Verdrehwinkel

Verdrehwinkel, or twist angle, is a measure of how much the orientation of a line field, cross-section, or object twists around its axis as one progresses along a curve or length. It captures the cumulative rotation of the local reference direction relative to a fixed frame.

In differential geometry and related fields, Verdrehwinkel describes the rotation of a director field attached to a space curve. If a narrow ribbon follows the curve, the twist refers to the angular rotation of the ribbon’s cross-section about the curve’s tangent as one moves along its length. The instantaneous rate of twist is the derivative of this angle with respect to arc length, often denoted dφ/ds, and the total twist is the integral of that rate along the curve.

Verdrehwinkel is closely connected to other geometric quantities such as torsion and writhe. Torsion measures the rate of change of the curve’s binormal, while writhe captures the spatial coiling of the curve itself. In certain ribbon models, the linking number Lk is related to twist and writhe by the Călugăreanu–White–Flanders formula Lk = Tw + Wr, linking the concepts of twist, writhe, and overall topology.

Applications of Verdrehwinkel appear in engineering, materials science, and biology. It is used to describe the twist of cables and fibers, the supercoiling of DNA, and the orientation of twisted nanotubes or textile structures. Measurements are typically expressed in radians or degrees and can be obtained from geometric analysis or experimental sensors that track rotational orientation along the length.

See also: Torsion, Twist, Writhe, Frenet–Serret frame, Călugăreanu–White–Flanders theorem.