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(presented in Seoul ICM 2014) We present and analyze several old and new methods for mapping the disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside a circular disc to a point (x, y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphasis on mappings with desirable properties. These include conformal, equiareal, and radially-constrained mappings. Finally, we present applications of such mappings to cartography, panoramic photography, and hyperbolic art.
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