Fast and Stable Conformal Mapping Between a Disc and a Square
Mapping between a square or rectangle to a disc or hemisphere, and vice versa, arises in many areas of computer graphics, including environment and reflection mapping, sampling, and BRDFs to name a few. Different maps have different properties: equal-area maps may be more applicable in sampling, while low-distortion or continuity might be preferable in other applications. Conformal mapping preserves angles and thereby locally preserves shape. Although it has been used for over a century, conformal mapping between a disc and a square involves extensive computation with complex numbers. This paper reviews the construction of a conformal map between the unit disc and the unit square, which is formulated as an elliptic integral, and reviews several computational methods. Efficient algorithms are presented for mapping the disc to the square, and from the square to the disc. An implementation is provided in compact C language source code that runs at speeds comparable to simple trigonometric maps.
Stark, Michael M.. (2009). "Fast and Stable Conformal Mapping Between a Disc and a Square". Journal of Graphics, GPU, and Game Tools, 14(2), 1-23.