空间曲线基于内在几何量的高质量采样和B样条拟合

为解决均匀参数采样在许多情况下得到质量不高的采样点,进而生成不理想的B样条拟合曲线,提出空间曲线基于内在几何量的均匀采样方法,以获得给定总数且具有代表性的采样点.首先定义基于弧长、曲率和挠率加权组合的特征函数,通过调整组合参数更好匹配不同的曲线形状;然后提出空间曲线基于内在几何量的自适应采样方法,迭代生成满足给定距离阈值的采样点.采用最大绝对误差和均方根误差作为评价指标,与均匀弧长采样方法和基于弧长和曲率平均的均匀采样方法进行对比,并通过实例进行验证.结果表明,文中方法在采样质量和B样条拟合结果上获得明显改善.     

逆向工程中自由曲面的自适应采样与建模

介绍了将逆向工程与正向设计结合实现自由曲面模型重构的新方法 .由激光线扫描法测量曲面获取截面轮廓数据 ,采用准均匀 B样条逼近算法对离散数据进行拟合 ,然后进行 B样条曲线基于曲率特征的自适应采样以实现数据压缩 ,再生成三维造型软件 Pro/ E的接口文件 *.ibl,最后在 Pro/ E中重构曲面模型 .     

G 2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G ²) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG ² blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG ² interpolation/blending methods in IR².     

散乱点云数据曲率估计方法

针对带有强噪声离散点云数据曲率计算问题,提出一种基于稳健统计的曲率估计方法。首先,用一个二次曲面拟合三维空间采样点处的局部形状;其次,随机地选择该采样点邻域内的子集,多次执行这样的拟合过程,通过变窗宽的最大核密度估计,就得到了最优拟合曲面;最后,将采样点投影到该曲面上,计算投影点曲率信息,就得到采样点曲率。实验结果表明,所提方法对噪声和离群点是稳健的,特别是随着噪声方差的增大,要明显好于传统的抛物拟合方法。     

Self-intersections of offsets of quadratic surfaces: Part I, explicit surfaces

Although offset surfaces are widely used in various engineering applications, their degenerating mechanism is not well known in a quantitative manner. Offset surfaces are functionally more complex than their progenitor surfaces and may degenerate even if the progenitor surfaces are regular. Self-intersections of the offsets of regular surfaces may be induced by concave regions of surface where the positive offset distance exceeds the maximum absolute value of the negative minimum principal curvature or the absolute value of the negative offset distance exceeds the maximum value of the positive maximum principal curvature. It is well known that any regular surface can be locally approximated in the neighborhood of a pointp by the explicit quadratic surface of the form r(x,y)=[x,y1/2(x²+y²)]^T to the second order where – and – are the principal curvatures at pointp. Therefore investigations of the selfintersecting mechanisms of the offsets of explicit quadratic surfaces due to differential geometry properties lead to an understanding of the self-intersecting mechanisms of offsets of regular parametric surfaces. In this paper, we develop the equations of the self-intersection curves of an offset of an explicit quadratic surface. We also develop an algorithm to detect and trace a small loop of a self-intersection curve of an offset of a regular parametric surface based on our analysis of offsets of explicit quadratic surfaces. Examples illustrate our method.     

基于曲率分析的三次Bezier曲线采样方法的研究

针对三维建模过程中数据量大的缺点,提出一种简单的基于曲率分析的三次Bezier曲线采样方法。该方法采用每个分段的三次Bezier曲线的特征点和该段曲率半径的极小值作为采样密度的判断标准,曲线采样主要分为多层轮廓和单一轮廓两种情况,对于多层轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,轮廓之间的间距,曲线的长度。而对于单一轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,曲线的长度。通过以上因素,计算出采样点的数目。实验结果证明,提出的方法可行有效,可用于三维建模的数据点采样。     

参数曲面上的插值与混合

如何表示曲面上的曲线,在处理诸如数控加工中的路径设计以及CAD/CAM等领域频繁出现的曲面裁剪问题时显得日益重要.给出了数据点的切方向(切方向及曲率向量或测地曲率值)指定而G¹连续(G²连续)插值曲面上任意点列的方法.作为曲面上曲线插值问题的特例,还讨论了曲面上曲线的混合问题.基本思想是借助于微分几何的有关结论,曲面上曲线的插值问题被转化为其参数平面上类似的曲线插值问题.该方法能够用二维隐式方程来表示曲面上的插值曲线,从而把在显示该曲线时所面对的曲面求交的几何问题转化为计算隐式曲线的代数问题.实验证明该方法是可行的,而且适用于CAD/CAM及计算机图形学等领域.     

Local shape of offsets to algebraic curves

In this paper we introduce the notion of local shape to describe the behavior of a real place of an algebraic curve around its center. We analyze how the local shape is affected by the offsetting process, and we relate this phenomenon to the curvature of the curve. Furthermore, we characterize the situations when the offsetting process behaves locally well, so that the local shape is preserved.     

Minimizing the maximum curvature of quadratic Bézier curves with a tetragonal concave polygonal boundary constraint

In curve design such as highway design and motion planning of autonomous vehicles, it may be important to minimize the maximum curvature. In this paper we address the problem of minimizing the maximum curvature of a quadratic Bézier curve within a boundary constraint determined by a tetragonal concave polygon. The curve is parameterized by lengths between its control points, called the “control lengths”. Finally, numerical results demonstrate applicability of the method to smooth a piecewise linear path resulting from a path search technique. The results apply whenever it is desired to have a smooth transition between intersecting straight lines.     

基于局部曲面拟合的散乱点云简化方法

随着数据获取手段的进步,散乱点云数据在三维重建中获得越来越广泛的应用,然而庞大的数据量往往影响重建的效率。现有简化算法中采用的曲率计算方法精度不高,导致模型特征模糊。本文在分析曲面特征的基础上给出了一种曲面特征的定量描述方法。该方法采用局部曲面拟合得到曲面在一点处的近似曲面,然后用法曲率在360度范围内的平均值代替平均曲率来描述曲面在一点处的特征。简化时采用K-D树剖分点云数据,根据子节点所包含的采样点数、空间区域大小和曲面特征大小控制简化过程。实验结果表明,该方法能够更好地保持曲面的几何特征,从而证明了算法的有效性。     

关键点选取的最小二乘渐进迭代逼近

目的 最小二乘渐进迭代逼近（LSPIA）方法多以均匀参数化或弦长参数化的形式均匀地确定初始控制点,虽然取得了良好效果,但在处理复杂曲线时,迭代速度相对较慢且误差精度不一定能达到预期设定值。为了进一步提高迭代效率和误差精度,本文提出了基于关键点（局部曲率最大点和极端曲率点）的最小二乘渐进迭代逼近方法。方法 首先计算所有数据点的离散曲率,筛选出局部曲率最大点;接着设定初始的曲率下限,筛选出极端曲率点;然后将关键点与均匀选取的控制点按参数顺序化,并将其作为迭代的初始控制点;最后利用LSPIA方法对数据点进行拟合。结果 对同一组数据点,分别采用LSPIA方法和基于关键点的LSPIA方法,本文方法较好地提高了收敛速度;在相同的控制点数目下,与LSPIA算法相比,本文方法的误差精度较小。结论 本文方法适合于比较复杂的曲线,基于曲率分布的关键点的选取,可以更好地反映曲线的几何信息。数值实例表明,结合关键点筛选策略的LSPIA算法提高了计算效率,取得了更好的拟合效果。     

Constructing PDE-based surfaces bounded by geodesics or lines of curvature

In order to explore a new approach to construct surfaces bounded by geodesics or lines of curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First, the corresponding biharmonic surface can naturally be derived under some degenerative conditions; second, the parameters in the PDE implicate some physical meaning, such as elasticity or rigidity; third, there are only a few parameters that need to be evaluated, and hence the computation is simple. In addition, this paper constructs two adjacent surfaces with C¹ continuity whose common boundary is the same given curve as well as respective geodesic (or line of curvature). Examples show that this method to construct PDE-based surfaces bounded by geodesics or lines of curvature is easy and effective.     

Identification of space curves from two-dimensional perspective views

This paper describes a new method to be used for matching three-dimensional objects with curved surfaces to two-dimensional perspective views. The method requires for each three-dimensional object a stored model consisting of a closed space curve representing some characteristic connected curved edges of the object. The input is a two-dimensional perspective projection of one of the stored models represented by an ordered sequence of points. The input is converted to a spline representation which is sampled at equal intervals to derive a curvature function. The Fourier transform of the curvature function is used to represent the shape. The actual matching is reduced to a minimization problem which is handled by the Levenberg-Marquardt algorithm [3].     

Marching along surfacesurface intersection curves with an adaptive step length

A method for generating points on the intersection of two C² smooth parametric surfaces is presented. In each generated point the tangent and the curvature of the intersection curve are obtained from the surface positions, first and second derivatives. Initial approximation of the next point lies on a parabola approximating the intersection curve in a vicinity of the last point found. The length of the parabola between the two points is evaluated so that its maximal deviation from the chord joining the points is not greater than a given deviation tolerance. The new point is relaxed to the intersection curve.     

基于适应度函数的无人船遗传算法航径规划

遗传算法通过适应度函数选取最优的路径,采用了无人船转弯半径来改进适应度函数,实现无人船遗传算法航径规划。考虑到无人船机动性能对航迹平滑性的要求,在初始种群中利用贝塞尔曲线优化方法,将原有的折线路径优化成光滑的曲线路径;在适应度函数中添加曲率判断,以无人船最小转弯半径为约束条件,设置曲线路径的最大曲率,最后通过适应度函数筛选出符合约束条件的光滑路径。仿真结果表明,所提出的方法能获得符合无人船最小转弯半径约束的光滑路径,相比于平滑算法,该方法的曲率更小,收敛速度更快。     

Two-dimensional curve synthesis using linear curvature elements

A new algorithm is presented for the generation of a smooth curve through two data points with specified curvature and tangent direction at those points. The curvature profile $\text{K(s)}$ is designed to be piecewise linear, constructed from linear curvature elements (linces). This gives a curve form consisting of arcs of Cornu spirals (clothoids). A trilince is the normal solution as this can be optimized to meet a suitable criterion for fairness of the curve. A FORTRAN program based on this algorithm is incorporated in a linear curvature spline package.     

Curvature continuous interpolation of curve meshes

This paper presents a method for local construction of a curvature continuous (GC²) piecewise polynomial surface which interpolates a given rectangular curvature continuous quintic curve mesh. First, we create a C² quintic basic curve mesh, which interpolates the same grid points, preserves the tangent slopes and curvatures but not derivative vectors at the grid points. After estimating twist and higher order mixed partial derivatives for each grid point, we generate locally the so-called C² biquintic basic patches, which serve to compute the first and second order cross-derivative functions of the final interpolation surface. In order to match the tangents and second order derivative vectors of the original boundary curves at the grid points, these basic patches are reparametrized by 5 × 3 and 3 × 5 functions, which lead to vector-valued first and second order cross-derivative functions of degrees 7 and 9 of the final surface patches, and eventually lead to a GC² piecewise polynomial surface of degree 9 × 9, which is then converted to a GC² Bézier composite surface. By arranging the surface patches in a chess-board fashion, the degrees of the final surface patches can be 9 × 5 and 5 × 9. An example for the construction of a GC² ship hull, together with its color-coded curvature maps, is given to illustrate the method. This method is attractive because the final surface has a much lower degree than other similar methods, it allows flexible local modification of the original curve mesh and local editing of the interpolation surface, and it is easily integrated into state-of-the-art geometric modeling systems.     

渐进迭代逼近方法在曲线变形上的应用

目的 随着几何造型、计算机动画等领域的快速发展,曲线的自由变形技术在近年来受到了广泛的关注。为了获得更多有趣、逼真的变形效果,提出基于渐进迭代逼近与主顶点方法的曲线局部变形算法。方法 给定数据点集,首先采用渐进迭代逼近方法或是基于最小二乘的渐进迭代逼近方法产生待变形曲线;其次对待变形区域使用延拓准则,基于主顶点方法与待变形曲线的形状信息选取控制顶点进行调整;最后对调整后的控制顶点运用局部渐进迭代逼近方法生成逼近曲线,得到期望的变形效果。结果 此变形操作借助于局部渐进迭代逼近方法,具有较好的灵活性。通过茶壶、面部轮廓、手等数值实例,表明了该方法可以得到良好的变形效果。进一步地,借助于叠加变形还可以得到整体的、周期的、伸缩的等各类更加丰富的变形效果。结论 本文研究渐进迭代逼近在曲线变形上的应用,将主顶点方法引入曲线的变形之中,把两者相结合提出了基于渐进迭代逼近与主顶点方法的曲线局部变形算法。该算法不仅具备渐进迭代逼近方法的收敛稳定性,且借助于主顶点方法,可以得到较好的变形效果。该方法适用于曲线的局部变形,丰富了曲线的变形效果。     

最大比率圆弧逼近轮廓的匹配方法

本文提出了一种新的圆孤逼近轮廓曲线进行目标匹配的方法—最大比率法。曲线上两点之间的圆弧和曲线夹成的面积与对应扇形的比值随曲线上点的曲率的变化而变化。通过设置一个阈值,算法可以检测曲率较大的特征点,用于圆弧逼近匹配。     

复杂曲面五轴加工局部干涉处理技术研究

针对复杂曲面环形刀五轴数控加工中的局部干涉问题,提出了一种基于曲 率匹配及网格点的干涉处理技术。首先,利用曲率匹配原则选出合理的刀具半径,以保证在 切触点处沿任何方向上刀具与被加工曲面之间不会发生干涉,然后在各个切触点处通过比较 刀具曲面最小主曲率与加工曲面最大主曲率确定出刀具的初始倾角。为了判断切触点邻近区 域是否存在干涉问题,采用了网格点来快速自动生成检测区域及初始检测点。文中对有效检 测点的筛选以及干涉的判断和处理技术分别进行了详细论述。最后,以非均匀有理B 样条 曲面为加工实例,对上述算法进行了测试和验证。