Abstract: In this paper, the conformal mapping problem on the transformation from the interior of a unit circle to the interior of the simply connected region or exterior with an arbitrary curvilinear boundary (including an arbitrary curvilinear cut crack) is discussed.The boundary of the simply connected region is approximated by a polygon.The mapping function from a unit circle to a polygon is founded by using the Schwartz-Christoffel integral.A numerical calculation method to determine the unknown parameters in the Schwartz-Christoffel integral is given.

Key words: arbitrary curve, mapping, Schwartz-Christoffel integral