Saddle surface of a hyperbolic paraboloid

A hyper­bolic parab­o­loid is a surface, which looks like a saddle. It is formed during such a move­ment of a parabola open down that its top glides on the other static parabola open up. Planes containing parabolas are perpen­dic­ular at every moment, while the axes are parallel.

When a hyper­bolic parab­o­loid crosses any hori­zontal plane, a hyper­bola is formed. If a plane passes through the centre of a saddle, a hyper­bola degen­er­ates into a pair of inter­secting straight lines (if we project a hyper­bola from a parallel section on this plane, the straight lines will be an asymp­tote of a hyper­bola-projec­tion).

It turns out that a hyper­bolic parab­o­loid is a ruled surface, which can be also formed by move­ment of a straight line!

Let’s draw a set of segments between two parallel lines at equal distances. Then let’s rotate straight lines around the central segment in oppo­site direc­tions (lengths of all segments, except the central, will change). So the segments posi­tioned in space this way all lie on a hyper­bolic parab­o­loid.

This surface can be beau­ti­fully real­ized as a model made of thin tubes.

The Gaussian curva­ture is nega­tive at every point of a hyper­bolic parab­o­loid. These surfaces are also called saddle, as they are visu­ally similar to saddles for horse-riding.

Other models in “Conic sections”