Stereographic Projection Unifies Rational Circle Parametrization and Hyperbolic Half-Angle Substitution — E8 Intelligence Research

FINDING: Stereographic projection unifies rational parametrization of the circle (Pythagorean triples) with the hyperbolic tangent half-angle substitution, revealing that Euclidean and hyperbolic trigonometry share a single projective mechanism. MATH: - Circle parametrization: (x, y) = ((1−t²)/(1+t²), 2t/(1+t²)), t = tan(θ/2) ∈ ℚ → rational points on x²+y²=1 - Pythagorean triple generation: for t = p/q, (x,y) = ((q²−p²)/(q²+p²), 2pq/(q²+p²)) → (a,b,c) = (q²−p², 2pq, q²+p²) - Hyperbolic analogue: cosh u = (1+t²)/(1−t²), sinh u = 2t/(1−t²), t = tanh(u/2) - Key identity: tanh(u/2) = sinh u/(1+cosh u) = (cosh u − 1)/sinh u - Area interpretation: for unit hyperbola, u = 2·(sector area); the hyperbolic angle equals twice the area swept - Poincaré disk: stereographic projection of hyperboloid x²+y²−z²=−1 from (0,0,−1) onto z=0 yields ds² = 4(dx²+dy²)/(1−r²)² - Cross-ratio invariance: stereographic projection preserves angles (conformal), maps circles↔circles CONNECTION: - The golden ratio φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com