Rumkurver

Rumkurver are smooth curves embedded in three-dimensional Euclidean space. In mathematics they are described by a vector-valued function r(t) = (x(t), y(t), z(t)) defined on an interval I ⊂ R. Often one reparameterizes by arc length s, in which case ||dr/ds|| = 1.

If the image of r lies entirely in a plane, the curve is planar; otherwise it is a space curve (rumkurve). The departure from planarity is measured by the torsion; for planar curves the torsion is zero.

Two basic quantities are curvature κ(s) and torsion τ(s). With respect to arc length, T(s) = r'(s) is the unit tangent; κ(s) = ||dT/ds||. The Frenet-Serret frame (T, N, B) is defined by N = (1/κ) dT/ds and B = T × N. The Frenet-Serret formulas describe how the frame changes along the curve: dT/ds = κ N; dN/ds = -κ T + τ B; dB/ds = -τ N.

Examples: a circular helix r(t) = (R cos t, R sin t, h t) is a space curve with constant curvature and torsion; a circle r(t) = (R cos t, R sin t, 0) has zero torsion; a straight line r(t) = v t has κ = 0.

Applications and relevance: space curves arise in differential geometry and the study of curves in 3D space; they model physical phenomena such as DNA structures or magnetic field lines, and are used in computer graphics, robotics, and kinematics.

History and terminology: the concept is foundational in differential geometry; the Frenet-Serret frame is named after Jean Frédéric Frenet and Joseph Serret.