Universal Helical Geometry: Spherical Coordinates, Earth Rotation, and DNA Topology

This paper derives the complete transformation of a standard helix r(t) = (R cos t, R sin t, ct) from Cartesian to spherical coordinates (r, θ, φ), providing explicit formulas for r(t), θ(t), φ(t), and their derivatives.We establish the constant angular velocity θ ′ (t) = 1 through rigorous calculation using the two-argument arctangent and Cartesian velocity components.The analysis reveals the helix's uniform rotation about the z-axis with monotonic polar angle evolution, connecting to Cuculescu's [1] geometric constructions of helical motion on developable surfaces.