Extending cubic uniform B-splines by unified trigonometric and hyperbolic basis

In this paper, the trigonometric basis {sin t, cos t, t, 1} and the hyperbolic basis {sinh t, cosh t, t, 1} are unified by a shape parameter C (0 less than or equal to C < infinity). It yields the Functional B-splines (FB-splines) and its corresponding Subdivision B-splines (SB-splines). As well, a geometric proof of curvature continuity for SB-splines is provided. FB-splines and SB-splines inherited nearly all properties of B-splines, including the C continuity, and can represent elliptic and hyperbolic arcs exactly. They are adjustable, and each control point b(i) can have its unique shape parameter C-i. As C-i increases from 0 to proportional to, the corresponding breakpoint of bi on the curve is moved to the location of b(i), and the curvature of this breakpoint is increased from 0 to proportional to too. For a set of control points and their shape parameters, SB-spline and FB-spline have the same position, tangent, and curvature on each breakpoint. If two adjacent control points in the set have identical parameters, their SB-spline and FB-spline segments overlap. However, in general cases, FB-splines have no simple subdivision equation, and SB-splines have no common evaluation function. Furthermore, FB-splines and SB-splines can generate shape adjustable surfaces. They can represent the quadric surfaces precisely for engineering applications. However, the exact proof of C-2 continuity for the general SB-spline surfaces has not been obtained yet. (C) 2004 Elsevier Inc. All rights reserved.