Abstract
In this article we present a new formula for the length of a closed curve involving a double integral of a certain potential function. This formula is based on the construction of pedal curves, and it turns out that the integral taken over the interior of a pedal curve does not depend on the choice of a pedal point. First, using the notion of support function we derive the expression for the perimeter of oval. Next, we extend our formula to arbitrary smooth simple plane closed curves of the class C^2, but to this aim we needed to use an appropriate notion of an interior of pedal curve. Since the pedal curve of a smooth simple plane closed curve can be fairly complicated and have self-intersections and overlapping parts, we introduce and essentially use a notion of its interior based on the winding numbers and orientations of certain closed parts of the pedal curve forming its partition. At the end of the paper we propose a number of topics for further considerations, also conjecturing that our formula holds for any plane curve having its pedal curve with respect to some point.
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