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There is a theory of conformal map for circular polygons (polygons bounded by arcs of circles). But in this case, instead of an integral in the Schwarz-Christoffel formula, you obtain a linear differential equation. In the case of a circular triangle, the equation is hypergeometric and you have an explicit representation of your mapping. The paper of Harmer and Martin mentioned in the previous answer deals mainly with the case of a triangle. The most comprehensive treatment of triangles is in the second volume of Caratheodory's textbook on complex variables, and in other books on hypergeometric functions. The case of quadrilateral is the simplest case when there is no explicit formula. It was subject of much research. See, for example, arXiv:1110.2696, arXiv:1111.2296, and references in these papers.