Recursive algorithms for the representation of parametric curves and surfaces

Recursive algorithms for the representation of parametric curves and surfaces are presented which are based upon a geometric property of the de Casteljau algorithm. The algorithms work with triangular and pyramidal arrays that provide an easy handling of the curve and the surface ‘in a large’ design.     

Knot removal for parametric B-spline curves and surfaces

This paper is the third in a sequence of papers in which a knot removal strategy for splines, based on certain discrete norms, is developed. In the first paper, approximation methods defined as best approximations in these norms were discussed, while in the second paper a knot removal strategy for spline functions was developed. In this paper the knot removal strategy is extended to parametric spline curves and tensor product surfaces. The method has been implemented and thoroughly tested on a computer. We illustrate with several examples and applications.     

Splat representation of parametric surfaces

Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. In this work, we present an approach to convert a parametric surface into a splat-based representation. It works in parametric space, performs an adaptive sampling based on the surface curvature and a given error tolerance and uses power Voronoi diagrams. The goal is to approximate the surface with an optimized set of elliptical splats.     

Construction of flexible blending parametric surfaces via curves

The main purpose of this paper is to provide a method that allows to solve the blending problem of two parametric surfaces. The blending surface is constructed with a collection of space curves defined by point pairs on the blending boundaries of given primary surfaces. Bézier and C-cubic curves are used to interpolate the blending boundaries. The blending surface is Gⁿ continuously connected to the primary surfaces.     

基于能量优化G2连续插值三次样条曲线

给出了一种在能量优化意义下构造G2连续保形插值三次参数样条曲线的方法.具体步骤如下:(1)以曲线应变能最小为目标构造目标函数,通过解线性方程组,求出优化意义下的每个插值点处的最优切矢方向;(2)用文中给出的简易公式求出各插值点的曲率,进而计算出插值点处的切矢模长,使曲线满足G2连续、保形插值的条件;(3)用Hermite插值方法求出相邻两插值点间的曲线.实验结果显示了方法的有效性.     

Analytic approach to intersection of all piecewise parametric rational cubic curves

Elimination theory is applied to develop an analytic approach to the intersection of any two piecewise parametric rational cubic curves. By splitting the general intersection problem into several simple cases, this algorithm reduces the intersection problem to the problem of finding the roots of a single polynomial in one variable of minimal degree. This technique is fast, automatic, efficient and robust.     

How many rational parametric cubic curves are there? I. Inflectionpoints

In his never-ending quest to build intuition about the relationship between algebra and geometry, the author turns his attention to cubic curves. The basic question is this: what sorts of shapes can a given symbolic expression generate? The ultimate goal is to transform an arbitrary parametric cubic curve both geometrically and parametrically to match one of a set of canonical simple algebraic forms     

Shape classification of the parametric cubic curve and parametric B-spline cubic curve

The conditions leading to a point of inflexion, loop cusp for parametric cubic curves and parametric B-spline cubic curves are investigated. Some useful conclusions are obtained.     

基于能量优化G2连续插值三次样条曲线

给出了一种在能量优化意义下构造G²连续保形插值三次参数样条曲线的方法。具体步骤如下：（1）以曲线应变能最小为目标构造目标函数，通过解线性方程组，求出优化意义下的每个插值点处的最优切矢方向；（2）用文中给出的简易公式求出各插值点的曲率，进而计算出插值点处的切矢模长，使曲线满足G²连续、保形插值的条件；（3）用Hermite插值方法求出相邻两插值点间的曲线。实验结果显示了方法的有效性。     

How many different rational parametric cubic curves are there? 3.The catalog

The author is interested in rational parametric cubic curves, particularly in finding out how many essentially different shapes the given equation can make. Types of homogeneous cubic polynomials are considered     

Refining cubic parametric B-splines

We present refinement properties of cubic parametric B-splines: a family of functions introduced some years ago in the context of shape-preserving approximation of curves. The refinement equations determine a related subdivision scheme which turns out to be irregular, nonstationary and nonuniform.     

Degenerate parametric curves

This paper discusses two degenerate cases of polynomial parametric curves for which the degrees of the defining polynomials can be reduced without altering the curve. The first case is the improperly parametrized curve for which each point on the curve corresponds to several parameter values. The second case, which can only occur for rational polynomial parametric curves, exists when the defining polynomials all have a common factor.

This paper describes how to detect and correct each type of degeneracy. Examples are given which demonstrate that seemingly innocuous Bézier curves may suffer from either of these degeneracies.     

Techniques for cubic algebraic surfaces

A survey of some techniques that may have potential for free-form modeling with algebraic surfaces is continued. Classical results as well as several recent innovations are included. Specific attention is paid to cubic algebraic surfaces, although many of the ideas presented have application to algebraic surfaces of any degree. Topics addressed include piecewise constructions, interpolation to points and space curves, and parameterization     

Computing all parametric solutions for blending parametric surfaces

In this paper we prove that, for a given set of parametric primary surfaces and parametric clipping curves, all parametric blending solutions can be expressed as the addition of a particular parametric solution and a generic linear combination of the basis of a free module of rank 3. As a consequence, we present an algorithm that outputs a generic expression for all the parametric solutions for the blending problem. In addition, we also prove that the set of all polynomial parametric solutions (i.e. solutions that have polynomial parametrizations) for a parametric blending problem can also be expressed in terms of the basis of a free module of rank 3, and we prove an algorithmic criterion to decide whether there exist parametric polynomial solutions. As a consequence we also present an algorithm that decides the existence of polynomial solutions, and that outputs (if this type of solution exists) a generic expression for all polynomial parametric solutions for the problem.     

Chebyshev economization for parametric surfaces

The communication of polynomial curve and surface data between various CAD systems frequently requires that a degree reduction be performed. The most common way of accomplishing this reduction at present is via ‘black box’ methods. In this note we extend the notion of Chebyshev economization from real polynomials to parametric polynomials in three dimensions, thus providing an analytical approach to degree reduction. More importantly, we propose a generalization for parametric surfaces which enjoys many of the properties associated with Chebyshev economization.     

一种三次非均匀B样条曲线的细分算法

近几年来,以B样条曲线为代表的曲线细分已成为计算机图形学领域的一项重要研究内容。提出一种基于对分方式的细分算法,能均匀地细分曲线,并用较少的细分次数得到对曲线较好的逼近效果。采用该细分算法,方便而快速地在计算机上绘制B样条曲线,对给定参数做出更加优良的控制动作,并提高控制系统的运动速度和曲线的显示速度,实例表明了该算法的有效性。     

Smoothing of cubic parametric splines

The most common curve representation in CADCAM systems of today is the cubic parametric spline. Unfortunately this curve will sometimes oscillate and cause unwanted inflexions which are difficult to deal with. This paper has developed from the need to eliminate oscillations and remove inflexions from such splines, a need which may occur for example when interpolating data measured from a model. A method for interactive smoothing is outlined and a smoothing algorithm is described which is mathematically comparable to manual smoothing with a physical spline.     

Blending two parametric curves

Segments of two given curves can be blended to produce a segment of a new curve. Blending can provide a smooth transition from one curve to another and can give various degrees of smoothness at the endpoints of the blend, where the smoothness is measured analogously to parametric continuity C⁽ⁿ⁾ and geometric continuity G⁽ⁿ⁾. Blending can provide an approximation to a given curve segment. The accuracy of the approximation to short segments obtained by different blending formulas is compared via asymptotic analysis. Finally, it is shown how to find a blend that is a parametric polynomial whose parameter is approximately an arc length parameter.