Let a topological sphere be formed from | z | ⩽ 1 |z| \leqslant 1 by dissecting the circumference into finitely many pairs ( I j , I j ′ ) ({I_j},{I’_j}) of disjoint arcs, identifying I j {I_j} and I j ′ {I’_j} in opposite directions and making further identifications among the endpoints. If there exists a meromorphic function Q ( z ) Q(z) , real and non-negative on | z | = 1 |z| = 1 and satisfying certain consistency conditions with respect to the dissection (given in detail in our Introduction), then one forms a Q-polygon by using the element of length i d s = Q ( z ) d z / z ds = \sqrt {Q(z)} \;dz/z to effect the metric identification of the pairs I j , I j ′ {I_j},{I’_j} . In a natural way, Q-polygons become Riemann surfaces, thus can be mapped conformally onto the number sphere. When Q is of the form Q ( z ) = Σ j = − N N B j z j Q(z) = \Sigma _{j = - N}^N{B_j}{z^j} , then the corresponding mapping functions, suitably normalized, become the extremal schlicht functions for the coefficient body V N + 1 {V_{N + 1}} [3, p. 120]. Suppose that for a given dissection of | z | = 1 |z| = 1 there is a family Q ( z , t ) Q(z,t) of consistent meromorphic functions. For Q sufficiently smooth as a function of ε \varepsilon , we study the variation of the corresponding normalized mapping functions f ( p , ε ) f(p,\varepsilon ) , using results of [2], and show smoothness of f as a function of ε \varepsilon . Specializing Q to the form above, we deduce the existence of smooth submanifolds of ∂ V N + 1 \partial {V_{N + 1}} and obtain a variational formula for the extremal schlicht functions corresponding to motion along these submanifolds.