Rotation number

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Let be a regular curve, i.e. is smooth and for all . Then there is a continuous piecewise-differentiable function such that , the normalized velocity vector at , is equal to . Moreover, the difference is independent of the choice of .

Figure: r082650a

Now, let be a piecewise-smooth regular closed curve and let partition into intervals such that restricted to is differentiable for all . Let be the exterior angle between the tangent vectors at the corner at , i.e. is the angle between and (with ). The number

is called the rotation number of the curve .

If is identified with the complex plane and is smooth (so that all are zero), then is the winding number of the closed curve with respect to the origin.

Let be piecewise-smooth, regular, closed, and simple (i.e. no self-intersections), and suppose that the exterior angles are always in absolute value. Then the so-called Umlaufsatz says that , depending on the orientation. From this it is easy to calculate the 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 -gon is . There also result the various formulas for triangles (and other figures) made up of circle segments, such as in the case of the circle segment triangle depicted on the left in Fig.a2; and for the circle segment triangle depicted on the right in Fig.a2. Here the denote the number of degrees of the circle segments in question, , .

Figure: r082650b

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


[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
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