I am trying to review, and learn about how to compute and gaussian and mean curvature. Given $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, how can I compute the gaussian and mean curvatures?
This is what I have so far,
$$K(u, v) = \frac{a^2 b^2 c^2}{[c^2 \sin^2(v) (a^2 \sin^2(u)+b^2 \cos^2(u))+a^2 b^2 \cos^2(v)]^2}$$
Please help me out.
Best Answer
The parametric equations of an ellipsoid can be written as \begin{align} x &=a \cos u \sin v\\ y &=b \sin u \sin v\\ z &=c \cos v \end{align} for $u \in [0,2\pi)$ and $v \in [0,\pi]$. In this parametrization, the coefficients of the first fundamental form are \begin{align} E &= (b^2\cos^2u+a^2\sin^2u)\sin^2v\\ F &= (b^2-a^2)\cos u\sin u\cos v\sin v\\ G &= (a^2\cos^2u+b^2\sin^2u)\cos^2v+c^2\sin^2v \end{align} and of the second fundamental form are \begin{align} e &= \frac{abc\sin^2v}{\sqrt{a^2b^2\cos^2v+c^2(b^2\cos^2u+a^2\sin^2u)\sin^2v}}\\ f &= 0\\ g &= \frac{abc}{\sqrt{a^2b^2\cos^2v+c^2(b^2\cos^2u+a^2\sin^2u)\sin^2v}}. \end{align} In this parametrization, the Gaussian curvature is $$ K(u,v)=\frac{eg-f^2}{EG-F^2}=\frac{a^2b^2c^2}{[a^2b^2\cos^2v+c^2(b^2\cos^2u+a^2\sin^2u)\sin^2v]^2 } $$ and the mean curvature is \begin{align} H(u,v) &=\frac{eG-2fF+gE}{2(EG-F^2)}\\ &=\frac{abc[3(a^2+b^2)+2c^2+(a^2+b^2-2c^2)\cos(2v)-2(a^2-b^2)\cos(2u)\sin^2v]}{8[a^2b^2\cos^2v+c^2(b^2\cos^2u+a^2\sin^2u)\sin^2v]^{3/2}}. \end{align}