Scientists have produced the first analytical formula that calculates how arrays of elliptical metal patches respond to electromagnetic waves. The model uses a Galerkin technique that works directly on the shape of each patch, without needing a fine mesh, and an affine transformation that turns any rotated ellipse into a unit circle. This single mathematical recipe therefore covers every orientation and aspect ratio of the ellipse. The calculation is extremely fast – evaluating the full‑band spectrum takes only a few tens of milliseconds on a standard workstation, about 1,000 times faster than conventional full‑wave solvers. Its predictions agree with rigorous method‑of‑moments simulations up to the fundamental resonance and correctly capture substrate‑guided mode anomalies. For 200 eigenvalues spanning five aspect ratios, five gap values and four common substrate designs, the model matches the reference within 1 % for 92 % of the values and within 0.5 % for nearly 76 %. The few outliers – the small‑eigenvalue modes when the ellipse is very stretched and the gap is tiny – are mostly due to the benchmark solver’s own convergence limits. The formulation applies only to periodic arrays on grounded dielectric slabs and to elliptical patches; it cannot directly handle isolated or non‑elliptical shapes, or predict propagating surface waves beyond the first Brillouin zone.