Turning number

English

Noun

A curve of turning number 3 (but winding number 2)

1. Template:Lb 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.
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   • 2002, Journal of Physics: Mathematical and general, page 6187,

     For any oriented curve $\beta (\tau ),\tau \in I,I=[0,\mathrm{\Lambda }]$ of class $C^1$ ($\tau$ is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number ${𝒯}_{n_{\beta }}$ may be defined as

     ${𝒯}_{n_{\beta }}=\frac{1}{2\pi }\underset{0}{\overset{\mathrm{\Lambda }}{\int }}{\kappa }_{\beta }(\tau )d\tau$

     where ${\kappa }_{\beta }(\tau )$ 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

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