Universal Helical Geometry: Spherical Coordinates, Earth
Rotation, and DNA Topology
This paper derives the complete transformation of a standard helix r(t) = (R cost, 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.
