turning number

plural turning numbers

1. geometryA 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.

   This program illustrates the Whitney-Graustein theorem, that two closed curves (immersions of the circle) in the plane can be deformed into one another without letting go of the immersion property if and only if their turning numbers are the same.

   1989, Wayne Sewell, Weaving a Program: Literate Programming in WEB, page 516

   2 more examples

   Moreover, l gives the turning number of the k-gon one sees in the horizontal symmetry plane when the axes of the ends are deleted.[…]Moreover, every even k ≥ 30 except for 32, 36, 40, 44, 48 52 an 56 gives an example, while for every even k > 102 there is more than one example since different turning numbers are possible.

   1998, Karsten Groβe-Brauckmann, Robert B. Kusner, John M. Sullivan, “Constant Mean Curvature Surfaces with Cylindrical Ends”, in Hans-Christian Hege, Konrad Polthier, editors, Mathematical Visualization: Algorithms, Applications and Numerics, Springer, page 114

   2002, Journal of Physics: Mathematical and general, page 6187, For any oriented curve β(τ),τ∈I,I=[0,Λ] of class C¹ (τ is the arclength parametrization) lying in an oriented Euclidean plane E, the turning number 𝒯_(n_β) may be defined as 𝒯_(n_β)=1/(2π)∫₀^Λκ_β(τ)dτ where κ_β(τ) is the signed curvature of β.