The theory of Schwarz-Christoffel transformations in complex analysis provides a conformal way of mapping a circular disc to a square and vice versa. Here is a fundamental diagram for this mapping in the complex plane.
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| Conformal map from unit circular disc to square & vice versa |
□→ ◯
To map from square to circular disc, one needs the complex-valued Jacobi elliptic function cn.
http://mathworld.wolfram.com/JacobiEllipticFunctions.html
To map from circular disc to square, one needs the complex-valued Legendre elliptic integral of the 1st kind F.
http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html |
For a mathematical derivation of this diagram, see Section 8 of my paper in
https://arxiv.org/ftp/arxiv/papers/1509/1509.06344.pdf
It is relatively easy to verify this diagram through Mathematica. One can also do it online through Wolfram Alpha. Mathematica has built-in functions JacobiCN and EllipticF that can take-in complex-valued inputs.
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| Using the Jacobi cn function in Wolfram Alpha |
Unfortunately, this fundamental diagram has the square off-center and tilted by 45ยบ . For many applications, it is preferable to have the square centered and oriented correctly. My previous blog entry has equations for this
https://squircular.blogspot.com/2015/09/schwarz-christoffel-mapping.html
