A 4-point interpolatory subdivision scheme for curve design
The following interpolatory subdivision scheme is analyzed: Given control points \(\{p_ i\in {\mathfrak R}^ d\); \(i=-2,...,n+2\}\), intermediate points are added by the scheme \(p_ i'=(1/2+w)(p_ i+p_{i+1})- w(p_{i-1}+p_{i+2}),\) for -1\(\leq i\leq n\). Iterating this scheme infinitely many times, typically an infinite set of points is obtained. Properties of this set depend on the value of the parameter w. It is shown that for \(| w| <1/4\) the set forms a continuous curve, and for \(0<w<1/8\) the curve is \(C^ 1\). The role of the parameter w is demonstrated by a few examples. It is mentioned, that for some values of w outside the range \(| w| <1/4\) the limiting curve seems to be continuous, but it has many loops and sharp bends and probably can be of Hausdorff dimension \(>1\).
scientific article; zbMATH DE number 4041139