A Piecewise Exact Anchor Model for the Perimeter Approximation of Superellipses

Calculating the exact perimeter of a superellipse defined by x/aⁿ + y/bⁿ = 1 for a general degree n ≥ 1 remains a classic challenge in analytical geometry, as the arc length integral cannot be expressed in terms of elementary functions. Existing single-term algebraic approximations fail to maintain exact accuracy across all key geometric boundary points simultaneously. In this paper, we present the Piecewise Exact Anchor Model (PEAM), a hybrid mathematical framework that locks classical analytical formulas for exact boundary conditions (Rhombus at $n = 1$, Ellipse at $n = 2$, and Square at n → ∞) while employing a high-order power-decay algebraic fitting function for higher-order squircles (n ≥ 3). Numerical verification demonstrates that PEAM limits maximum relative errors below $0.25%$ for all squircle configurations while guaranteeing $0.00%$ exact matches at all fundamental geometric limits.