Suppose that a surface has equation z = f(x,y). A point P on the surface is a saddle-point if the tangent plane at P is horizontal and if P is a local minimum on the curve obtained by one vertical cross-section and a local maximum on the curve obtained by another vertical cross-section. It is so called because the central point on the seat of a horse’s saddle has this property. The hyperbolic paraboloid, for example, has a saddle-point at the origin. See also hessian, stationary point (in two variables).