Let S be the torus obtained by revolving about the z-axis, the circle in the xz-plane with radius 1 centered at (2,0,0). The inside curvature is the region where $K<0$ given by $x^2+y^2<4$ and the outer curvature where $K>0$ given by $x^2+y^2>4$.
I need to give an alternate proof of this description using the equation $K={-x''/x}$, where K is the Gaussian curvature.
I'm completely stuck on where to begin. The surface obtained by revolving the trace of $\gamma(t)=(x(t),0,z(t))$ about the z-axis. Using the surface patch $\sigma(\theta,t)=(x(t)cos(\theta), x(t)sin(\theta),z(t))$.
I know this equation for K comes from parametrizing by the arc length letting $(x')^2+(z')^2=1$ and evaluating the equation $K=\frac{eg-f^2}{EG-F^2}$ after finding the 1st and 2nd fundamental forms.
Any help or direction would be great. Thanks