Curves, length, fractal dimension
type: Item
instance of: scholarly article
title: Curves, length, fractal dimension
zbMATH Open document ID: 0773.28005
author: Claude Tricot
published in: Real Analysis Exchange
publication date: 1993-06-29T00:00:00Z
review text: This paper surveys a book of the same author entitled ``Curves and fractal dimension'' (to appear). The fractal dimension of a curve \(\Gamma\) is defined as \[ \Delta(\Gamma)=\limsup_{\varepsilon\to 0}\left(2-{\log A_ \varepsilon\over\log\varepsilon}\right), \] where \(A_ \varepsilon\) is the area of the \(\varepsilon\)-sausage around the curve \(\Gamma\). The relation of the dimension to the length of inscribed polygonal curves of \(\Gamma\) is discussed. For example he gives examples that the formula \[ \Delta(\Gamma)=\limsup_{\varepsilon\to 0}\left(1+{\log L_ \varepsilon\over|\log\varepsilon|}\right) \] is no longer true in cases of curves which are not self-similar, where \(L_ \varepsilon\) is the length of a polygonal curve whose vertices belong to \(\Gamma\), and such that every steplength (except perhaps the last one) is equal to \(\varepsilon\). He introduces the notion of expansive curves which now allow such a kind of dimension formula. Further topics are velocity and length, the so-called \((\alpha,\beta)\) characterization of curves as well as fractal dimension and local velocity.
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 222111
zbMATH Keywords: uniform deviation curves
zbMATH Keywords: fractal dimension
zbMATH Keywords: expansive curves
zbMATH Keywords: velocity
zbMATH Keywords: length
reviewed by: Hermann Haase
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 222111