Poincaré checkerboard

Poincaré disk: two crossing families of hyperbolic geodesics partition the disk. A Cartesian grid in Klein coordinates K=2P/(1+|P|²) becomes circular arcs meeting the rim orthogonally. Colors alternate across each line.

This is a finite irregular geodesic checkerboard, not a regular {4,q} tiling. Interior cells are quadrilaterals; cells at the rim are clipped. The rim is infinitely far away in the hyperbolic metric.

Hemisphere: a longitude–latitude checkerboard on the upper half of a sphere. Meridians meet at the pole, so polar cells narrow to triangular tips. Latitudes are generally not geodesics. Shading is optional.

The two modes are different geometries, not distance-preserving transformations of one another.