Torsion & the Frenet-Serret Formulas

The TNB frame, curvature, torsion, and the Frenet equations.

The idea

The Frenet-Serret formulas describe how the natural frame of a space curve turns as the curve is traversed. Let $\gamma$ be a space curve traced at unit speed, with unit tangent $T = \gamma'$, principal normal $N$ (the unit vector in the direction of $T'$), and binormal $B = T \times N$; the curvature is $\kappa = |T'|$.

Theorem (Frenet-Serret formulas).

For a unit-speed space curve with frame $T, N, B$ and curvature $\kappa$, there is a function $\tau$, the torsion, such that $T' = \kappa N, \qquad N' = -\kappa T + \tau B, \qquad B' = -\tau N.$

The vectors $T, N, B$ are orthonormal, so they form a frame that rides along the curve, and an orthonormal frame can only rotate: each derivative is a combination of the other two vectors. The formulas say that the rotation has just two independent rates. Curvature is the rate at which the curve bends within the plane spanned by $T$ and $N$, the plane it momentarily lies in; torsion is the rate at which that plane itself tilts, measured by the motion of its normal $B$. A curve with $\tau = 0$ everywhere stays in one plane.

Both rates are measured per unit of arc length, so neither depends on how fast the curve is traced, and both survive rotations and translations of space. They are a complete set of invariants: two unit-speed curves with the same $\kappa$ and the same $\tau$ differ by a rigid motion. A reflection, however, reverses the sign of $\tau$ and leaves $\kappa$ unchanged, so torsion records the curve's handedness.

Ways to work on it

• Walkthrough. The Frenet frame, the Frenet-Serret formulas, and the helix's curvature and torsion.
• Practice. Compute the curvature or torsion of a circular helix.
• Hardest. Recover torsion from the scalar-triple-product formula for an arbitrary curve.

Not sure where to start? Take the ten-question placement test.