Elliptic Integrals: From the Geometry of φ to the AGM Algorithm

This paper presents a comparative analysis between two fundamentally different approaches to computing complete elliptic integrals of the first kind K(k): 1. The classical AGM (Arithmetic-Geometric Mean) method - a fast iterative algorithm with quadratic convergence.2. The Sapri Aurea formula - a direct geometric approximation based on the golden ratio φ = (1+√5)/2 ≈ 1.618, proposed by the author. The Sapri Aurea formula requires only 7 operations (compared to 20-30 for AGM) and achieves:- 0.00004% precision for small curvature (k < 0.1) with δ = 0- 0.1-0.3% precision for k < 0.8 with δ = 0.23- ~1% precision for k < 0.95 with calibrated δ The paper includes:- Complete mathematical derivation- Comparative analysis with AGM- Empirical δ(k) values- Geometric sampling theory based on φ- JavaScript implementation of all formulas- Precision scaling from real-time graphics to theoretical physics The work reveals a deep geometric connection between elliptic integrals and the golden ratio, suggesting φ as a "natural curvature optimizer" that appears consistently in problems involving ellipses, optics, wave dynamics, and perspective projection.