In differential geometry, the Frenet–Serret formulas describe thekinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-calledtangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret in 1851. Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery.
The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame, together form anorthonormal basis spanning R3, and are defined as follows:
- T is the unit vector tangent to the curve, pointing in the direction of motion.
- N is the normal unit vector, the derivative of T with respect to thearclength parameter of the curve, divided by its length.
- B is the binormal unit vector, the cross product of T and N.
The Frenet–Serret formulas are
where d/ds is the derivative with respect to arclength, κ is the curvature and τ is the torsion of the curve. The two scalars κ and τ effectively define the curvature and torsion of a space curve. The associated collection, T, N, B, κ, and τ is called theFrenet–Serret apparatus. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.