Lecture Notes for 120 - UCLA Department of Mathematics

2.5. TWO SURPRISING RESULTS 47
(7) Let q :[0,L] ! R 2 be a closed curve parametrized by arclength. Show
that if ´ L
0 appleds =2⇡, thenapple ± cannot change sign and the rotation index is
±1. Later we will show that this implies that the curve is simple as well.
(8) Let q (t), t 2 [a, b] be a regular planar curve and ✓ (t) 2 [✓ 0 ,✓ 1 ] an arclength
parameter for T. Define v (t) as the distance from the origin to
the tangent line through q (t).
(a) Show that
v (t) = q (t) · N ± (t)
(b) Show by an example (e.g., a straight line) that q is not necessarily a
function of ✓.
(c) Define the curve
q ⇤ (✓) = dv
d✓ T
vN ± = dv (cos ✓, sin ✓) v ( sin ✓, cos ✓)
d✓
and show that
dq ⇤ ✓ d 2 ◆
d✓ = v
d✓ 2 + v (cos ✓, sin ✓)
(d) Show that when q ⇤ is a regular curve then it is a reparametrization
of q.
(e) Under that assumption show further that
v + d2 v
d✓ 2 = 1 apple
L (q) =
ˆ ✓1
✓ 0
v (✓) d✓
(f) How is this related to Crofton’s formula?
2.5. Two Surprising Results
Definition 2.5.1. A vertex of a curve is a point on the curve where the curvature
is a local maximum or a local minimum.
Theorem 2.5.2. (Mukhopadhyaya, 1909 and Kneser, 1912) A simple closed
curve has at least 4 vertices.
We start with the following observation.
Proposition 2.5.3. Suppose we have a curve q that is tangent to a circle and
lies inside (resp. outside) the circle, then its curvature is larger (resp. smaller)
than or equal to the curvature of the circle at the points where they are tangent.
Proof. Assume the curve q is tangent to the circle of radius R centered at c
at s = s 0 . This implies that
|q (s) c| 2 apple R 2 and |q (s 0 ) c| 2 = R 2
Thus the function s 7! |q (s) c| 2 has a (local) maximum at s = s 0 . This implies
that its derivative at s 0 vanishes, which is the fact that the curve is tangent to
the circle, and that its second derivative is nonpositive. Assume that both circle
and curve are parametrized to run counter clockwise. Thus they have the same