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This figure shows a finite portion of hyperboloid of one sheet, together with a portion of 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. At least, this is true in situations where where the tangent plane contains some real points of the surface other than the point of contact. And, in fact, this condition is satisfied for the hyperboloid of one sheet as well as the hyperbolic paraboloid. Since this condition holds for every tangent plane, 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. Note: Viewers are strongly encouraged to rotate this figure (drag it with the mouse, as usual). Indeed, this is the only way to really see what's happening here. Click here to go back to the main drawing of the hyperboloid of one sheet. |
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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