Saddle surface z = x² − y²
Equation: z = x^2 - y^2
z = x² − y² is a hyperbolic paraboloid. At the origin both partial derivatives vanish, yet the Hessian has opposite-sign eigenvalues — a saddle.
Level sets z = h are hyperbolas. The surface appears in thin-shell roofs, differential geometry, and critical-point classification.