Hyperbolic checkerboard

This saddle is a hyperbolic paraboloid, not the constant-curvature hyperbolic plane.

S(u,v) = (u, v, k·u·v)

Holding either parameter fixed gives a straight line. Two crossing families of these lines bound four-sided cells, colored by the parity of their grid indices.

These cells are quadrilaterals, not generally intrinsic squares. Each colored cell follows the curved surface. The saddle has variable negative Gaussian curvature when k ≠ 0.