On continuity conditions for C-B-spline curves and surfaces

C-B-spline curves are an extension of cubic B-spline curves. In addition to having many properties that B-spline curves have, they have the adjustable shape parameter, and can represent arcs and surfaces more precisely yet require less memory space in computer. As, to our knowledge, there does not exist any paper in the open literature on the continuity condition of C-B-spline curves and surfaces, we now present such a paper. In this paper, we first present condition of C-B-spline curves based on the analysis of C-B-spline basis functions and terminal properties. Then, the geometric model of C-B-spline surfaces is built and condition for C-B-spline surfaces in u and v directions is derived and simplified by choosing the control parameters properly. The result has its specific geometric significance. As a case in point, condition for C-B-spline surfaces is implemented by assigning check points along their common boundary curve and modifying the adjacent surface to meet the requirements of sharing common tangent plane. Finally, from what has been discussed above, we describe in detail how to construct surfaces of revolution, ellipsoids and sweeping surfaces. Moreover, due to the similar structures between C-B-splines and B-splines, the continuity condition of C-B-spline curves and surfaces can be introduced to the CAD/ CAM modeling system.