Trigonometric extension of cubic B-spline curves

A kind of cubic algebraic trigonometric B-spline base functions with a shape control parameter is presented, and the corresponding curves are defined by the introduced base functions. The curves inherit some properties with traditional cubic B-spline curves and can interpolate directly some control points without solving system of equations or inserting some additional control points. The curves can be used to represent exactly straight line segments, circular arcs, elliptic arcs, parabola and some transcendental curves such as circular helix. The shape of the curves can be adjusted globally through changing the parameters. For shape modeling freely, the method of localization with respect to parameters is proposed, which makes the resulted curves can be locally controlled. In addition, the curves are C² continuous in proper condition. These ideas can be also extended to produce the corresponding tensor product surfaces.