Frenet–Serret Frame: Curvature & Torsion of a Space Curve

Space Curve

Position along curve 0.00

Play speed 0.25

Show osculating circle

Live Readouts

Curvature κ

—

Torsion τ

—

Speed |r′(t)|

—

How it works

For a smooth space curve r(t), the Frenet–Serret frame is the orthonormal triad (T, N, B) that travels with the curve:

T = r′ / |r′|                (unit tangent)
B = (r′ × r″) / |r′ × r″|    (unit binormal)
N = B × T                    (unit principal normal)

Curvature measures how fast T turns per unit arc length; torsion measures how fast the curve twists out of its own osculating plane (the plane spanned by T and N):

κ(t) = |r′ × r″| / |r′|³
τ(t) = [(r′ × r″) · r‴] / |r′ × r″|²

κ = 0 everywhere means a straight line. τ = 0 everywhere means a planar curve — a circle sitting entirely in one plane, for instance, has zero torsion but nonzero curvature. The osculating circle shown here has radius 1/κ and lies in the plane spanned by T and N — it is the circle that best approximates the curve at that single point, matching its position, tangent and curvature.

Curve dropdown — switches between a helix (constant κ and τ), a trefoil knot and a (2,3) torus knot (both with κ, τ varying along the curve).
Position slider — moves the frame to any point on the curve; derivatives are evaluated numerically by central differences at that point.
Play / speed — animates the frame continuously along the curve at the chosen rate.

This is the same machinery used to bank a roller-coaster car, orient a camera flying along a path, or compute the local geometry of DNA's double helix.

Tangent T Normal N Binormal B

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.