How to find the curve and axis?

Question

We consider the following one-sheeted hyperboloid:

$y^2-4x^2+4z^2=4$

This is also a surface (or solid) of revolution. So it must be generated by rotating a curve about an axis.

What curve and axis could be those? How can I find them?

Thanks.

Answer 1

Perhaps you wanted to find out about

$$ 4 x^2 - y^2 + 4 z^2 = 4 $$

This one is a surface of a 1-sheet hyperboloid of revolution because,

if the coefficient of y^2 is 0, you immediately recognize unit circle

$$ x^2 + z^2 = 1 $$

Inspection of coefficients and their sign is important.

It has the y-axis as axis of symmetry.Note the coefficients of $x^2$ and $ z^2 $ are same.

Take sections parallel to x-z plane at a distance or height of $ y =y_1 $

$$ x^2 + z^2 = 1 + y_1^2 /4 $$

The right hand side has a radius increasing above 1 whether y1 is > 0 or y1 < 0.

Next, take section $ z=0. $ This gives you

$$ 4 x^2 - y^2 = 4 $$ which is the hyperbola which is rotated about y-axis.