The two families of lines on a smooth quadric

This figure shows some lines from each of the two families of lines on the hyperbolic paraboloid  z = xy.
The lines from one family are shown in red, and the lines from the other family are shown in blue.
 
It is fairly easy to use the equation of the surface to find the lines.  Indeed, if we make the substitution  x = a
then we get the line that is given by the system of linear equations    x = a,     z = ay.  In a similar way, the
substitution  y = b  leads to the line that is given by the system of linear equations    y = b,     z = bx
 
As usual, you can rotate the figure by grabbing it with the mouse. To return to the home position
at any time, just type "h".
 
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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