Page 243 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Since the curves e(0, t) and e(t, L) are related via a rigid motion, i.e., e(t, L) = Re(0, t) where R is
rotation by , it follows that (t)  (t,L)  (0,L)  (0,t)  (0,0)  is a constant. Since clearly

(0) 0, we get (0, L) – (0, 0) = (L,L) – (0,L), and we conclude:


 (L,L)   (0,0)   (t,L)   (0,L)   (0,t)    (0,0)    2 .

n
Definition 8. A piecewise smooth curve is a continuous function :[a,b]   such that there is
a partition of [a, b]:
a = a < a < · · · < b = b
n
1
0
such that for each 1  j  n the curve segment     a  j 1 ,a  is smooth. The points (a ) j are
j 
j

called the corners of . The directed angle      from '(a ) to '(a ) is called the exterior




j
j
j
angle at the j-th corner. Define  j :[a j 1 ,a ]  as in Proposition 2, i.e., so that '   cos ,sin j   j .
j
j

The rotation number of  is given by:
1 n 1 n
n   2     (a )   (a  j 1 )   2    . j
j
j
j
 j 1
 j 1
Again, n is an integer, and we have:

n
n   2 1  [a,b] k ds  2 1    . j

 j 1
The Rotation Theorem can be generalized to piecewise smooth curves provided corners are
taken into account.
2
Theorem 6. Let :[0,L]   be a piecewise smooth, regular, simple, closed curve, and assume
that none of the exterior angles are equal to . Then n   1.

20.3.3 Convexity
2
Definition 9. Let :[0,L]   be a regular closed plane curve. We say that  is convex if for each
t  [0, L] the curve lies on one side only of its tangent at t , i.e., if one of the following inequality
0
0
holds:
   (t0)  e 2  0,

   (t0)  e 2  0,

2
Theorem 7. Let :[0,L]   be a regular simple closed plane curve, and let k be its curvature.
Then  is convex if and only if either k  0 or k  0.
We note that an orientation reversing reparametrization of  changes k  0 into k  0 and vice
versa. Thus, ignoring orientation, those two conditions are equivalent. We also note that the
theorem fails if  is not assumed simple.

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