平面NURBS曲线及其Offset的双圆弧逼近

除直线、圆弧、速端曲线等少数几种曲线外,平面参数曲线的offset曲线通常不能表示成有 理参数形式,因此在实际应用中,为了方便造型系统中数据结构和几何算法的统一表示,offse t曲线通常用低次曲线逼近来表示.通过用双圆弧逼近表示NURBS(non-uniform rational B -spline)曲线及其offset,并利用双圆弧逼近的特有性质,把offset的双圆弧逼近转化为原 曲线的双圆弧逼近,简化了问题的求解.同时考虑了双圆弧逼近算法中分割点的选取、公切点 的确定以及误差估计等主要问题.具体算     

Bounding the Distance between 2D Parametric Bézier Curves and their Control Polygon

Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control polygon based on the Greville abscissae. Several of the norms appearing in these bounds are orientation dependent. We next present algorithms for finding the optimal orientation angle for which two of these norms become minimal. The use of these bounds and algorithms for constructing polygonal envelopes of planar polynomial curves, is illustrated for an open and a closed composite Bézier curve.     

Planar piecewise algebraic curves

An approach is described for piecing together segments of planar algebraic curves with derivative continuity. The application of piecewise algebraic curves to area modelling (the two-dimensional analogue of solid modelling) is discussed. A technique is presented for expressing a planar rational parametric curve as an algebraic curve segment. An upper bound is derived for the farthest distance between two algebraic curves (one of which may also be a parametric curve) within a specified region.     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.     

The method of resolvents: A technique for the implicitization, inversion, and intersection of non-planar, parametric, rational cubic curves

Let P(t) be a non-planar, parametric, rational cubic curve. The method of resolvents is applied to: (1) construct three quadric surfaces whose intersection is equal to P(t) (implicitization); (2) solve for the parameter t as the ratio of two linear expressions in the coordinates x, y, z (inversion). The results of these two operations are then applied to construct an optimal, robust, intersection algorithm for any two non-planar rational cubic curves, and it is shown that two such curves can intersect in at most five points. Specializations of these results for non-planar, integral, cubic curves are derived, and extensions of these techniques to non-planar, rational cubic, Bézier curves are also discussed.     

A locally controllable spline with tension for interactive curve design

We describe a new type of parametrically defined space curve. The parameters which define these curves allow for the convenient control over local shape attributes while maintaining global second order geometric continuity. The coordinate functions are defined by piecewise segments of rational functions, each segment being the ratio of cubic polynomials and a common quadratic polynomial. Each curve segment is a planar curve and where two segments meet the curvature is zero. This simple mathematical representation permits these curves to be efficiently manipulated and displayed.     

A Direct Approach to Computing theμ-basis of Planar Rational Curves

This paper presents an O(n²) algorithm, based on Gröbner basis techniques, to compute the μ -basis of a degree n planar rational curve. The prior method involved solving a set of linear equations whose complexity by standard numerical methods was O(n³). The μ -basis is useful in computing the implicit equation of a parametric curve and can express the implicit equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric equations.     

平面三次混合双曲多项式曲线的特征图判别

根据文献[9](Wang G Z,Yang Q M.Planar cubic hybrid hyperbolic polynomial curve and its shape classification.Progress in Natural Science,2004,14(1):41-46)中提出的H-曲线带奇点或拐点的条件,利用H-曲线奇点、拐点的仿射不变性,给出H-曲线几何特征图的判别法,并找到了不同特征图在三维空间中的关系.该判别法完善了H-曲线的奇异点检测理论,提升了几何特征图维数.     

参数多项式曲线的快速逐点生成算法

给出了参数多项曲线（包括Ｂｅｚｉｅｒ曲线、Ｂ样条曲线等）的一种快速逐点生成算法．在曲线的逐点生成过程中,只用到加减法,故效率极高．而且,此方法可在两３方面加以推广,一是推广到有理参数曲线（包括非均匀有理Ｂ样条曲线）,一是推广到多项式参数曲面以及更高维的多项式参数函数．     

Surface-to-surface intersections

Techniques for computing intersections of algebraic surfaces with piecewise rational polynomial parametric surface patches and intersections of two piecewise rational polynomial parametric surface patches are discussed. The techniques are classified using four categories-lattice evolution methods, marching methods, subdivision methods, and analytic methods-and their principal features are discussed. It is shown that some of these methods also apply to the general parametric surface-intersection problem