= Fundamental forms of an elliptic ring torus
{title2=$I=D^2du^2+R^2dv^2$}
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 $D^2=a^2\cos^2u+b^2\sin^2u$. 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 $II_{ij}=\langle\sigma_{ij},N\rangle$:
$$
I=D^2du^2+R^2dv^2,\qquad II=-\frac{ab}{D}du^2-\frac{bR\sin u}{D}dv^2.
$$
Reversing the <unit normal> reverses $II$, but the <Gaussian curvature> $K=ab^2\sin u/(RD^4)$ is unchanged.
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