| HN Mirror

If you want to avoid weird seam artifacts, using 2 hemispheres of stereographic projection is probably better still. Each hemisphere projects to a disk but you can just fill out the texture to the boundaries of a square, duplicating a bit of each hemisphere into the corners of the other's texture (or you could leave those parts blank if you want). There's a 1:2 difference in scale from the center to the edge of each disk, so you could argue this is slightly wasteful of pixels for a given minimum required level of detail, but the projection is conformal so it's considerably less tricky to figure out how to sample it to decide the color for destination pixels drawn at steep perspective, and the stereographic projection is very cheap to compute in both directions (1 division + some additions and multiplications per projected point), even cheaper than the gnomonic projection used for a cubemap.

If you want something conformal that has less scale variation and wastes fewer corner pixels than a pair of stereographically projected hemispheres and is still not too conceptually tricky, you can use a pair of slightly overlapping Mercator projections, at right angles to each-other, covering the sphere like the two pieces of leather covering a baseball. Each one can have a rectangular texture. There are some NOAA papers suggesting this approach for the grids for solving differential equations needed in weather simulation of the Earth.

The most pixel-efficient projection I know starts by breaking the sphere into an octahedron, then taking each octant to be covered in a grid of hexagonal pixels, using "spherical area coordinates" in each octant to determine the grid. Each octant can then be represented in an ordinary square-pixel image by a half square ("45–45–90 right triangle"), so the result is something like this <https://observablehq.com/@jrus/sac-quincuncial> with a hexagon grid like <https://observablehq.com/@jrus/sphere-resample> (scroll a few examples down from the top of the page). But figuring out the details about how to sample the texture when you need to cross edge boundaries, etc., makes using this quite a bit more fiddly than the 2 stereographic projection version. And there will be some seam artifacts.