In this paper, CB-splines replace the polynomial basis functions {t3, t2, t, 1} for cubic B-spline segments with the circular basis functions {sin t, cos t, t, 1}, and HB-splines use the hyperbolic basis functions {sinh t, cosh t, t, 1}. Both of these spline classes are incomplete, in the sense that each is located only on one side of the B-splines. The curvature of a CB-spline at each breakpoint cannot be larger than that of the B-spline with the same control points, whereas an HB-spline cannot have smaller curvature. These two classes are unified here to functional B-splines (FB-splines) by use of a shape parameter Ci at the i-th control point, which can take any value in the range 0 <= Ci < ∞. The circular basis is used when 0 <= Ci <= 1, and the hyperbolic basis is used when 1 < Ci < ∞. (Note that, as Ci → 1, we get a cubic B-spline segment.)
It is stated (mostly without proof) that FB-splines inherit most of the properties of B-splines: C2 continuity, geometric invariance, the convex hull property, the local approximation scheme, and the variation diminishing property. Furthermore, they can exactly represent elliptic or hyperbolic arcs. However, unless the FB-spline is of uniform shape (Ci = C, for all i), there is no subdivision equation. This shortcoming is rectified by the introduction of subdivision B-splines (SB-splines), which are defined by the obvious generalization of the subdivision equation for uniform-parameter FB-splines. The similarities and differences between FB- and SB-splines are discussed (again, mostly without proof). The only actual proof in this paper establishes the C2 continuity of SB-splines.
The results here evidently assume planar breakpoints, but this is never explicitly stated. A final section discusses the use of FB- and SB-splines to construct shape-adjustable surfaces of revolution and tensor product surfaces. A few examples of the latter are displayed. The authors conjecture that these surfaces should have C2 continuity, but this has not been proven in general.
Although this paper describes potentially useful new tools for curve and surface design, it suffers from having too many unstated assumptions and unproven assertions. The language is a bit awkward at points as well, reflecting the fact that English is not the native language for either author.
