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Excerpt from Conformal Image Warping
The Riemann mapping theorem guarantees existence and uniqueness of conformal mappings between regions. Given a region, there exists a conformal mapping of this region onto the unit disk. The mapping is made unique by fixing the mapping of a single point in the region onto the center of the unit disk, and fixing the orientation of the unit disk (see appendix 1 for a discussion of the Riemann mapping theorem).
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