By which I mean, is there anyway that the intersetion of a plane and a hyperboloid of one sheet will be a parabola?
I know that intersecting a plane and a cone so that the plane is parallel to the edge of the cone will result in a parabola, but is there any way to do something similar with a hyperboloid of one sheet?
I am a little confused because I thought a hyperbola extended in a straight line as it goes to infinity (kind of like the graph of $y^2=x^2$), so I thought that a hyperboloid of one sheet would act like a cone (in that the edge is a straight line that heads to infinity).
Try $x^2+y^2-z^2=-1$, $z>0$ (one component of a two sheeted hyperboloid) or $x^2+y^2-z^2=1$ intersected with $x= z-1$.