This figure shows some of the lines on a hyperboloid of one sheet, and a quadric cone that is inside the hyperboloid. In fact the hyperboloid is asymptotic to the cone at infinity. More specifically, the equation of the cone can be written in the form:
while the equation of the hyperboloid can be written in the following form:
The figure was drawn with = 0.5 and = 0.134.
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. Some lines from each family are shown in the figure.
The lines in one ruling are shown in red, and the lines in the other ruling are shown in blue.
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