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Velocity, length, dimension. (English) Zbl 0942.28010

Fisher, Yuval (ed.), Fractal image encoding and analysis. Proceedings of the NATO ASI, Trondheim, Norway, July 8-17, 1995. Berlin: Springer. NATO ASI Ser., Ser. F, Comput. Syst. Sci. 159, 243-259 (1998).
The author studies the fractal approach to the length and dimension of a curve. The fractal approach consists of defining a length at precision \(\varepsilon\), for all \(\varepsilon> 0\). In some sense, the curve is approximated, more and more closely, by a sequence of finite-length curves. There are different ways to create such sequences. Some are purely geometrical, others rely on analysis. In latter case, the curve must be parametrized, and an \(\varepsilon\)-length is defined with help of sums or integrals. Being given a parametrization of the curve and a covering of the defining interval, a general notion of length is given and studied in the paper with two applications, one with arcs of equal diameter (compass method) and the other with arcs of equal breadth (the method of constant breadth). The relation between \(\varepsilon\)-lengths and the fractal dimension as well as the relationships among the efficient \(\varepsilon\)-lengths are then investigated in the paper. As particular cases, the \(\varepsilon\)-lengths and fractal dimension of several typical classes of curves such as spirals, self-similarity curves, self-affine graphs, and graphs of continuous functions are studied as well.
For the entire collection see [Zbl 0910.00050].

MSC:

28A80 Fractals
28A78 Hausdorff and packing measures
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