Method for evaluating the fractal dimension of curves using convex hulls

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Abstract
A local analysis for simple curves is performed using the geometric notions of size and deviation of an arc. For typical parametrized curves, the rate of decrease of these averaged measurements is related to the value of the fractal dimension. A method for calculating this dimension is derived, generalizing the variation method used for graphs of continuous functions to a large family of curves. Our method, the constant-deviation variable-step method, is tested on various types of curves (fractal Brownian motion, Gosper curve, and geographical coastline). A comparison is made with other methods such as the divider-step method, variation method, and diameter method. We show that our method offers the same level of performance while exhibiting much more general applications.

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