摘要：
目的 整体曲线包括传统有限闭区间(比如[0,α])上的内部段和该区间外的延拓段.在计算机辅助设计(CAD)中,构造整体曲线常用分段表示,存在冗余数据——为了减少冗余,需知道各分段间的关系,并判断它们是否在同一整体曲线上.由此,本文研究当整体C-Bézier曲线原参数域[0,α]在(-∞,+∞)上缩放变动时,曲线的控制顶点的变化情况.方法 通过基函数的递推比较,寻找运动前后控制顶点之间的关系.首先考虑特殊细分情下线性插值.因插值后生成的NUAT-B样条基分段且具有支撑区间,它无法适应整体情况.因此用其与t轴间的区域面积取代它;接着进一步讨论了一般情况下沿整体C-Bézier运动的线性插值.由于C-Bézier参数区间长度要小于π,特殊细分情况下线性插值不能直接推广.不过虽然参数区间在变化,整体曲线上每点位置却不变.针对这点,使用两次递归,寻求得到以线性插值形式沿整体C-Bézier曲线运动的结果.结果 只要保持参数区间的长度在(0,π)上,运动的曲线都可以写成传统的C-Bézier内部段的形式,且控制顶点可以表示为原始控制顶点直接的线性组合,或者逐步地线性插值(包括内插和外插)的形式.结论 考虑整体曲线及沿整体曲线的运动,可以改变C-Bézier曲线的造型区间,减少造型过程中的冗余数据.不过,C-Bézier基由递归积分定义,其运动过程较慢.所以今后可以考虑加速运动的方法,也可以考虑其他类型的拟-Bézier曲线.%Objective The parameter of a conventional C-Bézier curve is often limited in a closed interval.In this study,we focus on an integral one made up of the traditional inner segment in a finite closed interval (such as [0,α]) and a part out of the interval.However,in computer-aided design,the modeling of an integral curve is often expressed as different stages and results in redundant data.In fact,when modeling an entire curve,the control points of different segments may have relations with one another.Therefore,if the control points of one segment and some shape parameters are stored,then the entire curve may be obtained,and the curve data may be decreased.We need to determine the relations among different segments and judge whether they are on the same integral curve to decrease the final redundancy.We raise two questions:1) Given an inner curve,can any segment of its integral curve be presented as an inner form? and 2) Are two neighboring inner C-Bézier curves on the same integral curve? We select a C-Bézier curve for our research.The focus of this study is to consider the changes in control vertices for the C-Bézier curve when the original parametric region [0,α] is scaled on (-∞,+ ∞).Method Any C-Bézier curve is divided into two arcs from geometric point of view:a center Bézier curve and a trigonometric part.On the basis of their movements,any segment of an integral C-Bézier curve can be represented as an inner form.We can analyze relations of control vertices from algebra perspective and give three forms (direct,subdivision,and linear interpolation forms) between newly produced control points in the movement and old ones.First,we represent certain segments of the integral C-Bézier curve as an inner form,consider basis functions recursively,and compare them to obtain the direct form of original control points.Second,one endpoint of the moving segment is considered,which relates to a subdivision scheme.The scheme subdivides the inner curve into two neighboring C-Bézier segments.Similar to the direct form,expressions can be easily worked out by using recursive evaluation.Third,we consider a corner-cutting form under special subdivision situation to identify linear interpolation from easy to difficult.The corner cutting is an alternative of the direct form,and the corner-cutting form can be obtained by the knot-inserting process.However,the NUATB-spline generated after interpolation cannot adapt to the integral case because it is piecewise and is zero out of the interval.We use the area between a corner-cutting scheme and t-axis to extend the scheme.Subdivision with a corner-cutting form is obtained on the basis of recursion and the relations between a basis and the subdivision scheme.The linear interpolation form is considered to move along an integral C-Bézier curve for general case.The length of the parameter interval of an inner C-Bézier curve needs to be less than π;thus,the corner-cutting form under special subdivision situation cannot be directly extended.Although the parameter interval of the C-Bézier curve changes,the position of each point on the integral curve never changes.We utilize an evaluation scheme to solve the extension problem.Consistent with Bézier curve,results of the movement along an integral C-Bézier curve with a linear interpolation form are obtained by using recursive evaluation twice.Finally,we establish an algorithm to judge if two given inner C-Bézier curves are on the same integral curve by considering that integral curve can be used to reduce redundant data.The error of the C-Bézier curve can be limited by the error of its control points;hence,we use an error item to control the judgment accuracy after calculating control points by direct form.Result This study focuses on C-Bézier curves and regards the traditional inner part and the extended part out of an interval as integrals.An inner C-Bézier curve can be moved along the integral curve while its parameter integral length is less than π,and motion curves can be represented as an inner C-Bézier form when its parameter interval length is in (0,π).New control points can be obtained by a direct linear combination or stepwise linear interpolation (including traditional interpolation and extrapolation) form of the old ones.A subdivision scheme,including direct and corner-cutting forms,of the inner C-Bézier curve is included as a subcase.The integral curve and the movement along it may be considered to reduce redundant storage data.Conclusion The applications are as follows:First,the movement along an integral C-Bézier can be used to scale the parameter interval of a given C-Bézier curve.Second,integral curve can be considered to reduce redundancy by focusing on the part and extending the parameter interval.Third,two neighboring C-Bézier curves are judged on whether they are on the same integral curve under permissible error.If they are on the same curve,then data of one curve may be reduced while storing,whereas data of the other one can be saved.However,the movement process is slow because of the recursive integral definition of the C-Bézier basis.In the future,we may consider the acceleration of the movement method or other types of Bézier-like curves.
