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 ≤ C < ∝). 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 C2 continuity, and can represent elliptic and hyperbolic arcs exactly. They are adjustable, and each control point bi can have its unique shape parameter Ci. As Ci increases from 0 to ∝, the corresponding breakpoint of bi on the curve is moved to the location of bi, and the curvature of this breakpoint is increased from 0 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 C2 continuity for the general SB-spline surfaces has not been obtained yet.