Second-order polygonal discretization of the area functional and explicit error asymptotics

Abstract

We study a geometric polygonal discretization of the area functional
\[
A(\gamma) = \frac{1}{2} \int_a^b (x,dy - y,dx)
\]
for planar parametrized curves. The method is based on a chordal (shoelace-type) construction and depends only on endpoint values of the parametrization.

We establish convergence for $C^1$-curves and prove second-order accuracy under $C^2$ regularity. For $C^3$-curves, we derive a complete asymptotic expansion of the discretization error and explicitly identify the leading term
\[
A(\gamma) - S_n(\gamma) = \frac{h^2}{12} \int_a^b \det(\gamma'(t), \gamma''(t)),dt + O(h^3),
\]
revealing a direct dependence on curvature.
The analysis highlights a nontrivial cancellation mechanism inherent in the geometric structure of the method, distinguishing it from classical quadrature rules applied to the corresponding integrand.

As an application, we consider a rational parametrization of the unit circle and obtain an explicit asymptotic formula for the associated rational approximation of $\pi$. Numerical experiments confirm the theoretical convergence rate and demonstrate the efficiency of Richardson extrapolation.
The results establish a precise connection between geometric discretizations and classical numerical integration theory.