A hyperboloid of one sheet

   This figure shows a finite portion of hyperboloid of one sheet. The hyperboloid of one sheet is a quadric ruled surface, i.e., a surface of degree 2 that contains infinitely many lines. In fact, there are two 1-parameter families of lines on this surface.
Click 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
    Note that the intersection of the hyperboloid and the tangent plane is a reducible plane conic -- accordingly, the union of two lines in the tangent plane. {At least, 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.

   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

Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu