Consider the ellipsoid S of equation $x^2 + \frac{y^2}{4} + \frac{z^2}{9} = 1$. For each of the six points of intersection with the coordinate axes compute: the principal curvatures and the principal directions. Use symmetry to reduce the number of computations.
2026-03-31 18:42:30.1774982550
How to calculate the Principal Curvature of an ellipsoid with given equation?
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Parameterize the surface (for instance, by solving for one of the variables -- which one will depend on the point), compute the shape operator using the Weingarten equations (see http://en.wikipedia.org/wiki/Shape_operator#Shape_operator), and diagonalize it. The eigenvalues are the principal curvatures and the eigenvectors the principal directions.
To parameterize the surface, solve for one of the variables. For example,
$$x = \sqrt{1 - y^2/4-z^2/9}$$ is valid for the part of the surface near $(1,0,0)$. Then $$r(u,v) = \left(\sqrt{1 - u^2/4-v^2/9}, u, v\right)$$ describes the surface near $(1,0,0)$.