Frenet Frame

The Frenet-Serret frame attaches a moving orthonormal coordinate system to every point of a smooth space curve. At each point the frame consists of the tangent vector T (unit velocity direction), the normal vector N (the direction the curve is bending toward), and the binormal vector B = T \times N. The Frenet-Serret formulas describe how these vectors change along the curve in terms of two scalar quantities — curvature \kappa and torsion \tau — and are the foundation of differential geometry of curves, used in robotics, computer graphics tube extrusion, and structural analysis.

The visualization plots a closed reference curve and samples the frame at nine evenly spaced positions. Red arrows show the tangent T, green arrows show the normal N, and blue arrows show the binormal B. Watching the frame as it follows the curve makes the geometric meaning of curvature and torsion concrete: the normal swings into the bending direction, and the binormal tilts when the curve leaves the local osculating plane.