Frenet–Serret Frame: Curvature & Torsion of a Space Curve

Every smooth curve through 3D space carries its own local coordinate system — the Frenet–Serret frame — made of three mutually perpendicular unit vectors: the tangent T pointing along the direction of travel, the normal N pointing toward the center of curvature, and the binormal B completing the right-handed triad. This simulation traces that frame along a helix, a trefoil knot and a (2,3) torus knot, computing curvature κ and torsion τ directly from the curve's own first, second and third derivatives at every point, and drawing the osculating circle of radius 1/κ that best hugs the curve locally. Drag the position slider or hit play to watch how a knotted curve's twist and bend evolve as you travel along it.