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。     

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     

二次均匀B样条曲线的双圆弧逼近方法*

提出了一种用双圆弧对二次均匀B样条曲线的分段逼近方法。首先,对一条具有n 1个控制顶点的二次均匀B样条曲线按照相邻两节点界定的区间分成n-1段只有三个控制顶点的二次均匀B样条曲线段;然后对每一曲线段构造一条双圆弧进行逼近。所构造的双圆弧满足端点及端点切向量条件,即双圆弧的两个端点分别是所逼近的曲线段的端点,而且双圆弧在两个端点处的切向量是所逼近的曲线段在端点处的单位切向量。同时,双圆弧的连接点是双圆弧连接点轨迹圆与其所逼近的曲线段的交点。这些新构造出来的双圆弧连接在一起构成了一条圆弧样条曲线,即二次均匀B样条曲线的逼近曲线。另外给出了逼近误差分析和实例说明。     

一种用圆弧逼近三次平面Bézier曲线的算法

本文给出一种用圆弧逼近三次平面Beziter曲线的算法,该算法的特点是能保持曲线的整体光滑性,所用圆弧数量少,并可对逼近精度进行控制,该算法稍加变化后也适用于用圆弧逼近其它类型的平面曲线。     

一种用圆弧逼近三次平面Bezier曲线的算法

本文给出一种用圆弧逼近三次平面Ｂｅｚｉｅｒ曲线的算法。该算法的特点是保持曲线的整体光滑性，所用圆弧数量少，并可对逼近精度进行控制。该算法稍加变化后也适用于圆弧逼近其它类型的平面曲线。     

数字曲线切点检测技术

针对数字曲线的切点识别问题,提出一种基于转角累加的数字曲线切点检测技术。该技术对数字曲线进行光顺处理,构造转角累加线、转角累加线曲率线和曲率波,通过对相邻的分段逼近直线求交确定切点位置。实验结果表明,该方法切点识别准确、迅速,对于由直线、圆弧组成的轮廓边界能实现准确分段。     

平面碎片匹配算法的研究

在分析平面曲线的几何特性的基础上,提出了一种基于曲率等不变量的平面非规则边界曲线匹配的算法,该方法通过提取平面非规则曲线的角点和匹配角点来寻找初始匹配点,同时利用对应点的曲率相等或者等价的几何特性来匹配平面非规则曲线,并且在理论和实验上对方法的可行性进行了证明。     

三次Bézier 曲线与圆弧有重合点时的Hausdorff 距离

Hausdorff 距离常用来度量两条曲线的匹配程度,因此,它可以用来度量 三次Bézier 曲线与圆弧之间的逼近程度。论文给出了三次Bézier 曲线与圆弧在中点重合时, 它们之间的Hausdorff 距离表达式;以及三次Bézier 曲线与圆弧在一般情况重合（除端点外） 时的Hausdorff 距离表达式。通过这些表达式可以直接得出三次Bézier 曲线与圆弧之间的 Hausdorff 距离。     

Bézier曲线的等距曲线的同次多项式逼近

等距曲线广泛应用工数控机床加工过程、机器人行走路线、刺绣针法生成等工业领域中,与基曲线相比,其表示更为复杂,基本小能用有理曲线来精确表示.为了使等距曲线与CAD/CAM系统更好地相容,基于圆弧的Bézier多项式逼近,提出一种Bézier曲线的等距曲线的同次多项式逼近方法.首先利用Tchebyshev多项式逼近圆弧,并由此得到圆弧的任意次数的Bézier多项式逼近;然后利用上述圆弧逼近的方法去逼近等距曲线的基圆.进而推导出了一种Bézier曲线的等距曲线多项式逼近方法,得到等距逼近曲线是与基曲线次数相同的Bézier曲线.最后通过实例与其他基于圆弧逼近的等距曲线逼近方法进行了比较,结果表明,文中方法与其他方法具有相似的逼近效果,但大大降低了逼近次数.     

双曲线偶逼近三次平面曲线的迭代算法

1 引言在进行图形处理时,经常用到各种不同的三次平面曲线,如Bezier曲线,B样条曲线等,而由于计算机软件及图形输出设备不同,它们所支持的曲线类型也不相同,因此,实际绘制图形时,通常用一种类型的曲线逼近另一种类型的曲线,如用双圆弧逼近三次平面曲线的方法,已得到广泛应用并取得良好效果。但用双圆弧逼近三次平面曲线,在两节点间仅有5个交     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

平面列表点曲线的最优双圆弧拟合

本文利用相切的双圆弧拟合平面列表点曲线，最优原则取“应变能”与弧长加权之和为最小。这个方法克服了单圆弧样条拟合及其它优化原则方法的缺点，在数控加工中将得到很好的应用。     

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.     

变曲率对称圆弧曲线及其在圆弧样条拟合中的应用

针对数控加工的需要，对圆弧样条拟合曲线的形状进行局部修改和优化，提出了一种新的圆弧样条曲线的基本形式－变曲率对称圆弧曲线，并给出了其计算方法和具体应用，该方法可满足不同运算字长数控系统对拟合后圆弧样条曲线最大曲率半径的要求，同时还可满足随动控制加工对拟合曲率变动量的要求。     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

圆渐开线平面插值样条及其应用

利用拼接的圆渐开线实现对平面上的数据点及其切向的插值,通过解决两点及其切向的圆渐开线插值,以及在各种不同情况下的插值处理方法,提供了圆渐开线平面插值样条的生成算法,由于圆渐开线为凸曲线,其曲率与弧长成反比,因此其样条曲线对插值曲线的形状控制是有利的,并可作为圆弧样条插值方法的一种扩展。     

Precision rotationally invariant curve parameterization

The problem of rotationally invariant approximation of the curve given by points is considered. The arc length is chosen for the curve parameter. The algorithm of calculating it up to the fourth order of accuracy is developed. In addition, other characteristics of the curve are constructed, viz., the slope, the curvature, and its derivative. General requirements to parametric approximation of the curve are considered that provide for rotationally invariant approximation.     

On gray scale image measurements: I. Arc length and area

We investigate methods for approximating the arc length of a planar curve and the area of a region whose boundary is a closed curve where the data are taken from a rectangular lattice of gray scale values. The commonly used sample-count method is a simple idea. Arc length is estimated by counting pixels along the curve. Typically the relative error is larger than desired. Area is estimated by counting interior pixels. The relative error is usually smaller than that for arc length, but identifying interior pixels may be a difficult geometric problem and adds computational overhead. The sample-distance methods for measuring arc length is also standard. This method requires the curve pixels to be ordered geometrically. The distances between consecutive pixels are summed to give an estimate of arc length. Although the relative errors are small, the required geometric ordering adds complexity to the problem. We introduce the sample-normal method for estimating arc length and area from gray level data. This method requires construction of unit normal vectors for the sampled curve points. Local estimates of arc length at the pixels are made from the normal vectors. The area can be approximated by using the normal vectors in the divergence theorem from calculus. Two major advantages for this method are that no geometric ordering of data points is required and that the algorithms are easily implemented. We compare the sample-normal method to the sample-count and sample-distance methods using both artificially created data and actual 8-bit digital images.