Let $X^n\subset \Bbb C\Bbb P^{n+1}$ be a hypersurface defined as the zero set of a degree $d$ polynomial with $d\leq n$. Such hypersurfaces have lines through each point $x\in X$. Let $\mathcal C_x\subset \Bbb P(T_xX)$ denote the set of tangent directions to lines on $X$ passing through $x$. Jun-Muk Hwang asked how $\mathcal C_x$ varies as one varies $x$. The answer turns out to be interesting, with two natural exterior differential systems governing the motion. In addition to describing these EDS and some immediate consequences, I will also discuss applications to questions in computational complexity and algebraic geometry. This is joint work with C. Robles.