Page 222 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 18: Theory of Space Curves

18.8 Plane Convex Curves and the Four Vertex Theorem Notes

A closed curve in the plane is called convex, if it lies at one side of every tangent line.

Proposition 8: A simple closed curve is convex, if and only if it can be so oriented that its
curvature k is  0.
The definition of a convex curve makes use of the whole curve, while the curvature is a local
property. The proposition, therefore, gives a relationship between a local property and a global
property. The theorem is not true if the closed curve is not simple. Counter examples can be
easily constructed.

d
Let (s)  be the function constructed above, so that we have k = . The condition k  O is,
ds
therefore, equivalent to the assertion that (s))  is a monotone non-decreasing function. We can
assume that (O) O.   By the theorem on turning tangents, we can suppose C so oriented that


  (L) 2 .
Suppose (s),  O  s  L, be monotone non-decreasing and that C is not convex. There is a point
A = x(s ) on C such that there are points of C at both sides of the tangent to C at A. Choose a
0
positive side of k and consider the oriented perpendicular distance from a point x(s) of C to .
This is a continuous function in s and attains a maximum and a minimum at the points M and N
respectively. Clearly M and N are not on and the tangents to C at M and N are parallel to x.
Among these two tangents and k itself there are two tangents parallel in the same sense. Call
s < s the values of the parameters at the corresponding points of contact. Since (s)  is monotone
2
1
non-decreasing and O  (s)   2, this happens only when (s)    (s ) for all s satisfying s  s
 
1
1
 s . It follows that the arc s  s  s is a line segment parallel to . But this is obviously
2
1
2
impossible.
(s ) ,
Next let C be convex. To prove that (s)  is monotone non-decreasing, suppose (s )    2
 
1
s < s . Then the tangents at x(s ) and x(s ) are parallel in the same sense. But there exists a tangent
2
1
1
2
parallel to them in the opposite sense. From the convexity of C it follows that two of them
coincide.
We are, thus, in the situation of a line  tangent to C at two distinct points A and B. We claim that
the segment AB must be a part of C. In fact, suppose this is not the case and let D be a point on
AB not on C. Draw through D a perpendicular  to in the half-plane which contains C. Then 
intersects C in at least two points. Among these points of intersection let F be the farthest from
 and G the nearest one, so that F  G. Then G is an interior point of the triangle ABF. The tangent
to C at G must have points of C in both sides which contradicts the convexity of C.
It follows that under our assumption, the segment AB is a part of C, so that the tangents at A and
B are parallel in the same sense. This proves that the segment joining x(s ) to x(s ) belongs to C.
1
2
Hence, (s)  remains constant in the interval s  s  s . We have, therefore, proved that (s)  is
1 2
monotone and K  O.
A point on C at which k’ = 0 is called a vertex. A closed curve has at least two vertices, e.g., the
maximum and the minimum of k. Clearly a circle consists entirely of vertices. An ellipse with
unequal axes has four vertices, which are its intersection with the axes.

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