voxel spheres in block games
Building a perfect sphere out of blocks is impossible: a block is a cube, and no arrangement of cubes reproduces a curved surface without approximation. What rasterization does is test every block on a cubic grid against the sphere's equation and decide inside or outside, layer by layer, so the result looks round enough to the eye, even though it is, underneath, a stack of steps.
Rasterization always adds a bit of excess at the edge; the overage percentage shrinks as the diameter grows.
The shell keeps only blocks touching at least one empty neighbor across the 6 orthogonal directions, about 29% of the filled total.
The circle uses the same logic in 2D that the sphere uses in 3D: distance from the block's center to the shape's center, compared to the radius squared.
Generate the sphere at the diameter you want and build level by level, bottom to top, following the layer shown for each height. Start from the middle layer (the widest) to orient yourself and close in toward the top and bottom. The hollow version saves many blocks.
Because the 4/3·π·r³ formula describes a smooth continuous sphere, while rasterization includes a whole block whenever its center falls within the radius, even if only a fraction of the cube would technically be inside; the gap typically runs between 1% and 5%, either direction depending on the diameter.
64 for sphere and dome; the flat circle, being computationally lighter, accepts up to a 128 diameter.
Because no block's own center lands exactly on the shape's geometric middle when the diameter is even; each block's center always sits at an integer position plus 0.5, so only odd diameters align a block with the exact center.
It depends on the diameter, but for the diameter-16 sphere the shell uses about 29% of the filled version's total blocks (632 versus 2,176), since the entire core is left empty.
The shape is computed locally in the browser; nothing is sent to servers.