形状可调插值曲线曲面的参数选择

目的 因大多数插值基函数中的参数都是全局参数，从而导致插值曲线曲面的形状无法进行局部调整。另外，当插值曲线曲面形状可调时，也存在如何选择参数才能获得形状较为理想的曲线曲面的问题，为此给出一种无需反求控制顶点、包含局部形状调整参数、具有显式表达式、能重构部分二次曲线曲面的插值曲线曲面构造方法，同时给出易于使用的形状参数确定方案。方法 基于经典3次Hermite插值曲线的Bernstein基函数表达形式，将其中的Bernstein基换成已证明具有全正性的一组三角基函数，根据三角基的端点性质调整曲线表达式以保证其插值性，然后设定插值数据点处的导向量，在其中引入参数，并保证相邻曲线段之间的连续性，得到了一种新的三角基插值曲线。结果 新曲线可以整理成以待插值数据点为控制顶点与一组插值基函数的线性组合形式，插值基表达式简单，插值曲线含一组局部形状调整参数，一个参数的改变只影响一条曲线段的形状，相邻曲线段之间G¹连续，曲线可以重构椭圆。根据不同目标给出了3种用于确定曲线中形状参数的准则，每种准则都提供了可以直接使用的公式。相应的插值曲面具有与插值曲线类似的性质。结论 形状参数选取准则的给出使含参数插值曲线曲面的设计由随意变为确定，这使得采用本文方法更易于得到满意的结果。本文所给插值基函数的构造方法具有一般性，可以采用相同的思路构造其他函数空间上性质类似的插值基。

Parameter selection of shape-adjustable interpolation curve and surface

Objective For the parameters in most of the interpolation basis functions are global parameters, resulting in the shape of the interpolation curves and surfaces cannot be adjusted locally. In addition, when the interpolation curves and surfaces are shape adjustable, we need to consider how to choose the parameters to obtain ideal shape. For this, this paper proposes a new construction method for interpolation curve and surface. This method has the following advantages:it requires no reverse calculation of control points, it contains a local shape parameter, it has explicit expression, and it can reconstruct certain conic sections. We also aim to present a shape parameter selection scheme that can be easily applied. Method The method is based on the expression of the classical cubic Hermite interpolation curve in Bernstein basis form. The Bernstein basis functions are substituted by a set of trigonometric basis functions that are proven to be completely positive in the literature. To ensure interpolation property, the expression of the curve is adjusted according to the endpoint property of the trigonometric basis. The derivate vectors at the interpolation data are assigned, and parameters are incorporated in them. The continuity between the adjacent curve segments is also considered. A new interpolation curve based on trigonometric basis is obtained. Result The new curve can be rearranged as the linear combination of the interpolation data and a set of interpolation basis functions. The interpolation basis has a simple expression. The interpolation curve contains a set of local shape parameters. The change of one parameter can only affect the shape of one curve segment. The adjacent two curve segments are G¹ continuous. The curve can reconstruct an ellipse. According to a different goal, three criteria for the selection of the shape parameter are provided, and each criterion has a formula that can be used directly. The corresponding interpolation surface has a similar property with the interpolation curve. Conclusion The parameter selection scheme transforms the design of the interpolation curve with parameter change from random to determinate. A satisfactory result can be obtained through this method. The construction method of the interpolation basis is general and can be used to construct other basis functions with similar properties.