Equal-Area and Quasi-Conformal Mappings between Squares, Discs and Fernández-Guasti Squircles

A new family of 2D mappings tackles one of graphics’ old shape-conversion problems: moving cleanly between squares, discs, and Fernández-Guasti squircles. The key change is that these mappings drop the usual radial constraint and instead use carefully chosen polar-angle behavior, letting them prioritize either exact area preservation or near-angle preservation.

For developers, the practical appeal is in the tradeoff space. The equal-area versions are mathematically expensive, relying on transcendental equations and incomplete elliptic functions, but they preserve shape much better along diagonals and reduce central angle distortion by roughly 1000x compared with Shirley mapping. On a modern laptop, a 1000 x 1000 image still finishes in under 2 seconds, though the method is 11–17x slower than Shirley’s.

The quasi-conformal versions are more immediately usable in production-style code. They use only elementary real-valued functions, and one tuned variant averages under 0.8 degrees of angle distortion while looking nearly identical to a true Schwarz-Christoffel transform. That version also runs up to 12% faster, which makes it interesting for real-time or batch warping pipelines where quality matters but full conformal math is too costly.

There’s also a broader animation angle here: by chaining square-to-disc and disc-to-square transforms, the same approach can morph between any two squareness values, or even create tunable warps by intentionally mismatching parameters on the forward and reverse passes. C++ and Octave implementations are available, so teams experimenting...