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

We study a geometric polygonal discretization of the area functional\[A(γ) = 1/2 ∫_a^b (x,dy - y,dx)\]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 C¹-curves and prove second-order accuracy under C² regularity. For C³-curves, we derive a complete asymptotic expansion of the discretization error and explicitly identify the leading term\[A(γ) - S_n(γ) = h^2/12 ∫_a^b (γ'(t), γ''(t)),dt + O(h^3),\]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 π. 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.