Abstract: To make the extended cubic trigonometric non-uniform B-spline curves possess not only shape adjustability, high order continuity, and exact representation of ellipse, but also variation diminishing, a class of non-uniform cubic trigonometric B-spline basis functions based on totally positivity is constructed. Firstly, we assume that the non-uniform cubic trigonometric B-spline basis functions to be constructed have C2 continuity and partition of unity at each knot, and accordingly the expressions of the basis functions are determined. Then it is proved that the basis functions have total positivity and other important properties. The definition of non-uniform cubic trigonometric B-spline curves are given, and its important properties such as variation diminishing are proved. It is also proved that the curve has C(2n–1) order continuity when taking special parameter values. The example shows that the curve constructed in this paper effectively solves the problems existing in the traditional method and is suitable for geometric design.

Key words: non-uniform B-spline, high order continuity, totally positivity, parameter, variation diminishing