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A 4-point interpolatory subdivision scheme for curve design. (English) Zbl 0638.65009
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 $$>1$$.
Reviewer: J.Krč-Jediný

##### MSC:
 65D05 Numerical interpolation 53A04 Curves in Euclidean and related spaces
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##### References:
 [1] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer aided geometric design, 1, 1-60, (1984) · Zbl 0604.65005 [2] Dyn, N.; Levin, D., Smooth interpolation by bisection algorithms, (), 335-337 [3] Micchelli, C.A.; Prautzsch, H., Uniform refinement of curves, (1987), to appear · Zbl 0651.65003 [4] Weissman, A., (), to appear
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