Bézier Curve with Shape Parameter

Through heightening the degree of polynomial function,a class of polynomial function of(n+1) degree that containing an adjustable constant parameter λ is presented in this paper.They are an extension of n degree Bernstein basis functions.Properties of this new basis are analyzed,which have symmetry,linear independence,weighting property and nonnegative property when the parameter λ is between-2 and 1,based on which a(n+1) degree polynomial curve with a shape parameter λ is defined.The curve,to be called λ-Bezier curve not only inherits the most properties of n-degree Bezier 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.When λ=0,the curve degenerates to n-degree Bezier Curve.Using tensor product approach,a surface with parameter λ is constructed,whose properties are similar to the curve's.At last,examples illustrate the method of constructing curve is very useful for curve/surface design.