ruimtecurves

In mathematics, ruimtecurves (space curves) are curves that lie in three-dimensional Euclidean space. They are typically given parametrically as r(t) = (x(t), y(t), z(t)) with t in some interval, where r is differentiable and regular, meaning r'(t) ≠ 0 for all t in the interval. The geometric properties of ruimtecurves are often studied in terms of their curvature κ(t) and torsion τ(t). Curvature measures how quickly the curve departs from a straight line, and torsion measures how quickly the curve departs from being planar.

For a parameterization r(t) with r' ≠ 0, curvature is κ = |r' × r''| / |r'|^3, and torsion is τ = ((r' × r'') · r''') / |r' × r''|^2. Reparameterizing by arclength s, one obtains T(s) = dr/ds, and the Frenet-Serret frame {T, N, B} with T' = κ N, N' = -κ T + τ B, and B' = -τ N. Planar ruimtecurves have τ ≡ 0; space curves with nonzero torsion do not lie in any plane. A common example is the circular helix, which has constant κ and τ.

ruimtecurves can be classified as regular or singular, polynomial or rational, and can exhibit knotted behavior in three-space. They have broad applications in computer graphics, animation, robotics path planning, and knot theory.