Pythagorean Hodograph C-曲线的几何构造方法

本文研究具有Pythogorean Hodograph (PH)性质的C Bézier曲线的几何性质.以PH C-曲线的代数性质为基础,应用平面参数曲线的复表示方法,本文证明一条C Bézier曲线是PH C-曲线的充分必要条件是其控制多边形的两内角相等,且其第2条边长为首末边长的等比中项.该性质与三次多项式PH曲线相类似,可以用于PHC-曲线的判别.此外,该性质可以很好地应用于解决PH C-曲线的Hermite插值问题,本文构造了PH C-曲线的G¹ Hermite插值实例,指出对于给定的G¹ Hermite端点条件,存在不超过2条PH C-曲线满足约束.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

三次Pythagorean Hodograph 曲线在给定总弧长下的$G^2$拼接

Pythagorean-hodograph (PH)曲线因其在弧长和等距线计算方面的优势而被广泛应用于曲线建模中.本文讨论了在总弧长约束下的三次PH曲线$G^2$连续拼接问题.具体地说,给定两个端点和一个拼接点,构造两条三次PH曲线,使其在指定总弧长下插值两个端点,并且在连接点处是$G^2$连续的.这也可以看作是一个曲线延拓问题.根据三次PH曲线的弧长公式和$G^2$连续条件,最终将问题转化为了一个带有约束的极小值问题,同时我们给出了几个具体例子来说明该方法.     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

曲率变化率最小约束下的五次Hermite插值曲线

本文提出了在曲率变化率最小约束条件下的五次Hermite插值曲线算法,与传统的Hermite插值曲线算法相比,利用该算法获得的插值曲线具有更均匀的曲率分布,曲线更光顺,质量更好。     

基于面积约束的统计直方图曲线拟合

我们提出用分段三次Hermite插值曲线拟合统计直方图的新方法.先根据统计直方图的特点选取Hermite插值曲线在插值点处的导数值和可调整的插值点,然后根据面积约束确定调整值,从而得到拟合曲线.所得拟合曲线与统计直方图有面积相等的约束,并且拟合曲线是C1连续的光滑曲线.所给方法简单、实用.     

C^1有理三次可控保形插值曲线

一、引言给定插值数据点集{(x_i,y_i)}_(i-0)~n,在许多实际应用中(VLSI,CAD/CAM等),要求插值曲线除满足一定的光滑性条件外,还必须反映插值点集的整体几何性质。例如,通常要求单调(凸)数据产生的插值曲线是单调(凸)的。分段三次Hermite插值多项式是外形     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

分段三次保形插值法

1 引言 计算机图形学的一个基本问题就是寻找一条光滑曲线过一组型值点{x_i,y_i}(i=0,1,…n+1),解决这一问题最简单的办法是用分段三次Hermite插值,这种插值构造容易,绘图简单. 分段三次Hermite插值的关键是估计型值点处的导数,只要估计出一组导数值,就对应一个分段三次Hermite插值.但在实际应用中,必须考虑插值曲线对型值点组某些特征的继承性,如曲线的保凸性,保形性等. [1—2]研究了分段三次Hermite插值的保单调性.[3]导出了分段三次Hermite插值保形的一个充要条件,这一条件表明并非任何型值点组都存在保形插值.正因为如此,许多文献采用了不同的方法解决保形插值问题.[4—5]用分段有理三次,但计算量增加较大;[6]     

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.     

Identification of Planar Sextic Pythagorean-Hodograph Curves

Pythagorean-hodograph (PH) curves offer computational advantages in Computer Aided Geometric Design, Computer Aided Design, Computer Graphics, Computer Numerical Control machining and similar applications. In this paper, three methods are utilized to construct the identifications of planar regular sextic PH curves. The first exhibits purely the control polygon legs'' constraints in the complex form. Such reconstruction of a PH sextic can be elaborated by $C^1$ Hermite data and another one condition. The second uses polar representation in two cases. One of them can produce a family of convex sextic PH curves related with a quintic PH curve, and the other one may naturally degenerate a sextic PH curve to a quintic PH curve. In the third identification, we use some odd PH curves to construct a family of sextic PH curves with convexity-preserving property.     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC ¹ Hermite data, we construct a spatial PH curve on a sphere that is aC ¹ Hermite interpolant of the given data as follows: First, we solveC ¹ Hermite interpolation problem for the stereographically projected planar data of the given data in ?³ with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ?³ using the inverse general stereographic projection.     

Pythagorean-hodograph preserving mappings

We study the scaled Pythagorean-hodograph (PH) preserving mappings. These mappings make offset-rational isothermal surfaces and map PH curves to PH curves. We present a method to produce a great number of the scaled PH preserving mappings. For an application of the PH preserving mappings, we solve the Hermite interpolation problem for PH curves in the space.     

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.     

Curve design with rational Pythagorean-hodograph curves

The dual Bézier representation offers a simple and efficient constructive approach to rational curves with rational offsets (rational PH curves). Based on the dual form, we develop geometric algorithms for approximating a given curve with aG ² piecewise rational PH curve. The basic components of the algorithms are an appropriate geometric segmentation andG ² Hermite interpolation. The solution involves rational PH curves of algebraic class 4; these curves and important special cases are studied in detail.