Rotation number

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Let $ c: [ a, b] \rightarrow \mathbf R ^ {2} $ be a regular curve, i.e. $ c( t) $ is smooth and $ \dot{c} ( t) \neq 0 $ for all $ t \in [ a, b] $. Then there is a continuous piecewise-differentiable function $ \theta ( t) $ such that $ \dot{c} ( t) / | \dot{c} ( t) | $, the normalized velocity vector at $ c( t) $, is equal to $ ( \cos \theta ( t), \sin \theta ( t)) $. Moreover, the difference $ \theta ( b) - \theta ( a) $ is independent of the choice of $ \theta $.

Figure: r082650a

Now, let $ c: [ 0, A] \rightarrow \mathbf R ^ {2} $ be a piecewise-smooth regular closed curve and let $ 0= b _ {-} 1 = a _ {0} < b _ {0} = a _ {1} < \dots < b _ {k} = A $ partition $ [ 0, A] $ into intervals such that $ c $ restricted to $ [ a _ {j} , b _ {j} ] $ is differentiable for all $ j $. Let $ \alpha _ {j} $ be the exterior angle between the tangent vectors at the corner at $ c( b _ {j-} 1 ) = c( a _ {j} ) $, i.e. $ \alpha _ {j} $ is the angle between $ \dot{c} ( b _ {j-} 1 - ) $ and $ \dot{c} ( a _ {j} +) $( with $ - \pi < \alpha _ {j} \leq \pi $). The number

$$ n _ {c} = \frac{1}{2 \pi } \sum _ { j } ( \theta _ {j} ( b _ {j} ) - \theta _ {j} ( a _ {j} )) + \frac{1}{2 \pi } \sum _ { j } \alpha _ {j} $$

is called the rotation number of the curve $ c $.

If $ \mathbf R ^ {2} $ is identified with the complex plane $ \mathbf C $ and $ c $ is smooth (so that all $ \alpha _ {j} $ are zero), then $ n _ {c} $ is the winding number of the closed curve $ t \mapsto \dot{c} ( t) / | \dot{c} ( t) | $ with respect to the origin.

Let $ c: [ 0, A] \rightarrow \mathbf R ^ {2} $ be piecewise-smooth, regular, closed, and simple (i.e. no self-intersections), and suppose that the exterior angles are always $ \neq \pi $ in absolute value. Then the so-called Umlaufsatz says that $ n _ {c} = \pm 1 $, depending on the orientation. From this it is easy to calculate the $ n _ {c} $ of closed curves with self-intersections. For instance, the rotation number of the figure eight curve is zero.

It readily follows from these results that, e.g., the sum of the interior angles of a convex $ n $- gon is $ ( n- 2) \pi $. There also result the various formulas for triangles (and other figures) made up of circle segments, such as $ \alpha _ {1} + \alpha _ {2} + \alpha _ {3} + \beta _ {1} + \beta _ {2} + \beta _ {3} = 180 \circ $ in the case of the circle segment triangle depicted on the left in Fig.a2; and $ - \alpha _ {1} + \alpha _ {2} + \alpha _ {3} - \beta _ {1} + \beta _ {2} + \beta _ {3} = 180 \circ $ for the circle segment triangle depicted on the right in Fig.a2. Here the $ \beta _ {i} $ denote the number of degrees of the circle segments in question, $ 0 \leq \beta _ {i} \leq 360 \circ $, $ i= 1, 2, 3 $.

Figure: r082650b

For more on the planar geometry of circle segment triangles and such, see, e.g., [a2], [a3].

References

[a1]	W. Klingenberg, "A course in differential geometry" , Springer (1978) pp. §2.1 (Translated from German)
[a2]	L. Bieberbach, "Zur Euklidischen Geometrie der Kreisbogendreiecke" Math. Ann. , 130 (1955) pp. 46–86
[a3]	W.K.B. Holz, "Das ebene obere Dreieck. Eine Aufgabestellung" , Selbstverlag Hagen (1944)
[a4]	H. Hopf, "Über die Drehung der Tangenten und Sehen ebener Kurven" Compositio Math. , 2 (1935) pp. 50–62