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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field $N$ near a point $p \\in S$.\nLet $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$.\nSince $S_p$ is [[self-adjoint|0MB8HQW495]], it has two real [[eigenvalues|0ZND7NQ970]] $k_1 \\ge k_2$, counted with multiplicity.\nThese are the *principal curvatures* of $S$ at $p$.\nA unit eigenvector of $S_p$ is a *principal direction* at $p$.\n\n## Remarks\n\nThe principal directions can be chosen orthonormal, by the spectral theorem.\nEquivalently, $k_1$ and $k_2$ are the maximum and the minimum of the [[second fundamental form|9KHQR16P1B]] $I\\!I_p(w)$ over unit vectors $w \\in T_p S$.\nReplacing $N$ by $-N$ changes $k_1, k_2$ into $-k_2, -k_1$.\nExample: for the saddle $z = xy$ at the origin, with $N = (0, 0, 1)$, one has $I\\!I_0(a, b) = 2ab$ on $T_0 S = \\R^2 \\times \\set{0}$.\nSo $k_1 = 1$ and $k_2 = -1$, with principal directions $(1, 1, 0)/\\sqrt{2}$ and $(1, -1, 0)/\\sqrt{2}$.\nNon-example: on the same saddle, the normal curvature in the direction $(1, 0, 0)$ is $I\\!I_0(1, 0) = 0$, which is neither $k_1$ nor $k_2$, so $(1, 0, 0)$ is not a principal direction.\n",
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