Arc Geometry I: Differential and Topological Structure of Arc Helices

We introduce a class of space curves called Arc Helices, generated by a two-frequency parametrization combining circular motion with periodic arc modulation. Unlike classical helices, which cannot adequately describe structures with internal systemic tension or non-spacetime topological mappings, the double-frequency parametrization of Arc Helices provides a fundamental geometric mechanism to model these complex physical realities. We show that Arc Helices naturally embed in a toroidal tube around a central circle. When the frequency ratio satisfies Ω/ω=p/q, the curve becomes periodic and forms a torus knot $T(p,q)$ with a higher-order internal Lissajous entanglement. A simple condition √a²+b²<R guarantees that the curve is embedded and free of self-intersections. These results establish the differential and topological foundations of Arc Geometry.