Moving under the influence of a vector field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:18:47Z http://mathoverflow.net/feeds/question/95304 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95304/moving-under-the-influence-of-a-vector-field Moving under the influence of a vector field Joseph O'Rourke 2012-04-26T21:18:17Z 2012-04-27T01:47:08Z <p>I have a continuously varying vector field $v(p)$ on $\mathbb{R}^2$, and a particle at point $p$ in the plane that can move in a direction $u(p)$ as long as $u(p)$ is turned at most $\pi/2$ left of $v(p)$. So at any point $p$, the particle can move in a quarter-circle of directions: from $v(p)$ to $v(p)$ rotated $90^\circ$ counterclockwise.</p> <p>I would like to identify the points in $\mathbb{R}^2$ reachable from a given start point $p_0$ under this constraint. For example, suppose the vector field is determined by a rotation about a fixed center $c$. Then the reachable points are just those in the disk centered on $c$ with radius $|p_0 - c|$: <br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="http://cs.smith.edu/~orourke/MathOverflow/VecFieldQuestion.jpg" alt="Vector Field"><br /> I can write down equations, in terms of dot- and cross-product, but they are not revealing to me.</p> <blockquote> <p><b>Q</b>. Is there some clean formulation of this problem that suggests a computationally feasible identification of the reachable points?</p> </blockquote> <p>Thanks for any insights/ideas!</p> http://mathoverflow.net/questions/95304/moving-under-the-influence-of-a-vector-field/95317#95317 Answer by fuzzytron for Moving under the influence of a vector field fuzzytron 2012-04-27T01:47:08Z 2012-04-27T01:47:08Z <p>Sounds like a distribution except that instead of having linear subspaces you have cones. There's this paper: <a href="http://www.springerlink.com/content/p5650r4744324821/" rel="nofollow">Langerock, "Conic Distributions and Accessible Sets,"</a> but it sounds an awful lot like your question (and I wonder if that's where you're starting from in the first place!). It also doesn't say anything about the computability of the accessible set, though they do provide some characterization.</p>