curvature and torsion of a curve
Two numbers tell you almost everything about how a curve sits in space at a point: how sharply it bends, and how much it twists out of the plane it momentarily lives in. Drive a car around a tight roundabout and you feel strong curvature; drive up a spiral parking ramp and on top of the bending you feel a screwing, out-of-plane twist — that is torsion. Curvature kappa and torsion tau are the precise measures of these two effects.
Curvature kappa measures how fast the unit tangent direction turns per unit length traveled: kappa = magnitude of dT/ds, where s is arc length. A straight line has kappa = 0; a circle of radius R has constant kappa = 1/R, so a smaller circle is more curved. The reciprocal 1/kappa is the radius of curvature — the radius of the circle that best hugs the curve at that point (the osculating circle). Torsion tau measures how fast the plane of the curve (the osculating plane spanned by T and N) tilts as you move; equivalently how fast the binormal B = T cross N rotates, via dB/ds = -tau N. A planar curve has tau = 0 everywhere because it never leaves its plane. A clean computational formula for a curve r(t): kappa = magnitude of (r' cross r'') / magnitude of (r')^3, and tau = (r' cross r'') dot r''' / magnitude of (r' cross r'')^2.
These invariants are the heart of the local theory of curves: the fundamental theorem of curves says that two functions kappa(s) > 0 and tau(s) determine a space curve uniquely up to rigid motion, so curvature and torsion are a complete fingerprint of shape. Practically, curvature controls comfort and safety in road and rail design (sudden changes in kappa cause lurches, so engineers insert transition curves where kappa ramps smoothly), it sets bending stress in beams and the focusing of lenses and mirrors, and torsion describes the supercoiling of DNA and the twist of cables and springs. The honest caveat: kappa and tau are pointwise (local) quantities — they describe the curve in an infinitesimal neighborhood and say nothing directly about global features like whether the curve closes up or knots.
A circle of radius R drawn flat on a table has constant curvature kappa = 1/R and zero torsion (it never leaves the table's plane). Lift it into a helix by adding steady vertical rise and the torsion becomes nonzero — the curve now screws out of every momentary plane. Curvature answers 'how tight is the bend?', torsion answers 'how much does it leave the page?'.
Curvature is bending within the plane; torsion is escape from the plane. A circle has only the first.
Do not confuse the torsion of a curve (a twisting-out-of-plane number) with the Gaussian curvature of a surface or the curvature of space — they are different concepts. And the sign of tau encodes a right- versus left-handed twist, so it carries orientation information, unlike kappa which is taken non-negative.