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  "body": "\n## Definition\n\nLet $S \\subset \\R^3$ be a [[regular surface|6080X3FAD4]] with [[Gaussian curvature|SQVQ5DQT6W]] $K \\colon S \\to \\R$.\nThe surface $S$ is *developable* if $K(p) = 0$ for every point $p \\in S$.\n\n## Remarks\n\nThe cylinder $x^2 + y^2 = 1$ is developable.\nIts principal curvatures are $1$ along the horizontal circles and $0$ along the vertical lines, so $K = 1 \\cdot 0 = 0$.\nThe sphere of radius $r$ is not developable, since $K = 1/r^2 \\neq 0$ at every point.\n\nBy the [[Theorema Egregium|YKSA8SJ81V]], $K$ depends only on the first fundamental form.\nA surface with $K = 0$ is locally isometric to the plane, and conversely.\nA developable surface without planar points is a ruled surface.\nAway from a small exceptional set, it is locally a piece of a cylinder, a cone or a tangent developable.\nSome authors define a developable surface as a ruled surface with $K = 0$.\n\n## Explanations\n\n### L1 human\n\nA developable surface is one with zero Gaussian curvature everywhere.\nAt each point one of the two principal curvatures is zero, so the surface bends in only one direction at a time.\n",
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