A one-parametric method for determining parameters in the Schwarz–Christoffel integral

We propose a method for determining parameters in the Schwarz–Christoffel integral. The desired mapping embeds into a one-parametric family of conformal mappings of the upper half-plane onto the family of polygons which was obtained by shifting one or several vertices of some initial polygon with angle preservation. We consider the case when the family of polygons and the initial polygon have the same number of vertices; the case when the family of polygons has two mobile vertices coinciding at the initial moment and not coinciding with other vertices; and the other case that the family of polygons is a polygon with mobile cut. The problem of finding the parameters of a family of mappings is reduced to integrating some system of ordinary differential equations.