2D Frenet Frame: Signed Curvature & the Evolute of a Plane Curve

Every smooth curve drawn on a flat page carries its own local coordinate frame — a moving pair of perpendicular unit vectors: the tangent T pointing along the direction of travel, and the normal N rotated a quarter turn from it. This simulation traces that 2D Frenet frame along an ellipse, a self-crossing limaçon and a 3-petal rose, computing signed curvature κ directly from the curve's own first and second derivatives at every point, drawing the osculating circle of radius 1/|κ| that best hugs the curve locally, and tracing the evolute — the locus of every one of those circles' centers as you sweep the whole loop. A live numerical integral of curvature around the closed curve verifies the classical turning-number theorem, the 2D counterpart to how 3D curves are governed by curvature and torsion together.