the torsion of a curve
Take a wire and bend it: so far it stays flat, lying in a single plane like a drawing on paper. Now twist it so it lifts out of that plane, climbing like a spiral staircase or a corkscrew. Curvature measured the bending; torsion measures this extra twisting — the rate at which a space curve writhes out of the flat plane it is momentarily bending in. A curve that stays in one plane has zero torsion; a corkscrew has torsion that is nonzero.
Precisely, torsion is about the osculating plane — the plane spanned by the tangent T and principal normal N, the plane the curve is instantaneously bending within. The binormal B is the unit vector perpendicular to that plane, so watching how B changes tells you how the osculating plane is tipping in space. The torsion tau is defined (in the standard sign convention) by B'(s) = -tau N: as you move a unit of arc length, the binormal swings by an amount tau in the direction of N, and tau is exactly that rate of swing. If tau = 0 the binormal is constant, the osculating plane never moves, and the curve lies flat in that plane. A computational formula for a general parametrization is tau = (r' x r'') . r''' / |r' x r''|^2, the scalar triple product of the first three derivatives over the squared cross-product length.
Torsion is the second of the two numbers that pin down a space curve. Curvature alone cannot distinguish a flat circular arc from a rising helical arc that bends just as sharply — they have the same curvature but different torsion. Together, curvature and torsion as functions of arc length determine the curve uniquely up to a rigid motion: that is the fundamental theorem of curves. Honest cautions. First, torsion is genuinely a three-dimensional notion: a plane curve has tau = 0 always, so torsion only carries information for curves that leave the plane. Second, torsion (like the binormal it is built on) requires nonzero curvature to be defined, since a straight stretch has no osculating plane to twist out of. Third, the sign of torsion depends on conventions, so different textbooks may differ by a minus sign — check the convention before comparing formulas.
The helix r(t) = (a cos t, a sin t, b t) has constant curvature kappa = a/(a^2 + b^2) and constant torsion tau = b/(a^2 + b^2). The parameter b controls the climb: set b = 0 and the helix collapses to a flat circle of radius a, where tau = 0 — no climb, no twist. Make b larger and the spiral rises faster, increasing the torsion.
Torsion = rate the curve twists out of its osculating plane; tau = 0 means the curve is planar.
A curve has zero torsion everywhere if and only if it lies in a plane. Torsion needs nonzero curvature to be defined (no osculating plane otherwise), and its sign is convention-dependent, so textbooks may differ by a minus sign.