Noninterpolatory Hermite subdivision schemes

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.

A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.

     

复二次B样条曲线

本文在复平面单位圆弧上引进了复二次Ｂ样条曲线，讨论了它的一些几何性质．实质上它是分段帕斯卡蜗线段的Ｃ１合成曲线．调整控制点可使某段曲线为圆孤．     

Changeable degree spline basis functions

A B-spline basis function is a piecewise function of polynomials of equal degree on its support interval. This paper extends B-spline basis functions to changeable degree spline (CD-spline for short) basis functions, each of which may consist of polynomials of different degrees on its support interval. The CD-spline basis functions possess many B-spline-like properties and include the B-spline basis functions as subcases. Their corresponding parametric curves, called CD-spline curves, are like B-spline curves and also have many good properties. If we use the CD-spline basis functions to design a curve made up of polynomial segments of different degrees, the number of control points may be decreased.     

Piecewise polynomial spaces and geometric continuity of curves

Summary We say that a curve has geometric continuity if its curvatures and Frenet frame are continuous. In this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity. We show that the basic theoretical properties of ordinary spline functions also hold for these spaces. These results extend and unify recent work on Beta-splines and Nu-splines which are used as a design tool in computer-aided geometric design of free form curves and surfaces.This work was initiated when the first author was on Sabbatical at Thomas J. Watson IBM Research Center, and was partially supported by the U.S.-Israel Binational Foundation, grant no. 86-00243/1.     

Deriving the beta-constraints for geometric continuity of parametric curves

Parametric splines curves are typically constructed so that the firstn parametric derivatives agree where the curve segments abut. This type of continuity condition has become known asC ⁿ orn ^th orderparametric continuity. It has previously been shown that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves. We definen ^th ordergeometric continuity (Gⁿ), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity.G ⁿ continuity provides for the introduction ofn quantities known asshape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.     

The Fundamental Blossoming Inequality in Chebyshev Spaces—I: Applications to Schur Functions

A classical theorem by Chebyshev says how to obtain the minimum and maximum values of a symmetric multiaffine function of n variables with a prescribed sum. We show that, given two functions in an Extended Chebyshev space good for design, a similar result can be stated for the minimum and maximum values of the blossom of the first function with a prescribed value for the blossom of the second one. We give a simple geometric condition on the control polygon of the planar parametric curve defined by the pair of functions ensuring the uniqueness of the solution to the corresponding optimization problem. This provides us with a fundamental blossoming inequality associated with each Extended Chebyshev space good for design. This inequality proves to be a very powerful tool to derive many classical or new interesting inequalities. For instance, applied to Müntz spaces and to rational Müntz spaces, it provides us with new inequalities involving Schur functions which generalize the classical MacLaurin’s and Newton’s inequalities. This work definitely demonstrates that, via blossoms, CAGD techniques can have important implications in other mathematical domains, e.g., combinatorics.     

Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach.     

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries — II: Detailed algorithm description

Details of algorithms to construct the Voronoi diagrams and medial axes of planars domain bounded by free-form (polynomial or rational) curve segments are presented, based on theoretical foundations given in the first installment Ramamurthy and Farouki, J. Comput. Appl. Math. (1999) 102 119–141 of this two-part paper. In particular, we focus on key topological and computational issues that arise in these constructions. The topological issues include: (i) the data structures needed to represent various geometrical entities — bisectors, Voronoi regions, etc., and (ii) the Boolean operations (i.e., union, intersection, and difference) on planar sets required by the algorithm. Specifically, representations for the Voronoi polygons of boundary segments, and for individual Voronoi diagram or medial axis edges, are proposed. Since these edges may be segments of (a) nonrational algebraic curves (curve/curve bisectors); (b) rational curves (point/curve bisectors); or (c) straight lines (point/point bisectors), data structures tailored to each of these geometrical entities are introduced. The computational issues addressed include the curve intersection algorithms required in the Boolean operations, and iterative schemes used to precisely locate bifurcation or “n-prong” points (n ⩾ 3) of the Voronoi diagram and medial axis. A selection of computed Voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm.     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.     

一类非端点插值B样条曲线降阶的方法

降阶算法是B样条曲线和曲面设计的一个基本算法,它广泛应用于组合曲线,蒙皮或扫描曲面等设计中.Piegl与Tiller曾给出B样条曲线的降阶方法.本文给出了解决更一般的非端点插值B样条曲线降阶的方法.新的方法主要是通过对现有的节点插入方法进行分析,给出了一种端点插值递推公式,并利用此公式对Piegl与Tiller降阶方法加以改进,使之能够解决非端点插值均匀及非均匀B样条曲线的降阶问题.     

The dual bases for the Bézier-Said-Wang type generalized Ball polynomial bases and their applications

A unifying representation for the existing generalized Ball bases and the Bernstein bases are given. Then the dual bases for the Bézier-Said-Wang type generalized bases (BSWGB for short) are presented. The Marsden identity and the mutual transformation formulas between Bézier curve and Bézier-Said-Wang type generalized curve (BSWGB curve) are also given. These results are very useful for the applications of BSWGB curves and their popularization in CAGD. Numerical examples are also given to show the effectiveness of our methods.     

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown.     

Smoothness Analysis of Subdivision Schemes by Proximity

Linear curve subdivision schemes may be perturbed in various ways, for example, by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes "T" is based on their proximity to the linear schemes "S" which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities which together with C^k smoothness of S imply C^k smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in the case k = 2. The main result of the paper is that convergent geodesic/projection/Lie group analogues of a certain class of factorizable linear schemes have C² limit curves.     

基于几何连续的AT-β-Spline曲线曲面的构造

为了更好地修改给定的样条曲线曲面,构造了满足几何连续的带两类形状参数的代数三角多项式样条曲线曲面,简称为AT-β-Spline.这种代数三角曲线曲面不仅具有普通三角多项式的性质,而且具有全局的和局部的形状可调性.同时还具备较为灵活的连续性.当两类形状参数在给定的范围内任意取值时,这种带两类形状参数的AT-β-Spline曲线满足一阶几何连续性;如果给定两段相邻曲线段中的两类形状参数满足-1≤α≤1,μ_i=λ_(i+1)或μ_i=λ_i=μ_(i+1)=λ_(i+1)时,则带两类形状参数的AT-β-Spline曲线满足C~1∩G~2连续.另外利用奇异混合的思想,构造了满足C~1∩G~2插值AT-β-Spline曲线,解决曲线反求的几何连续性等问题.同时还给出了旋转面的构造,描述了两类形状参数对旋转面的几何外形的影响;当形状参数取特殊值时,这种AT-β-Spline曲线曲面可以精确地表示圆锥曲线曲面.从实验的结果来看,本文构造的AT-β-Spline曲线曲面是实用的有效的.     

Families of space curves with large cohomology

We investigate space curves with large cohomology. To this end we introduce curves of subextremal type. This class includes all subextremal curves. Based on geometric and numerical characterizations of curves of subextremal type, we show that, if the cohomology is “not too small,” then they can be parameterized by the union of two generically smooth irreducible families; one of them corresponds to the subextremal curves. For curves of negative genus, the general curve of each of these families is also a smooth point of the support of an irreducible component of the Hilbert scheme. The two components have the same (large) dimension and meet in a subscheme of codimension one.     

Curvature driven flow of a family of interacting curves with applications

In this paper, we investigate a system of geometric evolution equations describing a curvature-driven motion of a family of planar curves with mutual interactions that can have local as well as nonlocal character, and the entire curve may influence evolution of other curves. We propose a direct Lagrangian approach for solving such a geometric flow of interacting curves. We prove local existence, uniqueness, and continuation of classical Hölder smooth solutions to the governing system of nonlinear parabolic equations. A numerical solution to the governing system has been constructed by means of the method of flowing finite volumes. We also discuss various applications of the motion of interacting curves arising in nonlocal geometric flows of curves as well as an interesting physical problem of motion of two interacting dislocation loops in the material science.     

An ideal spline-wavelet family for curve design and editing

Subdivision schemes provide the most efficient and effective way to design and render smooth spatial curves. It is well known that among the most popular schemes are the de Rham–Chaikin and Lane–Riesenfeld subdivision schemes, that can be readily formulated by direct applications of the two-scale (or refinement) sequences of the quadratic and cubic cardinal B-splines, respectively. In more recent works, semi-orthogonal and bi-orthogonal spline-wavelets have been integrated to curve subdivision schemes to add such powerful tools as automatic level-of-detail control algorithm for curve editing and rendering, and efficient simulation processing scheme for global graphic illumination and animation. The objective of this paper is to introduce and construct a family of spline-wavelets to overcome the limitations of semi-orthogonal and bi-orthogonal spline-wavelets for these and other applications, by adding flexibility to the enhancement of desirable characters without changing the sweep of the subdivision spline curve, by providing the shortest lowpass and highpass filter pairs without decreasing the discrete vanishing moments, and by assuring robust stability.     

A geometric approach to Euler’s addition formula

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.