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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" />.
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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 $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082650a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082650a.gif" />
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Figure: r082650a
 
Figure: r082650a
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265011.png" /> be a piecewise-smooth regular closed curve and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265012.png" /> partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265013.png" /> into intervals such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265014.png" /> restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265015.png" /> is differentiable for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265017.png" /> be the exterior angle between the tangent vectors at the corner at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265018.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265019.png" /> is the angle between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265021.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265022.png" />). The number
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265023.png" /></td> </tr></table>
+
$$
 +
n _ {c}  =
 +
\frac{1}{2 \pi }
 +
\sum _ { j } ( \theta _ {j} ( b _ {j} ) - \theta _ {j} ( a _ {j} )) +
 +
\frac{1}{2 \pi }
 +
\sum _ { j } \alpha _ {j}  $$
  
is called the rotation number of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265024.png" />.
+
is called the rotation number of the curve $  c $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265025.png" /> is identified with the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265027.png" /> is smooth (so that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265028.png" /> are zero), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265029.png" /> is the winding number of the closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265030.png" /> with respect to the origin.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265031.png" /> be piecewise-smooth, regular, closed, and simple (i.e. no self-intersections), and suppose that the exterior angles are always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265032.png" /> in absolute value. Then the so-called Umlaufsatz says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265033.png" />, depending on the orientation. From this it is easy to calculate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265034.png" /> of closed curves with self-intersections. For instance, the rotation number of the figure eight curve is zero.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265035.png" />-gon is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265036.png" />. There also result the various formulas for triangles (and other figures) made up of circle segments, such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265037.png" /> in the case of the circle segment triangle depicted on the left in Fig.a2; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265038.png" /> for the circle segment triangle depicted on the right in Fig.a2. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265039.png" /> denote the number of degrees of the circle segments in question, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082650/r08265041.png" />.
+
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 $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082650b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082650b.gif" />

Latest revision as of 08:12, 6 June 2020


This page is deficient and requires revision. Please see Talk:Rotation number for further comments.

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
How to Cite This Entry:
Rotation number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rotation_number&oldid=12710