带三参数的三次B样条扩展曲线及其应用

Cubic B-Spline Extension Curve with Three Parameters and Its Application

摘要: 为满足曲线自由设计的需求，本文对λ-B样条基和三次T-B样条基进行扩展，构造了λαβ-T-B样条基函数，定义了由该基函数生成的λαβ-T-B样条曲线，其中含有三个参数λ、α、β，构造的曲线不仅具备样条曲线的特性，还同时具备三角和代数多项式的优良性质，且具有形状可调性和更精确的逼近性；在给定条件下，形状参数有明确的几何意义，λ、β取值越大，α值越小，越能更好地逼近控制多边形；调节参数的取值，改变曲线的形状；选则恰当的控制顶点，根据参数对曲线的调节规律，确定恰当的形状参数，生成的λαβ-T-B样条曲线能较好贴近圆和椭圆；介绍了生成插值于给定数据点的λαβ-T-B样条曲线的构造方法，并给出具体数例的计算，验证了方法的有效性和可行性。

Abstract: In order to meet the requirement of free curve design, this paper extends the λ-B-spline basis and cubic T-B-spline basis and constructs a group of λαβ-T-B-spline basis function. The λαβ-T-B-spline curve generated by the base function is defined, which contains three parameters λ , α, and β, has the characteristics of spline basis function, but also has the good properties of trigonometric and algebraic polynomial basis, and has the shape tunability and more accurate approximation; under given conditions, shape parameters have clear geometric meaning. The larger the value λ of and β, the smaller the value of , the better the approximation control polygon can be. By changing the value of shape parameters, the curve can be adjusted; by selecting the appropriate control vertex and the appropriate shape parameters, the generated λαβ-T-B-spline curve can approach the circle and ellipse very well; In the end, the method of constructing λαβ-T-B-spline curve interpolating in the data points is introduced, and the calculation of specific numerical examples is given, reflect the effectiveness and feasibility.