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  "body": "\n## Summary\n\nA thin plate spline is a smooth function of two variables that interpolates scattered data with the least bending energy.\nIt is a sum of translates of the radial function $r^2 \\log r$ and an affine function, the case $d = 2$, $m = 2$ of a [[polyharmonic spline|XJ0JGPBC8R]].\nThe name comes from the physical picture of a thin metal sheet bent to pass through given heights.\nThin plate splines are widely used for surface fitting and for smooth image warping.\n\n## Explanations\n\n### L1 human\n\nPin a thin flexible sheet at a few points at given heights; it settles in the shape that bends the least.\nThe thin plate spline is the mathematical version of that shape.\nAway from the pins it satisfies the [[biharmonic equation|AS8A9F5SKE]], the equation of a bent elastic plate with no load on it.\nWith three pins that are not on a line, the sheet is simply the tilted plane through them, because a plane needs no bending at all.\n",
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