Geometric Analysis and Physical Significance of the Chaotic Line

This paper introduces a new fundamental planar curve, named the “Chaos Line”, defined by the implicit equations x3y2 +x2y3 = 1 and x3y2−x2y3 =1. We derive its forms in polar, spherical, and cylindrical coordinates, revealing a core structure: static term × flow term = constant. This curve represents a third fundamental geometric type, distinct from circles and hyperbolas, encoding symmetry, conservation, duality, and directional flow in a single elementary expression