A Linear Variational System for Modelling From Curves

We present a linear system for modelling 3D surfaces from curves. Our system offers better performance, stability and precision in control than previous non‐linear systems. By exploring the direct relationship between a standard higher‐order Laplacian editing framework and Hermite spline curves, we introduce a new form of Cauchy constraint that makes our system easy to both implement and control. We introduce novel workflows that simplify the construction of 3D models from sketches. We show how to convert existing 3D meshes into our curve‐based representation for subsequent editing and modelling, allowing our technique to be applied to a wide range of existing 3D content.     

Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.     

Modifying CAD/CAM surfaces according to displacements prescribed at a finite set of points

This article presents a method for modifying CAD/CAM surfaces automatically in accordance with displacements prescribed at a finite set of points in R³, such as node displacements predicted by finite-element analysis. The method is based on the ‘morphing’ approach introduced by Sederberg and Parry in 1986. The input to the process consists of (a) a CAD/CAM model containing trimmed polynomial B-spline surfaces and (b) a set of points and associated displacement vectors in R³. These points are assumed to be close to, but not necessarily on, the objects of the CAD/CAM model. A rectangular volume, enclosing the CAD/CAM model and the input points in R³, is represented as a volume spline, i.e. a trivariate tensor-product spline. A modified volume spline is computed using (a) a least-squares fit based on the given point displacements, and (b) a smoothing functional. The modified CAD/CAM objects are defined as compositions of the original parametric functions and the modified volume spline (i.e. a morphing). In order to ensure compatibility with standard commercial CAD/CAM systems, the modified surfaces are fitted with appropriate splines using any standard, reasonably shape-preserving, fitting procedure applied in the parameter domains of the original surfaces.     

带切向控制的多结点曲线造型方法

在普通的多结点样条中加入相当于导数条件的可控参数,通过调节这些参数控制插值曲线在各型值点的切向量,从而达到满意的曲线造型效果．该方法保持了多结点样条的优越性（基数型,局部性）,因此可以只对插值曲线作局部调整而不影响整体,有助于计算机辅助几何设计领域的工程人员设计、调整曲线的形状．     

一类二次曲面上的参数样条

在空间四个有序数据点所确定的一个二次曲面上,可以构造一类特殊的曲线。给出了四个形状控制因子的有理基函数,以及通过研究其参数间的函数关系定义函数集,构造一类样条曲线,使得通过改变控制因子能任意精确地逼近控制多边形。这类样条曲线端点处满足一定切线方向和有界曲率,容易将它们拼接成一条逼近样条曲线。利用这些样条构造出逼近样条曲面,具有更多的自由度。     

两种带形状参数的曲线

本文构造了两种带参数的三角样条基,基于这两组基定义了两种三角样条曲线。与二次B样条曲线类似,这两种曲线的每一段都由相继的三个控制顶点生成。这两种曲线具有许多与二次B样条曲线类似的性质,但它们的连续性都比二次B样条曲线更好。对于等距节点,在一般情况下,这两种曲线都整体C3连续,在特殊条件下,它们都可达C5连续。两种曲线中的形状参数均有明确的几何意义,参数越大,曲线越靠近控制多边形。另外,当形状参数满足一定条件时,这两种曲线都具有比二次B样条曲线更好的对控制多边形的逼近性。运用张量积方法,将这两种曲线推广后所得到的曲面也具有较好的连续性。     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

一类基于三角函数的插值曲线

从三角函数出发,构造了一类插值于首、末端点及其切矢的参数样条曲线,称之为T—Ferguson,并研究了合成T—Ferguson曲线的算法。T—Ferguson曲线丰富了参数样条曲线,是一种可行的构造插值曲线方法。     

基于最佳圆弧样条逼近的快速距离曲面计算

距离曲面是一种常用的隐式曲面,它在几何造型和计算机动画中具有重要的应用价值,但以往往在对距离曲面进行多边形化时速较慢,为了提高点到曲线最近距离计算的效率,提出了一种基于最佳圆弧样条逼近的快速线骨架距离曲面计算方法,该算法对于一条任意的二维NURBS曲线,在用户给定的误差范围内,先用最少量的圆弧样条来逼近给定的曲线,从而把点到NURBS曲线最近距离的计算问题转化为点到圆弧样条最近距离的计算问题,由于在对曲面进行多边形化时,需要大量的点到曲线最近距离的计算,而该处可以将点到圆弧样条最近距离很少的计算量来解析求得,故该算法效率很高,该实验表明,算法简单实用,具有很大的应用价值。     

A Recursive Subdivision Algorithm for Piecewise Circular Spline

We present an algorithm for generating a piecewise G ¹ circular spline curve from an arbitrary given control polygon. For every corner, a circular biarc is generated with each piece being parameterized by its arc length. This is the first subdivision scheme that produces a piecewise biarc curve that can interpolate an arbitrary set of points. It is easily adopted in a recursive subdivision surface scheme to generate surfaces with circular boundaries with pieces parameterized by arc length, a property not previously available. As an application, a modified version of Doo–Sabin subdivision algorithm is outlined making it possible to blend a subdivision surface with other surfaces having circular boundaries such as cylinders.     

G2连续的低次避障代数样条曲线

为便于机器人在避障时能高速前进,把整体G2连续的低次避障曲线从参数形式拓展到代数样条形式上。首先,对导向折线段中除去首末线段的其他线段插入中点,以生成一组控制多边形;然后,根据各控制多边形和与之对应的障碍物,得到既能使曲线规避所有障碍物,又能使曲线在整体上保持G2连续的形状因子。低次避障代数样条曲线不仅能够直接得到与给定点之间的位置关系,还具有次数低、连续阶高、计算简单、保形性好和便于控制的优点。曲线在次数为3时更是具有局部可调性,其在设计时的灵活度得以增加。     

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.     

Algorithms on cone spline surfaces and spatial osculating arc splines

Developable surfaces are of considerable importance to many industrial applications, e.g., sheet metal forming processes. The objective of this paper is to provide algorithms on the approximation of developable surfaces with pieces of right circular cones. Special emphasis is devoted to practical choices of free parameters and to error estimation. Furthermore, a new algorithm for the approximation of spatial curves with a circular arc spline is presented which stands in close relation to above algorithms on developable surfaces. The proposed arc spline has contact of order 2 to the given curve in a series of curve points. The investigation includes a segmentation algorithm and error estimation.     

Adaptive quasi-interpolating quartic splines

We present an adaptive quasi-interpolating quartic spline construction for regularly sampled surface data. The method is based on a uniform quasi-interpolating scheme, employing quartic triangular patches with C ¹-continuity and optimal approximation order within this class. Our contribution is the adaption of this scheme to surfaces of varying geometric complexity, where the tiling resolution can be locally defined, for example driven by approximation errors. This way, the construction of high-quality spline surfaces is enhanced by the flexibility of adaptive pseudo-regular triangle meshes. Numerical examples illustrate the use of this method for adaptive terrain modeling, where uniform schemes produce huge numbers of patches.     

一类有理逼近参数样条

在二次曲面上构造一种带有形状因子的有理参数样条曲线,该样条曲线能逼近所在的控制多边形,且有较好的几何特性,并且可以作升阶和降阶处理。分析其端点性质,便于拼接成光滑曲线,如果选取合适的形状因子,可以使得曲线连接成G2连续。     

形状可调的5次组合样条及其参数选择

目的 为了克服3次参数B样条在形状调整与局部性方面的不足,提出带参数的5次多项式组合样条。方法 首先构造一组带参数的5次多项式基函数;然后采用与3次B样条曲线相同的组合方式定义带参数的5次多项式组合样条曲线,并讨论基于能量优化法的5次组合样条曲线参数最佳取值问题;最后定义相应的组合样条曲面,并研究利用粒子群算法求解曲面的最佳参数取值。结果 5次组合样条不仅继承了3次B样条的诸多性质,而且还比3次B样条具有更强的局部性及形状可调性。由于5次组合样条仍为多项式模型,因此方程结构相对较为简单,符合实际工程的需要。利用能量优化法可获得光顺的5次组合样条曲线与曲面。结论 所提出5次多项式组合样条克服了3次参数B样条在形状调整与局部性方面的不足,是一种实用的自由曲线曲面造型方法。     

Spline approximation of offset curves

This paper deals with the approximation of (regular) offset curves (of a given spline curve of degree n) by a spline curve of arbitrarily chosen degree m. The approximating spline curves are determined by geometric continuity conditions and by parameter optimization for minimizing the range of the approximation error.     

Intrinsic spline curve with local control

A new spline algorithm that uses arc length as parameter and generates a curve in terms of the quantities ‘relative curvature’ and ‘relative torsion’ is described. The tangent and curvature vectors at each datapoint are estimated and the spline is built up span by span. This allows local changes without affecting the whole curve and facilitates the incorporation of derivative discontinuities. Unless discontinuities are specifically requested, the solution obtained is curvature continuous.