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.     

Minimal quasi-Bézier surface

The quasi-Bézier surface is a kind of commonly used surfaces in CAGD/CAD systems. In this paper, we study how to find the quasi-Bézier surface of minimal area among all the quasi-Bézier surfaces with prescribed borders, i.e., the Plateau-quasi-Bézier problem. The prescribed borders can not only be polynomial curves, but also catenaries and circular arcs. Moreover, the harmonic and biharmonic quasi-Bézier surfaces are investigated.     

Cone Spline Surfaces and Spatial Arc Splines – a Sphere Geometric Approach

Basic sphere geometric principles are used to analyze approximation schemes of developable surfaces with cone spline surfaces, i.e., G ¹-surfaces composed of segments of right circular cones. These approximation schemes are geometrically equivalent to the approximation of spatial curves with G ¹-arc splines, where the arcs are circles in an isotropic metric. Methods for isotropic biarcs and isotropic osculating arc splines are presented that are similar to their Euclidean counterparts. Sphere geometric methods simplify the proof that two sufficiently close osculating cones of a developable surface can be smoothly joined by a right circular cone segment. This theorem is fundamental for the construction of osculating cone spline surfaces. Finally, the analogous theorem for Euclidean osculating circular arc splines is given.     

带形状参数的代数三角样条曲线曲面的构造（英文）

The construction of trigonometric B-spline curves with shape parameters has become the hotspot in computer aided geometric design.However,the shape parameters of the curves and surfaces are all global parameters and only meet with C~2 continuity in some previous papers.In order to provide more flexible approaches for designers,the algebraic and trigonometric spline(AT-spline) curves and surfaces are constructed as a generalization of the traditional cubic uniform B-spline curves and surfaces.AT-spline curves and surfaces not only inherit the properties of trigonometric B-spline curves,but also exhibit better performance when adjusting its local shapes through two shape parameters.Particularly,the AT-spline rotational surfaces with two local shape parameters are presented.When the shape parameters take special value,it can accurately represent the conic curve and surface.     

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.     

常挠率运动曲线生成曲面上的贝克隆变换

该文给出由常挠率运动曲线生成曲面上的贝克隆变换,其中运动曲线的曲率满足修正KdV方程,从而得到著名的对于修正KdV方程贝克隆变换的一个几何实现.作为应用,取圆柱面作为种子曲面,构造了一些由周期运动曲线生成的新曲面,其中周期运动曲线在xy平面上的投影是闭曲线.     

New minimal surfaces in the hyperbolic space

We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H~3 found by Mori(1981), we obtain 2-and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R~2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H~3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.     

Networks of polynomial pieces with application to the analysis of point clouds and images

We consider Hölder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in ‘good continuation’ of each other. Little known to the community, a similar construction was used by Kolmogorov and Tikhomirov in their proof of their celebrated entropy results for Hölder classes.We show how to use such networks in the context of detecting geometric objects buried in noise to approximate the scan statistic, yielding an optimization problem akin to the Traveling Salesman. In the same context, we describe an alternative approach based on computing the longest path in the network after appropriate thresholding.For the special case of curves, we also formalize the notion of ‘good continuation’ between beamlets in any dimension, obtaining more economical piecewise linear approximation networks for curves.We include some numerical experiments illustrating the use of the beamlet network in characterizing the filamentarity content of 3D data sets, and show that even a rudimentary notion of good continuity may bring substantial improvement.     

Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

This paper surveys some recent developments around the notion of a scalar partial differential equation describing pseudo-spherical surfaces due to Chern and Tenenblat. It is shown how conservation laws, pseudo-potentials, and linear problems arise naturally from geometric considerations, and it is also explained how Darboux and B?cklund transformations can be constructed starting from geometric data. Classification results for equations in this class are stated, and hierarchies of equations of pseudo-spherical type are introduced, providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2, R)-valued linear problems. Furthermore, the existence of correspondences between any two solutions to equations of pseudo-spherical type is reviewed, and a correspondence theorem for hierarchies is also mentioned. As applications, an elementary immersion result for pseudo-spherical metrics arising from the Chern?CTenenblat construction is proven, and non-local symmetries of the Kaup?CKupershmidt, Sawada?CKotera, fifth order Korteweg?Cde Vries and Camassa?CHolm (CH) equation with non-zero critical wave speed are considered. It is shown that the existence of a non-local symmetry of a particular type is enough to single the first three equations out from a whole family of equations describing pseudo-spherical surfaces while, in the CH case, it is shown that it admits an infinite-dimensional Lie algebra of non-local symmetries which includes the Virasoro algebra.     

Correspondence and (2,1)-Bézier Surfaces

Tom Sederberg's method of moving curves (surfaces) is a new and effective tool for implicitizing curves (surfaces). From our point of view, the curve (surface) can be defined by using moving curves (surfaces) which in algebraic geometry are called correspondences. It turns out that from this definition we can easily derive both parametric and implicit representations of the curve (surface). In this paper, we investigate the geometry of the bi-degree (2,1)-Bézier surface and study the relationship between singularities and correspondences. We also characterize all the possible singular curves in terms of the control points of the surface.     

Correction: On the Severi problem in arbitrary characteristic

In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.

     

Evolutes of dual spherical curves for ruled surfaces

E. Study found that there is a one‐to‐one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.     

Exemples de comptages de courbes sur les surfaces

Some examples of curves countings on surfaces Let $X$ be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. We show that the geometric Manin’s conjectures hold for some degrees lying in the dual of the effective cone of $X$ (in particular, for those degrees the moduli space of morphisms has the expected dimension). The result applies to a class of generalized del Pezzo surfaces which has been intensively studied in the context of the arithmetic Manin’s conjecture.     

On the genus of a cyclic curve over a finite field

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.     

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Summary. Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a ``crystalline' approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension. Received January 28, 1993     

Extending immersions of curves to properly immersed surfaces

Proper generic immersions of compact one-dimensional manifolds in surfaces are studied. Suppose an immersion γ of a collection of circles is given with an even number of double points in a closed surface G. Then γ extends to various proper immersions of surfaces in three-manifolds that are bounded by G. Some of these extensions do not have triple points. The minimum of the genera of the triple point free surfaces is an invariant of the curve. An algorithm to compute this invariant is given.Necessary and suffecient conditions determine if a given collection δ of immersed arcs in a surface F maps to the double points set of a proper immersion. In case the conditions are satisfied, an immersion of F into a three-manifold that depends on δ is constructed explicitly. In the process, the possible triple points of immersed surfaces in three-manifolds are categorized.The techniques are applied to find examples of curves in surfaces that do not bound immersed disks in any three-manifold.     

On a special case of the intersection of quadric and cubic surfaces

The intersection curve between two surfaces in three-dimensional real projective space RP³ is important in the study of computer graphics and solid modelling. However, much of the past work has been directed towards the intersection of two quadric surfaces. In this paper we study the intersection curve between a quadric and a cubic surface and its projection onto the plane at infinity. Formulas for the plane and space curves are given for the intersection of a quadric and a cubic surface. A family of cubic surfaces that give the same space curve when we intersect them with a quadric surface is found. By generalizing the methods in Wang et al. (2002) [6] that are used to parametrize the space curve between two quadric surfaces, we give a parametrization for the intersection curve between a quadric and a cubic surface when the intersection has a singularity of order 3.     

A non-local area preserving curve flow

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in \(C^{\infty }\) sense as time goes into infinity.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000