给出了含有参数λ的(n+1)次多项式基函数,其是n次Bernste in基函数的扩展;分析了这组基的性质,基于该组基定义了带有形状参数的(n+1)次多项式曲线。曲线不仅具有n次Bézier曲线的特性:如端点插值、端边相切、凸包性、变差缩减性、保凸性等,而且具有形状的可调性:在控制顶点不变的情况下,随着参数不同,可产生不同逼近控制多边形的曲线。当λ=0时,曲线可退化为n次Bézier曲线。运用张量积方法,可生成形状可调的曲面,曲面具有曲线类似的性质。应用实例表明,本文定义的曲线应用于曲线/曲面的设计十分有效。

Through heightening the degree of polynomial function, a class of polynomial function of(n+1)degree that containing an adjustable constantparameterλis presented in this paper. They are an extension ofndegree Bernstein basis functions. Properties of this new basis are analyzed, which have symmetry, linear independence, weighting property and nonnegative property when the parameter λis between -2 and 1, based onwhich a(n+1)degree polynomial curvewith a shape parameter λis defined. The curve, to be called λ Bézier curve not only inherits the most properties of n degree Bézier curve, such as endpoints’properties, symmetry, convex hull property, geometric invariability, affine invariance, convex preserving property, variation diminishing property and so on, but also can be adjusted in shape by changing the value of λ without changement of control points. Whenλ=0, the curve degenerates ton2degree BézierCurve. Using tensor product approach, a surfacewith parameterλis constructed, whose properties are similar to the curve’s. At last, examples illustrate themethod of constructing curve is very useful for curve/surface design.