Curves, length, fractal dimension
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.
scientific article; zbMATH DE number 222111