So I was reading the American Mathematical Monthly Feb 2011 (Volume 118, number 2), and in particular, I was interested in Ravi Vakil's article about mathematics of doodling. There is a question I cannot prove (or find the proof of anywhere).
First, here is the definition of the doodle (quoted from the article):
"Informal deļ¬nition. I walk around the outside of X counterclockwise, sticking my
right hand out and marking the path with a marker. By a remarkable coincidence, my
arm has length precisely $r$ . We call the resulting doodle $N_r(X)$."
For any convex polygons or closed curves with the maximum winding number of $1$, we have that $Perim(N_r(X)) = Perim(X) + 2r\pi$ and $Area(N_r(X)) = Area(X) + rPerim(X) + r^2\pi$.
In general, for any closed curve, whose winding number is $q$, the $Perim(N_r(X)) = Perim(X) + q(2r\pi)$ and $Area(N_r(X)) = Area(X) + rPerim(X) + q(r^2\pi)$.
I am wondering if anyone knows how to prove the fact: "for any closed curve, whose winding number is $q$, the $Perim(N_r(X)) = Perim(X) + q(2r\pi)$ and $Area(N_r(X)) = Area(X) + rPerim(X) + q(r^2\pi)$." Or explain why the winding number has such an effect on the Area and Perimeter formula for $N_r(X)$.
(Reference: http://math.stanford.edu/~vakil/files/monthly116-129-vakil.pdf pp120-122).
Thanks a lot in advance.
Also, what do you think about the "cool fact"? Theorem 3. The average length of the shadow of a convex region of the plane, multiplied by , is the perimeter. Is this a well-known fact? How could we prove it?
Best Answer
This problem is one of the easiest applications of Frenet formulas for planar curves and can be found in differential geometry textbooks.
Some minor corrections: First, $q$ is usually called "turning number" rather than "winding number". (The winding number is how many times a curve goes around a marked point; the turning number is how many times its velocity vector goes around the origin.) The turning number equals the integral of the curvature divided by $2\pi$. Second, as others noticed, $r$ should not be too large if the curvature attains negative values. More precisely, the result holds true for $r<1/\max(-\kappa)$ where $\kappa$ denotes the curvature.
The proof goes as follows. Let $t\mapsto s(t)$ be an arc-length parametrization of the original curve and $V(t),N(t)$ its Frenet frame. Then the $r$-shifted curve is parametrized by $$ s_r(t) = s(t) - rN(t) . $$ Then the velocity of $s_r$ is given by $$ s_r'(t) = V(t) + r\kappa(t)V(t) = (1+r\kappa(t)) V(t) $$ because $s'=V$ and $N'=-\kappa V$ by Frenet formulas. Then $$ Length(s_r) = \int |s_r'| = \int |1+r\kappa| = \int (1+r\kappa) = Length(s) + r\int\kappa = Length(s) + 2\pi q r . $$ The area formula is obtained from the length one by integration.