object is to study the matchings ${M}(f,\theta)$, as well as the pairs of matchings $({M}(f,\alpha),{M}(f,\beta))$.

Recall that $f$ is assumed to be monic, so that if $C = {C_{\theta}}(f)$ and $\theta \in [0,\pi)$, then for $r$ sufficiently large the point of $C \cap S_r$ which is labeled $k$ lies on the asymptote at angle $\frac{k\pi+\theta}{n}$.

\[rmk:tangency\] If $C = {C_{\theta}}(f)$ is nonsingular, we say a few words for future reference about when ${M}(C,r)$ is defined, i.e., when each connected component of $C(r)$ has precisely two points lying on the circle $S_r$. Each connected component of $C$ is diffeomorphic to ${{\mathbb R}}$; since a connected component of $C(r)$ is a bounded connected subset of $C$, it is either a single point, or diffeomorphic to the interval $[0,1]$. Certainly each connected component which passes through the interior of the disk $D_r$ must intersect the boundary at least twice. It follows that ${M}(C,r)$ is well-defined as long as $C$ has no point of tangency (either external or internal) to the circle $S_r$.

\[rmk:discriminant\] When $f(z)=z^2+bz+c$ is quadratic, the pair of matchings $({M}(f,0), {M}(f,\pi/2))$ is determined by the quadrant of the complex plane containing the discriminant $b^2-4c$. An analogous description must exist for polynomials 