C3 cubic trigonometric B-spline curves with a real parameter

This paper presents a class of cubic trigonometric B-spline curves with a real parameter α, which are called α-B-spline curves. The α-B-spline curves have properties similar to those of the standard cubic uniform B-spline curves. In particular, the α-B-spline curves are C3 and can achieve shape adjustment by altering the value of the parameter α even if the control points are kept unchanged. With proper conditions, the α-B-spline curves also can exactly represent arcs of ellipses and parabolas. In order to obtain smooth α-B-spline curves, a method for determining the value of parameter α is presented.