Bézier curves and surfaces with simple G~2 conditions

A group of basis function consisting of three quartic polynomials with parameter is given.Then based on it,a set of basis function which is composed of n+1(n≥3)algebraic trigonometric blending functions with parameter,to be calledλ-Bernstein bases of order n,is defined recursively.They have many basic properties of Bernstein basis functions,such as non-negativity,normalization,symmetry,etc.Theλ-Bezier curves defined by them possess the basic properties of Bezier curve.Furthermore,the new curves have two distinct advantages.Firstly,their shape can be adjusted freely without changing the control points.Secondly,when the control points of two adjacentλ-Bezier curves satisfy the G1 continuity conditions of Bezier curve,they can achieve G2 continuity.By using the tensor product method,λ-Bezier surfaces can be defined,which also show many good properties.