turning number

== English == === Noun === turning number (plural turning numbers) (geometry, topology, mathematical analysis) A version of winding number in which the number of rotations is counted with respect to the tangent of the curve rather than a fixed point. 2002, Journal of Physics: Mathematical and general, page 6187, For any oriented curve β ( τ ) , τ ∈ I , I = [ 0 , Λ ] {\displaystyle \beta (\tau ),\tau \in I,I=\left[0,\Lambda \right]} of class C 1 {\displaystyle C^{1}} ( τ {\displaystyle \tau } is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number T n β {\displaystyle {\mathcal {T}}_{n_{\beta }}} may be defined as T n β = 1 2 π ∫ 0 Λ κ β ( τ ) d τ {\displaystyle {\mathcal {T}}_{n_{\beta }}={\frac {1}{2\pi }}\int _{0}^{\Lambda }\kappa _{\beta }(\tau )d\tau } where κ β ( τ ) {\displaystyle \kappa _{\beta }(\tau )} is the signed curvature of β. ==== Usage notes ==== In the animation, the smaller loop that does not go around the origin is also counted. === Further reading === Immersion (mathematics) on Wikipedia.Wikipedia Total curvature on Wikipedia.Wikipedia