Lecture Notes for 120 - UCLA Department of Mathematics

As (
2.3. LENGTH AND AREA 38
sin ✓, cos ✓) is the unit normal to the line passing through q (s) we note that
|( sin ✓, cos ✓) · T| = |sin |
where denotes the angle between the line and T (s). Sinces is fixed we have that
the two angles , ✓ differ by a constant, so it follows that
ˆ 2⇡
0
|( sin ✓, cos ✓) · T| d✓ =
= 4
This in turn implies the result we wanted to prove.
ˆ 2⇡
Crofton’s formula will reappear later for curves on spheres and at that point
we will give a different proof.
Corollary 2.3.2. If we reparametrize OL using ( , s) instead of (✓, s) we
obtain ˆ
ˆ L ˆ 2⇡
f (l) n q (l) dl = f (l ( , s)) |sin | d ds
l2OL
0
Proof. This is simply the change of variables formula for functions on OL
given our analysis in the proof of Crofton’s formula.
⇤
Before presenting another well known and classical result relating area and
length we need to indicate a proof of an intuitively obvious theorem. This result
allows us to speak of the inside and outside of a simple closed planar curve.
Theorem 2.3.3. (Jordan Curve Theorem) A simple closed planar curve divides
the plane in to two regions one that is bounded and one that is unbounded.
Proof. Consider a simple closed curve q that is parametrized by arclength.
We construct the parallel curves
0
q ✏ = q + ✏N ±
The velocity is
dq ✏
dt =(1 ✏apple ±) T
So as long as ✏ is small they are clearly regular. We also know that they are closed.
Finally we can also show that they are simple for small ✏. This follows from a
contradiction argument. Thus assume that there is a sequence ✏ i ! 0 and distinct
parameter values s i 6= t i such that
0
|sin
q (s i )+✏ i N ± (s i )=q (t i )+✏ i N ± (t i )
By compactness we can, after passing to subsequences, assume that s i ! s and
t i ! t. Sinceq is simple it follows that s = t. Nowif|apple ± |appleK then the derivatives
of N ± are bounded in absolute value by K so it follows that
|q (s i ) q (t i )| = |✏ i ||N ± (t i ) N ± (s i )|
apple |✏ i | K |t i s i |
But this implies that the derivative of q vanishes at s = t which contradicts that
the curve is regular.
Fix ✏ 0 > 0 so that both q ✏0 and q ✏0 are closed simple curves. We think of
them as inside and outside curves, but we don’t know yet which is which. Every
point p not on q will either lie on a parallel curve q ✏ with |✏| apple✏ 0 , in which case we
| d
⇤