curvaturetorsion

Curvaturetorsion is not a standard single invariant in differential geometry. Instead, it is often used informally to refer to the pair formed by a space curve’s curvature and torsion, namely (kappa(s), tau(s)), which together describe how the curve bends and twists in three-dimensional space.

For a smooth, regular curve gamma: I -> R^3, with parameter t (arc length s is often used), curvature is defined as kappa = |d^2 gamma/ds^2| when the curve is parameterized by arc length. In general parameter form, kappa = |gamma' × gamma''| / |gamma'|^3. Torsion is defined by tau = [ (gamma' × gamma'') · gamma''' ] / |gamma' × gamma''|^2 for general parameterizations, or, under arc length, tau = det(gamma', gamma'', gamma''') / |gamma''|^2.

The Frenet-Serret apparatus describes how the unit tangent T = gamma'/|gamma'|, the principal normal N, and the binormal B = T × N evolve along the curve: T' = kappa N, N' = -kappa T + tau B, B' = -tau N. The fundamental theorem of space curves states that, up to rigid motions, a pair of functions kappa(s) > 0 and tau(s) determines a curve uniquely.

Special cases include planar curves, for which tau(s) = 0 identically, and general space curves such as helices, where kappa and tau are constant. In applications, curvature and torsion are used to describe and design space curves in computer graphics, robotics, biology (e.g., DNA modeling), and geometric modeling, often through their curvature-torsion profile rather than a single composite quantity.