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  "body": "\n## Statement\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field near a point $p \\in S$.\nLet $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] of $S$ at $p$, with orthonormal principal directions $e_1, e_2 \\in T_p S$.\nFor a unit vector $w \\in T_p S$, let $k_n(w) = I\\!I_p(w)$ be the normal curvature in the direction $w$, where $I\\!I_p$ is the [[second fundamental form|9KHQR16P1B]].\nIf $w = \\cos \\theta \\, e_1 + \\sin \\theta \\, e_2$, then\n$$k_n(w) = k_1 \\cos^2 \\theta + k_2 \\sin^2 \\theta.$$\n\n## Remarks\n\nThe proof is one line: $I\\!I_p(w) = \\ip{S_p w}{w}$ with the [[shape operator|DQCD11X1VM]] $S_p$, and $S_p e_i = k_i e_i$, so the cross terms vanish.\nIt follows that $k_1$ and $k_2$ are the maximum and the minimum of the normal curvature over all directions.\nExample: on a cylinder of radius $r$ with the inner normal, $k_1 = 1/r$ around the cylinder and $k_2 = 0$ along it.\nA direction at angle $\\theta$ to the circular sections has normal curvature $\\cos^2 \\theta / r$.\nThe normal curvature $k_n(w)$ equals the signed curvature of the normal section, the curve cut out of $S$ by the plane through $p$ spanned by $w$ and the normal.\n\n## Explanations\n\n### L1 human\n\nAt a point of a surface, the bending changes with the direction you look in.\nEuler's theorem says that this change follows a very simple rule.\nOnce you know the strongest and the weakest bending, and their directions, you know the bending in every other direction.\n\n### L2 human\n\nThe theorem is the diagonal form of the quadratic form $I\\!I_p$ in an orthonormal eigenbasis of the self-adjoint map $S_p$.\nIt explains why the Gaussian curvature $k_1 k_2$ and the mean curvature $(k_1 + k_2)/2$ capture the whole second-order shape, up to the choice of principal directions.\nThe sign of $k_1 k_2$ decides whether the normal curvature keeps one sign (elliptic point), changes sign (hyperbolic point), or vanishes in one direction (parabolic point).\nThe directions with $k_n(w) = 0$ at a hyperbolic point are the asymptotic directions, at angles $\\theta$ with $\\tan^2 \\theta = -k_1 / k_2$.\n",
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