Shortest "painting" of the sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:24:19Z http://mathoverflow.net/feeds/question/70803 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70803/shortest-painting-of-the-sphere Shortest "painting" of the sphere unknown (google) 2011-07-20T08:37:01Z 2011-07-20T11:09:52Z <p>Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing $C(t)$ and having normal vector $T(t)$.</p> <p>Given a size $d$ of the "paint brush" we define the "brush" $b:[0,1]\to \mathcal{P}(S)$ by letting $b(t)$ be the points of $S$ that are at most a distance $d$ (metric on the sphere) from $C(t)$ that are contained in $P(t)$.</p> <p>We can think of this as saying the "brush" $b(t)$ is an arc on the sphere that is "orthogonal" to the motion $C(t)$ of the "paint brush".</p> <p>Given $d$ what is the arclength of the shortest curves such that $\cup_{t\in[0,1]} b(t) = S$. This says that the "paint brush" covered the sphere.</p> http://mathoverflow.net/questions/70803/shortest-painting-of-the-sphere/70806#70806 Answer by Jean-Marc Schlenker for Shortest "painting" of the sphere Jean-Marc Schlenker 2011-07-20T09:19:10Z 2011-07-20T10:44:38Z <p>This question is somewhat related to <a href="http://mathoverflow.net/questions/69099/shortest-closed-curve-to-inspect-a-sphere" rel="nofollow">this recent one</a>. More precisely, the comment by Gjergji Zaimi in the earlier question gives a painting of length $2\sqrt{2}\pi$ for $d=\pi/4$, which, as explained in another comment there, is optimal for a path at constant distance from the sphere. So for $d=\pi/4$ the optimal length should be $2\sqrt{2}\pi$.</p> http://mathoverflow.net/questions/70803/shortest-painting-of-the-sphere/70807#70807 Answer by Joseph O'Rourke for Shortest "painting" of the sphere Joseph O'Rourke 2011-07-20T10:23:06Z 2011-07-20T10:23:06Z <p>The model is that used by Henryk Gerlach and Heiko von der Mosel in their 2010 paper "On sphere-filling ropes" <a href="http://arxiv.org/abs/1005.4609" rel="nofollow">arXiv:1005.4609v1 (math.GT)</a> may be relevant. Their question is different: What is the longest rope of a given thickness on a sphere? But their explicit solutions are packings, and it seems they could be converted to painting paths. Here is a piece of their Fig. 6: <br /> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img src="http://cs.smith.edu/~orourke/MathOverflow/RopeSphere.jpg" alt="Rope on sphere"></p>