Curves & Frenet-Serret Frames

Explore space curves, curvature, torsion, and moving frames

Curves & Frenet-Serret Frames

Ride along a curve in space carrying three arrows that turn with it. How fast they turn tells you everything about the curve's shape: its curvature and its torsion.

Frenet Frame Explorer

Pick a curve and watch T (red), N (green), and B (blue) travel along it. On the helix, notice the frame turning at a perfectly steady rate.

Frenet-Serret Frame
T - Tangent (velocity direction)
N - Normal (turning direction)
B - Binormal (T × N)
At Current Point
Curvature κ:0.0000
Torsion τ:0.0000
Helix

Constant κ and τ

A curve that spirals around the z-axis with constant curvature and torsion

Go deeper:The Frenet-Serret Equations

A space curve is a path through 3D space, parametrized asr(t) = (x(t), y(t), z(t)). At each point, we attach a natural coordinate system called the Frenet-Serret frame.

This moving frame consists of three orthogonal unit vectors: the tangent T (direction of motion), the normal N (direction the curve is turning), and the binormal B = T × N(perpendicular to both).

The frame vectors satisfy a beautiful system of differential equations:

dT/ds = κN
dN/ds = -κT + τB
dB/ds = -τN

Here κ (kappa) is the curvature — how sharply the curve bends — and τ (tau) is the torsion — how much the curve twists out of its osculating plane.

Go deeper:Curvature κ and Torsion τ
κ = |dT/ds| = |r' × r''| / |r'|³
  • κ = 0: Straight line (no bending)
  • κ = 1/R: Circle of radius R
  • Large κ: Sharp turns
τ = (r' × r'') · r''' / |r' × r''|²
  • τ = 0: Planar curve (lies in a plane)
  • τ > 0: Right-handed twist
  • τ < 0: Left-handed twist
Go deeper:The Helix: A Perfect Example

The helix r(t) = (cos t, sin t, t/2π) is special: it has constant curvature and constant torsion. In fact, the helix is the only curve (besides the circle) with both κ and τ constant.

Watch how the Frenet frame rotates smoothly and uniformly as it travels along the helix, never speeding up or slowing down its twisting motion.

Key Takeaways

  • The Frenet-Serret frame (T, N, B) is an orthonormal basis that moves with the curve.
  • Curvature κ measures bending: how fast T rotates toward N.
  • Torsion τ measures twisting, and a curve is planar exactly when τ = 0.
  • The helix has constant κ and constant τ.

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