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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\R^n$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ with [[curvature|7JQS6VKFEZ]] $\\kappa$.\nThe *total curvature* of $\\gamma$ is\n$$\\int_a^b \\kappa(t) \\, \\norm{\\gamma'(t)} \\, dt,$$\nthat is, the integral of $\\kappa$ with respect to arc length.\nFor a plane curve with signed curvature $k$, the *total signed curvature* is $\\int_a^b k(t) \\, \\norm{\\gamma'(t)} \\, dt$.\n\n## Remarks\n\nExample: the circle $\\gamma(t) = (r \\cos t, r \\sin t)$, $t \\in [0, 2\\pi]$, has $\\kappa = 1/r$ and $\\norm{\\gamma'} = r$, so its total curvature is $2\\pi$.\nNon-example: the boundary of a square is a [[closed curve|0D032NSZKY]] that is not of class $C^2$, so the definition does not apply.\nFor convex polygons one uses instead the sum of the exterior angles, which is again $2\\pi$.\nThe total curvature does not change under a regular reparametrisation or a rigid motion.\nFor a closed regular plane curve, the total signed curvature equals $2\\pi$ times the turning number of the curve.\n",
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