On the Arc Length of a Supercircle and a Hypergeometric Formulation of $\pi$

Abstract

We obtain an infinite-series representation for the arc length of a supercircle in terms of the scale parameter $a$ and the shape parameter $n$. The resulting expression is constructed by means of generalized binomial coefficients and Gauss hypergeometric functions, distinguishing two regimes associated with the value of $n$. We also analyze the absolute convergence of the resulting series. We verify the consistency of the formulation from limiting cases and particular configurations of the family of supercircles: when $n\to0^+$ and $n\to\infty$, the length converges to the value $8a$, corresponding to the limiting rectilinear geometries, whereas for $n=1$ we recover the perimeter of the rhombus with diagonals of length $2a$. In addition, as a validation against supercircles with exact arc length, the formulation reproduces with high numerical precision the arc length of the parabolic star, the astroid, and the circle. Finally, by specializing the circular case $n=2$ and normalizing the length by the diameter $2a$, we obtain a series representation, in terms of hypergeometric functions, for the constant $\pi$.