EinheitsTangente

EinheitsTangente, or unit tangent, is a vector tangent to a differentiable curve that has unit length. In differential geometry, for a smooth curve r(t) in Euclidean space, the unit tangent vector T(t) is defined as T(t) = r'(t) / ||r'(t)|| whenever r'(t) ≠ 0. If the curve is parameterized by arc length s, then T(s) = dr/ds and by construction ||T(s)|| = 1. The EinheitsTangente provides the instantaneous orientation of the curve along its path.

The rate of change of the unit tangent encodes curvature: dT/ds = κ(s) N(s), where κ is the curvature and N is the principal normal. Together with the binormal B = T × N, the Frenet-Serret frame {T, N, B} satisfies dN/ds = -κ T + τ B and dB/ds = -τ N, where τ is the torsion. These relations describe how the curve bends in space and are fundamental to the study of curve geometry.

Computation and parameterizations: Given a parameterization r(t) with r'(t) ≠ 0, the unit tangent is T(t) = r'(t)/||r'(t)||. If the curve is reparameterized by arc length s, then r'(s) = T(s) and ||T(s)|| = 1. The unit tangent is generally invariant under orientation-preserving reparameterizations, while orientation-reversing reparameterizations invert its direction.

Applications: The unit tangent vector is used to describe curve geometry, define curvature and normal directions, analyze particle motion along a path, and in fields such as computer graphics, robotics, and path planning.

See also: Tangent vector, Arc length, Curvature, Frenet-Serret formulas.