基于二叉树的曲线描述方法

为了更好地描述曲线,引入了一种有效的曲线编码和描述方法———曲线树。这种曲线树是采用树的结构来描述曲线,其内部元素是有向相对高度。由于任何一种曲线都与一个曲线树一一对应,因此从树的根部开始,取其前几层得到的树,都是对该曲线的粗略的描述,而且随着层数的增加,刻画曲线的精度就越高。这种方法最大的一个优点是它不随曲线平移、拉伸和旋转而变化。在这种曲线描述的基础上,可进一步定义两曲线的距离,以便用来衡量曲线间的相似程度。     

Control point adjustment for B-spline curve approximation

Pottmann et al. propose an iterative optimization scheme for approximating a target curve with a B-spline curve based on square distance minimization, or SDM. The main advantage of SDM is that it does not need a parameterization of data points on the target curve. Starting with an initial B-spline curve, this scheme makes an active B-spline curve converge faster towards the target curve and produces a better approximating B-spline curve than existing methods relying on data point parameterization. However, SDM is sensitive to the initial B-spline curve due to its local nature of optimization. To address this, we integrate SDM with procedures for automatically adjusting both the number and locations of the control points of the active spline curve. This leads to a method that is more robust and applicable than SDM used alone. Furthermore, it is observed that the most time consuming part of SDM is the repeated computation of the foot-point on the target curve of a sample point on the active B-spline curve. In our implementation, we speed up the foot-point computation by pre-computing the distance field of the target curve using the Fast Marching Method. Experimental examples are presented to demonstrate the effectiveness of our method. Problems for further research are discussed.     

A second-order algorithm for curve orthogonal projection onto parametric surface

Repeated use of point projection to find the projection of a curve on a surface is rather inefficient as the iteration procedures in point projection is typically slow. A novel curve projection scheme is proposed for computing the orthogonal projection of a progenitor curve onto a parametric surface. Under this scheme, the projection curve is parameterized using the parameter of the progenitor curve. Differential geometric characteristics of the projection curve are analysed. A marching method with error adjustment is used to calculate the projection curve. Several examples are presented and comparisons are made to demonstrate the effectiveness of the proposed scheme.     

保测地曲率的曲面曲线设计

给出一种在给定光滑曲面上进行曲线设计的算法．由于曲面上曲线的曲率向量可分解为测地曲率向量与法曲率向量的和,故可以通过设计平面源曲线,并将这条源曲线保测地曲率对应到给定曲面上,得到一条测地曲率和平面源曲线的曲率相同的目标曲线．取给定曲面上一点为初始点,逐次迭代跟踪得到整条目标曲线．该算法既可以用来进行曲面上的图案设计,也可用于求解光滑曲面上的测地线。     

平面代数曲线间最近距离的计算

通过几何观察,指出一条曲线上的最近点是另一条曲线的等距曲线与该曲线的切点这一事实,同时提出基于等距思想的方法来求解2条平面代数曲线间的最近距离．该方法几何意义明显,可同时用来计算代数曲线／参数曲线间的最近距离．对于平面二次曲线,采用文中方法得到的单变量多项式方程次数比已有类似方法中结果方程的次数更低,从而可以降低方程求解的计算复杂度或提高求解的稳定性．     

A surface blending approach for displacement features on freeform surfaces

Embedding a number of displacement features into a base surface is common in industrial product design and modeling, where displaced surface regions are blended with the unmodified surface region. The cubic Hermite interpolant is usually adopted for surface blending, in which tangent plane smoothness across the boundary curve is achieved. However, the polynomial degree of the tangent field curve obtained symbolically is considerably higher, and the reduction of the degree of a freeform curve is a non-trivial task. In this work, an approximation surface blending approach is proposed to achieve tangential continuity across the boundary curve. The boundary curve is first offset in the tangent field with the user-specified tolerance, after which it is refined to be compatible with the offset curve for surface blending. Since the boundary curve is offset in a three-dimensional (3D) space, the local self-intersection in the offset curve is addressed in a 2D space by approximately mapping the offset vectors in the respective tangent planes to the parameter space of the base surface. The proposed algorithm is validated using examples, and the normal vector deviation along the boundary curve is investigated.     

多项式混合曲线曲面方法构造

针对混合曲线表示及其求导和求积困难的问题,通过计算构造出一种多项式混合曲线曲面形式.当待混合曲线是多项式时,混合曲线也为多项式形式.该多项式混合公式可以推广得到任意参数连续C(n)和几何连续G(n)的混合曲线曲面.另外,在得到的混合曲线曲面族中构造出了新的更优能量光顺方程,通过设置参数可得到合适的混合曲线曲面.实验结果表明,文中提出的混合曲线曲面造型方法稳定、有效.     

基于流体静力学的区域分割活动轮廓模型

提出一种新的活动轮廓模型,应用于灰度图像的区域分割。对于定义在灰度图像 上的闭合连续简单曲线,该模型应用流体静力学理论直接驱动,使其不断地缓慢收拢,直至收 敛于区域边界。在这个过程中,闭合连续简单曲线所经历的像素都被该模型根据像素性质判定 其区域归属。重新初始化有关变量,激活已收敛于区域边界的闭合连续简单曲线,继续驱动闭 合连续简单曲线收拢,直至该曲线收敛于内嵌的新区域边界或者收敛于一个点。在该模型运行 过程中,一条闭合连续简单曲线可能会分裂成多条闭合连续曲线,以适应多区域分割。当一条 闭合连续的简单曲线经过模型持续驱动之后收敛于一个点时,被其包围的区域分割才告结束。 本文提出的模型能够分割多区域和嵌套区域。     

Cubic B-spline curve approximation by curve unclamping

A new approach for cubic B-spline curve approximation is presented. The method produces an approximation cubic B-spline curve tangent to a given curve at a set of selected positions, called tangent points, in a piecewise manner starting from a seed segment. A heuristic method is provided to select the tangent points. The first segment of the approximation cubic B-spline curve can be obtained using an inner point interpolation method, least-squares method or geometric Hermite method as a seed segment. The approximation curve is further extended to other tangent points one by one by curve unclamping. New tangent points can also be added, if necessary, by using the concept of the minimum shape deformation angle of an inner point for better approximation. Numerical examples show that the new method is effective in approximating a given curve and is efficient in computation.     

基于小波的曲线可控误差拟合

研究了利用小波变换对所拟合曲线进行分解和重构的方法,给出了进行曲线拟合的误差控制,采用Daubechies小波对所拟合曲线进行快速小波变换。通过分层近似对原曲线进行拟合,所产生的误差刚好为高频滤掉部分。在构造近似曲线前,通过对曲线细节的预先计算,得出可控制误差的范围,从而决定是否继续进行曲线的细化分解,达到在可控制误差下进行曲线拟合的目的。文中给出了基于小波的曲线可控误差拟合算法,并用实例进行了说明。