Two different forms of C-B-splines

In a recent paper (Zhang, 1996), C-B-splines are introduced as extensions of cubic uniform B-splines. A new reparametrized form of C-B-splines, which is defined on the interval [0,1], is proposed here. From this form, a third form that could have different parameters alpha in a curve is derived. These new forms give an efficient algorithm for C-B-splines with any parameter alpha (0 less than or equal to alpha less than or equal to pi), an easy method for tolerance control for subdivision, and a simple way for connecting C-B-spline curves or surfaces with different alpha's. Also this paper explains that the C-B-splines have V-D properties and any C-B-spline can be approximated by its uniform B-spline in high accuracy with an easy error control. So, in the representations of curves and surfaces, C-B-splines can get not only high precision for engineering, but also fast calculation speed for computer display.