Cube projection

A Cube projection is a projection of the world map on the surface of a cube.



A person making a cube projection has three arbitrary choices to make:
 * Where on Earth to put the corners of the cube?
 * What sort of projection to use to map the spherical squares to plane squares?
 * How to arrange the plane squares?

Where to put the corners of the cube?
Perhaps the simplest arrangement is to center the N. and S. poles in the center of the top and bottom squares; then put the 4 "vertical" edges along one meridian in the Atlantic, 2 meridians in the Pacific, and one meridian in the Indian ocean.

Others prefer to put the N. and S. poles at opposite corners of the cube, or some other oblique arrangement to minimize land cutting.

What sort of projection to use?
By far the most popular projection onto a cube is the gnomonic projection. The point scale near the corner is 3 times the scale at the center of the square, and the area distortion near the corner is $$3 \sqrt{3} \approx$$ 5.2.

A few people choose a projection with less distortion than gnomonic.


 * Laurence P. Lee developed a conformal mapping that can be used to map spherical squares to plane squares.
 * O'Neill and Laubscher use an equal-area mapping for each spherical square to a plane square for their "Quadrilaterized Spherical Cube". ( ... but what is that mapping? Is it the Irving Fisher/Snyder's equal-area projection? ).
 * A HEALPix map with H = 4 can be folded into a cube. It uses two different projections. A version of the equal-area Collignon projection to map two disks to two squares -- at the top and bottom of the globe. A different projection -- Lambert's equal-area cylindrical projection -- to map the remaining cylinder to 4 squares.
 * Some polygonal maps use the Chamberlin trimetric projection

How to arrange the plane squares?
The most natural arrangement is to fold the map into a cube.

By convention the "sky cube" is arranged in a "T" shape with four equatorial faces in a row and two polar faces above and below on the far right.

Chris Maynard chops one square into 4 triangles to reduce the cut length.