S_r$ lies on a different one of these asymptotes, so that which pairs of points $C_\theta(f) \cap
S_r$ are joined by arcs in $C_\theta(f) \cap D_r$ is determined entirely by which pairs of asymptotes lie on the same connected component of $C_\theta(f)$.

If $C$ is any real plane curve, write $C(r) = C \cap D_r$. Suppose that $C(r)$ is nonsingular and has $m$ connected components, each with precisely two points lying on the circle $S_r$. Label these points counterclockwise from $0$ to $2m-1$, starting from the positive real axis. We make the following exception to the labeling rule: if $C=C_0(f)$ for some polynomial $f$ and $r$ is sufficiently large that $C(r)$ meets the component of $C$ asymptotic to the positive real axis, then we begin labeling from the point of $C(r)$ furthest along this asymptote.

Then $C(r)$ induces a matching ${M}(C,r)$ on the set $\{0,\ldots,2m-1\}$, where $a,b$ are matched if and only if the points labeled $a$ and $b$ lie on the same connected component of $C(r)$. If $C = {C_{\theta}}(f)$ is nonsingular and $r$ is sufficiently large, then we have observed that the matching ${M}({C_{\theta}}(f),r)$ on $\{0,\ldots,2n-1\}$ does not depend on $r$; we denote this matching by ${M}(f,\theta)$. Our 