Abstract: This paper aims to construct a shape-preserving extended cubic uniform B-spline curve. Firstly, within the theoretical framework of quasi extended Chebyshev space, we prove that the existing extended basis of the cubic Bézier curve, λ-Bézier basis for short, is the normalized B-basis of the corresponding space. Then we use the linear combination of the λ-Bézier 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. The extended basis can be represented as the product of the λ-Bézier basis and a conversion matrix. We prove the totally positive property of the matrix and the extended basis. By using this basis, we define a curve based on 3-point piecewise scheme and analyze its properties. The totally positive property makes the curve can simulate the shape of the control polygon. The surface based on 16-point piecewise scheme is briefly introduced.

Key words: curve design, shape preserving property, totally positive basis, shape parameter