# Moving under the influence of a vector field

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.

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 $|p_0 - c|$:

I can write down equations, in terms of dot- and cross-product, but they are not revealing to me.

Q. Is there some clean formulation of this problem that suggests a computationally feasible identification of the reachable points?

Thanks for any insights/ideas!

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What should your constraint mean at a zero of the vector field? If you don't allow zeros, I'd say you have a Lorentz structure on the plane, and you're asking about the causal relationships between points. – Ryan Budney Apr 26 '12 at 21:21
@Ryan: Good question! I must allow zeros, for rotations about a point are among my vector fields. I guess then $u(p)$ must also be zero: the particle stops and stays there. – Joseph O'Rourke Apr 26 '12 at 21:31
@Ryan: Thanks for the "Lorentz structure" hint; that connection did not occur to me. – Joseph O'Rourke Apr 26 '12 at 21:34
Okay, then you're looking at the causal structure on the plane minus the zeros of the vector field / Lorentz structure. – Ryan Budney Apr 27 '12 at 6:45