A generalized curve subdivision scheme of arbitrary order with a tension parameter (Q631528)

The authors begin by defining generalized B-spline basis functions. For order two, these are translates \(N_{n,2}(l,t)\doteq N_{0,2}(l,t-nl)\), where \(N_{0,2}\) has support in \([0,2l)\) and is defined piecewise by \(N_{0,2}(l,t)\doteq\alpha\sin{\omega t}\) for \(t\in[0,l)\), and by its reflection on \([l,2l)\), with \(\alpha\) a normalization factor. Higher order basis functions are obtained by smoothing: \(N_{n,k}(l,t)\doteq\frac{1}{l}\int_{t-l}^tN_{n,k-1}(l,s)\,ds\). Such functions give a generalization of uniform, algebraic-trigonometric, and algebraic-hyperbolic B-splines; indeed, the latter are recovered by taking \(\omega=0,1,i\), respectively. The authors then obtain a (non-stationary) subdivision scheme for generalized B-splines for a given control polygon, and show that, for special values of the tension parameter \(u\doteq\cos(\omega l/2)\), this scheme reproduces other well-known schemes. Continuity of order \(k-2\) is demonstrated for the limit curve of the subdivision scheme in the case \(u\geq 0\) by comparing with an asymptotically equivalent stationary subdivision scheme. The remainder of the article is devoted to expressing some common analytic curves, such as Lissajous curves, in terms of generalized B-splines. The article is suitable for a general audience familiar with splines and subdivisions schemes. The article is organized, although the notation is inconsistent and confusing in one or two places.