## 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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Prof. Joel Roberts
School of Mathematics
University of Minnesota
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