Hyperbolic Paraboloids

Hyperbolic paraboloids are the canonical example of a surface with a "saddle point" - a stationary point which is neither a maximum nor a minimum. At such points on surfaces, the Gaussian curvature is negative. Hyperbolic paraboloids are ruled surfaces. They also belong to the more specific family of "doubly ruled surfaces", alongside only hyperboloids of one sheet and (trivially) planes. Doubly ruled surfaces have two distinct straight lines through each point on them, and can be "swept out" by lines in two different ways. This property is exhibited in model III 15 below. Algebraically speaking, the surface $$z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$$ can be seen as doubly ruled because it allows the parameterisation $$(s, t) \mapsto ( a(s+t), b(s-t), 4st),$$ which is a line parameterised by $s$ when the value of $t$ is fixed, and vice-versa when $s$ is fixed. Other models exhibit lines of curvature, and intersections with planes. Each horizontal plane of the form $z = c$ intersects in a hyperbola, as demonstrated by model III 14  below. Model III 16 displays some lines of curvature on a hyperbolic paraboloid. Barkow Leibinger's ‘Loom-Hyperbolic’ art installation features a collection of woven saddles. 

 

Models: III 13; III 15; III 16

Model III 13Model III 15Model III 16

 

Model III 14 

Model III 14 - With horizontal planar cutsModel III 14 - With horizontal planar cutsModel III 14