Frenet-Serret frame

Definition

Let $\gamma$ be a regular curve (for convenience, unit speed-parametrized) in $\mathbb {R} ^{3}$. The Frenet-Serret frame or Serret-Frenet frame of $\gamma$ associates, to each point on $\gamma$, an orthonormal basis at that point. The orthonormal basis comprises the followign unit vectors: the unit tangent, the unit normal and the unit binormal.

The Frenet-Serret frame keeps changing in direction as we move along the curve, and this change in direction is characterized by the Frenet-Serret equations, which show that the relative rate of change depends only on the curvature and torsion. Thus, the geometry of a unit-speed curve depends only on the values of curvature and torsion, as scalar functions.