Graphically, a conformal map transforms any pair of curves intersecting at a point in the region so that the image curves intersect at the same angle.
Conformal maps are both mathematically interesting and practically useful.
Because of the preservation of angles, a conformal map of a square grid in the plane results in a curvilinear orthogonal grid. Furthermore, the transplantation by a conformal map of a solution to Laplace's equation is still a solution in the image region, so that conformal maps can be used in heat transfer, electrostatics, steady fluid flows, and other phyiscal applications in which Laplace's equation arises. In general it's impossible to write down a simple formula for the conformal map from one region to another. There are two catches, however. The first is that except for some very simple polygons, the formula requires an integral that has no closed form.