Difference between revisions of "Rotation number"
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826501.png" /> be a regular curve, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826502.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826503.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826504.png" />. Then there is a continuous piecewise-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826505.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826506.png" />, the normalized velocity vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826507.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826508.png" />. Moreover, the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826509.png" /> is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265010.png" />. | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826501.png" /> be a regular curve, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826502.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826503.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826504.png" />. Then there is a continuous piecewise-differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826505.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826506.png" />, the normalized velocity vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826507.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826508.png" />. Moreover, the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r0826509.png" /> is independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265010.png" />. | ||
Revision as of 15:07, 6 May 2012
This page is deficient and requires revision. Please see Talk:Rotation number for further comments.
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].
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 |
Rotation number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_number&oldid=12710