This figure shows a finite portion of hyperboloid of one sheet. The hyperboloid of one sheet is a
a surface of degree 2 that contains infinitely many lines. In fact,
there are quadric ruled surface, i.e., 1-parameter families of lines on this surface.
twoClick here to see: - A drawing of the hyperboloid
in which some of these lines are explicitly shown.
- A drawing that shows some
of the lines on the hyperboloid, along with the quadric cone that is
asymptotic to the hyperboloid at
*z*= - A drawing of the hyperboloid
and one of its tangent planes
the intersection of the hyperboloid and the tangent plane is a reducible plane conic -- accordingly, the union of two lines in the tangent plane. {*Note that*this is true in situations where the tangent plane contains some real points of the surface other than the point of contact.} This explicitly shows why there are two families of lines on this surface.*At least,*
The other smooth quadric ruled surface, the hyperbolic paraboloid, also contains two 1-parameter families of lines. |

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*I made the figure on this page by substituting my own data in a
*Geometry Center* webpage.*

Prof. Joel Roberts

School of Mathematics

University of Minnesota

Minneapolis, MN 55455

USA

Office: 351 Vincent Hall

Phone: (612) 625-1076

Dept. FAX: (612) 626-2017

e-mail: `roberts@math.umn.edu
http://www.math.umn.edu/~roberts
`