10
$\begingroup$

Polya's orchard problem asks for what radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. The answer is known; $r$ should be about $1/R$. See this earlier question for details and references.

Suppose instead of straight rays, visibility may follow a circular arc (of any radius) from the origin. Let $B=B(r)$ be the furthest distance visible in the plane along any circular arc when all lattice points (excluding the origin) center disks of radius $r$. It is easy to see that $B > R$, where $R=\frac{\sqrt{1-r^2}}{r}$ is the Polya distance determined by $r$:

          Boomerangs
          $r=\frac{1}{4}$, $R \approx 3.9$.
          Two circular arc rays (of different radii) from the origin to beyond $R$ are shown.
$B$ is finite for any $r>0$: For otherwise the radius $a$ of the arc that shoots to $\infty$ must itself be infinite. But then it is a straight-line ray, and couldn't reach further than Polya's $R$.

Q. Can anyone see an estimate of, or a bound on $B$ as a function of the disk radius $r$?

Perhaps the furthest reaching arcs straddle the disks on the $x$- and $y$-axes?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ (entirely tangential comment: how the heck do you come up with these titles? most of the time I see one of your questions I open it because I cannot conceive how it is a mathematical question, and then I read it and realize that no other title can be better than the one you chose) $\endgroup$
    – Willie Wong
    Commented Nov 19, 2015 at 14:31
  • 1
    $\begingroup$ One possible (but really, really poor) bound is this. First make all the discs twice as big. Not you know that any curve that deviates from a straight line in sufficiently small manner will hit a disc in radius $R$. This contradicts a curve reaching distance $2\tilde{R}$ via a curve of radius of curvature $>\tilde{R}$ since within distance $R$ of origin the curve is "well approximated by a line". But this estimate is likely to be way bigger than what actually is there. (What? I am an analyst. :-p.) $\endgroup$
    – Willie Wong
    Commented Nov 19, 2015 at 14:39
  • 6
    $\begingroup$ @WillieWong: "O'Rourke quits faculty to pen tabloid headlines." $\endgroup$
    – Joseph O'Rourke
    Commented Nov 19, 2015 at 14:53

0

Reset to default

You must log in to answer this question.