Blocking visibility with cylinders - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:37:47Z http://mathoverflow.net/feeds/question/64396 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64396/blocking-visibility-with-cylinders Blocking visibility with cylinders Joseph O'Rourke 2011-05-09T14:31:07Z 2011-05-11T00:55:48Z <p>Suppose you have a supply of infinite-length, opaque, unit-radius cylinders, and you would like to block all visibility from a point $p \in \mathbb{R}^3$ to infinity with as few cylinders as possible. (The cylinders are infinite length in both directions.) The cylinders may touch but not interpenetrate, and they should be disjoint from $p$, leaving a small ball around $p$ empty. (Another variation would insist that cylinders be pairwise disjoint, i.e., not touching one another.)</p> <p>A collection of parallel cylinders arranged to form a "fence" around $p$ do not suffice, leaving two line-of-sight $\pm$ rays to infinity. Perhaps a grid of cylinders in the pattern illustrated left below suffice, but at least if there are not many cylinders, there is a view from an interior point to infinity (right below).</p> <p><img src="http://cs.smith.edu/~orourke/MathOverflow/CylindersCrossed.jpg" width="640" height="240" alt="Crossed Cylinders" /></p> <p>I feel like I am missing a simple construction that would obviously block all rays from $p$. Perhaps crossing the cylinders like the poles of a <a href="http://en.wikipedia.org/wiki/Tipi" rel="nofollow">tipi (teepee)</a> could help, but it seems this would at best lead to inefficient blockage. Suggestions welcome&mdash;Thanks!</p> <p><b>Addendum1.</b> Perhaps if the weaving above is rendered irregular by displacing the cylinders slightly by different amounts, so that cracks do not align, then a sufficient portion of the weaving will block all visibility.</p> <p>Here (left below) is the start of Gerhard's first suggested construction (a portion of the weaving above), which I don't see how to complete. But perhaps seeing this depiction will aid intuition. <br /> <img src="http://cs.smith.edu/~orourke/MathOverflow/CylindersTangent.jpg" width="300" height="225" alt="Crossed Tangent to sphere" /> <img src="http://cs.smith.edu/~orourke/MathOverflow/CylinderForest.jpg" width="300" height="225" alt="Cylinder forest" /> <br /></p> <p><b>Addendum2.</b> To the right above I added (three-quarters of) a forest along the lines (but not exactly as) Yaakov suggested.</p> http://mathoverflow.net/questions/64396/blocking-visibility-with-cylinders/64495#64495 Answer by Yaakov Baruch for Blocking visibility with cylinders Yaakov Baruch 2011-05-10T12:54:52Z 2011-05-10T14:13:31Z <p>Here is one construction. On the horizontal xy plane place a forest of vertical cylinders of radius r&lt;1/2 (or =1/2 if we allow contacts) centered at each point in $(\mathbb{Z} \backslash{\lbrace0\rbrace})\times(\mathbb{Z}\backslash{\lbrace0\rbrace})$; moreover place 2 similar cylinders parallel to x centered at (y=+/-1, z=0), 2 more parallel to y centered at (x=0, z=+/-1) and the last 2 parallel to z and centered at (x=+/-1, y=0). Then (0,0,0) is blocked by the forest in all directions except those in the xz and yz planes, which are blocked by the other 6 cylinders. The forest clearly does not need to be infinite and it should be easy to find an upper limit on its size.</p> <p>${\bf UPDATE}$ As pointed out by Mark in a comment, the forest should be based on $(\mathbb{Z} \backslash{\lbrace0\rbrace})\times(\mathbb{Z}\backslash{\lbrace-1,0,1\rbrace})$.</p>