A 4-point interpolatory subdivision scheme for curve design
type: Item
instance of: scholarly article
title: A 4-point interpolatory subdivision scheme for curve design
zbMATH Open document ID: 0638.65009
DOI: 10.1016/0167-8396(87)90001-X
author: B. George
published in: Computer Aided Geometric Design
publication date: 1987-01-01T00:00:00Z
review text: 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 \(>1\).
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 4041139
zbMATH Keywords: computer-aided curve and surface design
zbMATH Keywords: curves
zbMATH Keywords: surfaces
zbMATH Keywords: point insertion
zbMATH Keywords: interpolatory subdivision scheme
MaRDI profile type: Publication
cites work: A survey of curve and surface methods in CAGD
cites work: Q3752837
cites work: Q3796614
full work available at URL: 0167-8396(87)90001-x
OpenAlex ID: W2075492994
description: scientific article; zbMATH DE number 4041139