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For instance, the cylinder and cone are developable, but the general hyperboloid of one sheet is not.
The surface of a hyperboloid of one sheet.
The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces.
Ruled surfaces are surfaces that have at least one straight line running through every point; examples include the cylinder and the hyperboloid of one sheet.
In particular, when and move with constant speed along two skew lines, the surface is a hyperbolic paraboloid, or a piece of an hyperboloid of one sheet.
Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive.
The exterior of the imposing structure is strongly characterised by its circular plan and by the external wall, in the shape of a hyperboloid of one sheet, without any openings.
The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively.
A hyperboloid of one sheet is a doubly ruled surface; if it is a hyperboloid of revolution, it can also be obtained by revolving a line about a skew line.
Remark: A suitable stereographic projection shows: is isomorphic to the geometry of the plane sections on a hyperboloid of one sheet (quadric of index 2) in projective 3-space over field .
If one rotates a line L around another line L' skew but not perpendicular to it, the surface of revolution swept out by L is a hyperboloid of one sheet.
Like the hyperboloid of one sheet, the hyperbolic paraboloid has two families of skew lines; in each of the two families the lines are parallel to a common plane although not to each other.
Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid (which is often referred to as the saddle surface or "the standard saddle surface") and the hyperboloid of one sheet.
But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2).
For example, a hyperboloid of one sheet is a quadric surface in P ruled by two different families of lines, one line of each passing through each point of the surface; each family corresponds under the Plücker map to a conic section within the Klein quadric in P.
The one-sheet hyperboloid has two positive eigenvalues and one negative eigenvalue.
The proof makes use of the property that for every conic section we can find a one-sheet hyperboloid which passes through the conic.
