How to Find Access of Symmetry in Quadric Surface?

[Rapier]

Homework Statement

Construct a hyperboloid of one sheet whose axis of symmetry is the y-axis.

Homework Equations

Hyperboloid of One Sheet --> x^2 + y^2 - z^2 = c

The Attempt at a Solution

The relevant equation is the one given in the book and in my notes. Obviously I can have this shape in other axes. I know that I am supposed to try and visualise the shapes in 3space, but it's a skill that I find very difficult to grasp.

In the above equation, I've got in:
x/y plane: circle
x/z plane: hyperbola
y/z plane: hyperbola

I'm unsure how to determine the axis of rotation and would be grateful for any help in visualising the shape.

 

[LCKurtz]

Homework Statement

Construct a hyperboloid of one sheet whose axis of symmetry is the y-axis.

Homework Equations

Hyperboloid of One Sheet --> x^2 + y^2 - z^2 = c

The Attempt at a Solution

The relevant equation is the one given in the book and in my notes. Obviously I can have this shape in other axes. I know that I am supposed to try and visualise the shapes in 3space, but it's a skill that I find very difficult to grasp.

In the above equation, I've got in:
x/y plane: circle
x/z plane: hyperbola
y/z plane: hyperbola

I'm unsure how to determine the axis of rotation and would be grateful for any help in visualising the shape.

Well, you have noticed the trace in the z=0 plane is a circle. Or is it? Does it matter if c > 0 or not? But anyway, what about traces in z = constant planes other than the xy plane. You will get circles, or nothing, depending on the relative size of z and c and whether the hyperboloid has one or two sheets. But, to answer your question, the axis that gives the centers of the circular cross sections is the axis of rotation.