Abstract: NURBS curves and surfaces can be applied in the lineal and prevalent CAD systems.In this paper,a design of surface pencil through the given non-uniform B-spline curve as geodesic is constructed,and its explicit NURBS's expression and intact algorithm are got.First,using the character of the discrete B-spline and the derivative formula of B-spline,the expression of surface pencil with constraint is presented.Next,using randomicity of the marching-scale functions,the products of three marching-scale functions and the Frenet frames of the given curve have been precisely represented as the B-spline forms.Furthermore,preestablishing a series of weights in the pre-rational B-spline expressions for the marching-scale functions' factors,we improve the occasion and give a group of family-parameters of the surface pencil.It is convenient for adjusting the shapes of the resulting surfaces.Finally,the examples in the experiment show that the algorithm is efficient and applicable.