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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" />

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 $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082650b.gif" />

Figure: r082650b

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

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