A new fractal dimension for curves based on fractal structures

Abstract

In this paper, we introduce a new theoretical model to calculate the fractal dimension especially appropriate for curves. This is based on the novel concept of induced fractal structure on the image set of any curve. Some theoretical properties of this new definition of fractal dimension are provided as well as a result which allows to construct space-filling curves. We explore and analyze the behavior of this new fractal dimension compared to classical models for fractal dimension, namely, both the Hausdorff dimension and the box-counting dimension. This analytical study is illustrated through some examples of space-filling curves, including the classical Hilbert's curve. Finally, we contribute some results linking this fractal dimension approach with the self-similarity exponent for random processes.