Fundamental forms of an elliptic ring torus ($I=D^{2}d{u}^2+R^{2}d{v}^2$)

Rotate an ellipse with positive semiaxes $a,b$ about an axis a distance $r>a$ from its centre. For the parametrized surface $\sigma (u,v)=((r+a\sin u)\cos v,(r+a\sin u)\sin v,b\cos u)$, put $R=r+a\sin u$ and $D^2=a^2{\cos }^{2}u+b^2{\sin }^{2}u$. An outward unit normal is $N=(b\sin u\cos v,b\sin u\sin v,a\cos u)/D$. Direct differentiation gives the first fundamental form and second fundamental form, with the convention $I{I}_{ij}=⟨{\sigma }_{ij},N⟩$:Reversing the unit normal reverses $II$, but the Gaussian curvature $K=a{b}^2\sin u/(R{D}^4)$ is unchanged.