A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics

Special Relativity is traditionally formulated using hyperbolic spacetime geometry, where coordinate velocity is bounded by c and Lorentz boosts correspond to hyperbolic rotations parameterized by rapidity φ. While mathematically robust, the non-compact nature of hyperbolic geometry obscures the conservation laws governing coordinate motion and proper time progression. In this paper, we demonstrate that the standard Minkowski line element can be mapped isomorphically onto a compact Euclidean circle via the velocity relation: v² + 𝕋² = c² where 𝕋 ≡ c(dτ/dt) defines the chronokinetic velocity—the coordinate rate of proper time progression. By formalizing the Gudermannian bridge (θ = gd(φ)), we show that the kinematic pitch angle θ = arcsin(v/c) unifies: The Lorentz factor: γ = sec θ Relativistic momentum: p = mc tan θ Total energy: E = mc² sec θ Relativistic dispersion: (pc)² + (mc²)² = E² Furthermore, the relativistic longitudinal optical Doppler factor is shown to emerge as a direct stereographic half-angle projection: D = tan(π/4 + θ/2) This reformulation preserves Lorentz invariance, eliminates asymptotic divergences as v → c through a compact angular domain, and establishes a direct bridge between classical mechanics, relativistic dynamics, and quantum de Broglie clock frequencies.