Answer
In rectangular coordinates, the equation becomes
${\left( {\frac{x}{1}} \right)^2} + {\left( {\frac{z}{1}} \right)^2} = {\left( {\frac{y}{1}} \right)^2} + 1$
This is the equation of a hyperboloid of one sheet.
Work Step by Step
Write
${r^2}\left( {1 - 2{{\sin }^2}\theta } \right) + {z^2} = 1$
${r^2} - 2{\left( {r\sin \theta } \right)^2} + {z^2} = 1$
Since ${r^2} = {x^2} + {y^2}$ and $y = r\sin \theta $, so
${x^2} + {y^2} - 2{y^2} + {z^2} = 1$
${x^2} - {y^2} + {z^2} = 1$
${\left( {\frac{x}{1}} \right)^2} + {\left( {\frac{z}{1}} \right)^2} = {\left( {\frac{y}{1}} \right)^2} + 1$
By Eq. (2) of Section 13.6, this is the equation of a hyperboloid of one sheet.
