The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere. It is an oblique aspect of the Peirce quincuncial projection.
The projection was developed by Émile Guyou of France in 1887.
The projection can be computed as an oblique aspect of the Peirce quincuncial projection by rotating the axis 45 degrees. It can also be computed by rotating the coordinates −45 degrees before computing the stereographic projection; this projection is then remapped into a square whose coordinates are then rotated 45 degrees.
The projection is conformal except for the four corners of each hemisphere’s square. Like other conformal polygonal projections, the Guyou is a Schwarz–Christoffel mapping.
Its properties are very similar to those of the Peirce quincuncial:
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
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