Many studies have been conducted on the extension of the B-spline curve with parameters. These extended curves have similar local shape controlability as the B-spline curve and have shape adjustability that is independent of the control points. However

previous studies on this topic used global parameters

thus preventing the shape of the curves to be locally adjusted. The blending functions in several studies do not have total positivity

thus removing the variation-diminishing properties and convex-preserving properties of the curves. The purpose of this paper is to construct a curve with convexity-preserving property

local shape adjustment

and local shape control. By using the theoretical framework of quasi-extended function space

we first prove that the existing extended basis of the cubic Bézier curve

called -Bernstein basis

is exactly the normalized B-basis of the corresponding space. Thereafter

we use the linear combination of the -Bernstein basis to express the extended basis of the cubic uniform B-spline curve. According to the preset properties of the curve

we deduce the properties of the extended basis and then determine the coefficients of the linear combination and the expression of the basis. The extended basis can be represented as the product of the -Bernstein basis and a conversion matrix. We prove the totally positive property of the matrix. By using this basis

we define a piecewise curve with one local shape parameter with the same structure as the cubic B-spline curve. The total positivity of the conversion matrix determines the total positivity of the extended basis. The total positivity of the basis determines that the extended curve has a variation-diminishing property and convex-preserving property. The locality of the shape parameter determines that the shape of the curve can be adjusted locally. The piecewise structure indicates that the curve has local shape control ability. The method for constructing the extended B-spline basis with total positivity has generality. Compared with most extended curves in literature

the curve given in this paper has variation diminishing property and convex-preserving property; thus

providing an efficient method for conformal design.