For the parameters in most of the interpolation basis functions are global parameters

resulting in the shape of the interpolation curves and surfaces cannot be adjusted locally. In addition

when the interpolation curves and surfaces are shape adjustable

we need to consider how to choose the parameters to obtain ideal shape. For this

this paper proposes a new construction method for interpolation curve and surface. This method has the following advantages:it requires no reverse calculation of control points

it contains a local shape parameter

it has explicit expression

and it can reconstruct certain conic sections. We also aim to present a shape parameter selection scheme that can be easily applied. The method is based on the expression of the classical cubic Hermite interpolation curve in Bernstein basis form. The Bernstein basis functions are substituted by a set of trigonometric basis functions that are proven to be completely positive in the literature. To ensure interpolation property

the expression of the curve is adjusted according to the endpoint property of the trigonometric basis. The derivate vectors at the interpolation data are assigned

and parameters are incorporated in them. The continuity between the adjacent curve segments is also considered. A new interpolation curve based on trigonometric basis is obtained. The new curve can be rearranged as the linear combination of the interpolation data and a set of interpolation basis functions. The interpolation basis has a simple expression. The interpolation curve contains a set of local shape parameters. The change of one parameter can only affect the shape of one curve segment. The adjacent two curve segments are G continuous. The curve can reconstruct an ellipse. According to a different goal

three criteria for the selection of the shape parameter are provided

and each criterion has a formula that can be used directly. The corresponding interpolation surface has a similar property with the interpolation curve. The parameter selection scheme transforms the design of the interpolation curve with parameter change from random to determinate. A satisfactory result can be obtained through this method. The construction method of the interpolation basis is general and can be used to construct other basis functions with similar properties.