The second and fourth intersections occur as the limaçon passes through the origin. For the circle, however, the origin is the point $\left(r\,\mathrm{\θ}\right)\=\left(0\,0\right)$, whence there is no $\left(r\,\mathrm{\θ}\right)$-pair for which the equations defining both curves are simultaneously satisfied.

As the slider in Figure 7.1.11(a) is moved, keep in mind that the polar angle starts from $-\mathrm{\π}$, and that for the top half of the inner loop of the limaçon, $r$ is negative, so that $\mathrm{\π}$ is added to each such negative polar angle prior to graphing the corresponding positive $r$.