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  "body": "\n## Summary\n\nThe principal curvatures of a surface at a point are the largest and the smallest bending of the surface over all tangent directions.\nThey are the eigenvalues of the [[shape operator|DQCD11X1VM]], and the corresponding eigenvectors are the principal directions.\nTheir product is the Gaussian curvature and their average is the mean curvature.\n\n## Explanations\n\n### L1 human\n\nStand on a saddle-shaped surface and look in different directions.\nIn one direction the surface curves up the most, in another it curves down the most.\nThese two extreme amounts of bending are the principal curvatures.\nWhen they differ, the two directions are at right angles.\nOn a cylinder of radius $r$, one principal curvature is $1/r$, around the cylinder, and the other is $0$, along it.\n",
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