract from the realism in a scene, where one usually does not expect a solid object to have clouds of smaller objects floating around it. We present a topology correction technique, integrated in a ray casting algorithm for hypertextured implicit surfaces, that detects and removes all the surface components that have become disconnected from the main surface. Our method works with implicit surfaces that are C² continuous and uses Morse theory to find the critical points of the surface. The method follows the separatrix lines joining the critical points to isolate disconnected components.