  We present and discuss several old and new methods for mapping a circular
disc to a square. In particular, we present analytical expressions for mapping
each point (u,v) inside the 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
to logo design, panoramic photography, and hyperbolic art.
