In a recent paper by the author a new geometric definition of deficient values for a function $\omega(z)$ meromorphic in $|z|<\infty$ was introduced, and with its aid a connection between the geometric structure of $F_r=\{\omega(z):|z|\leqslant r\}$ and the distribution of values of $\omega(z)$ was established. In the present paper definitions characterizing the structure of $\partial F_r$, more delicately are introduced, and a more detailed study of these connections is carried out. As a by-product a theorem of Miles is obtained as a corollary. This theorem complements, in a sense, Ahlfors' second fundamental theorem of the theory of covering surfaces. Bibliography: 3 titles.