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  "body": "\n## Definition\n\nLet $\\varphi(r) = r^2 \\log r$ for $r \u003e 0$ and $\\varphi(0) = 0$.\nLet $x_1, \\dots, x_N \\in \\R^2$ be distinct points, not all on one line, and let $y_1, \\dots, y_N$ be real numbers.\nThe *thin plate spline* interpolant of these data is the function\n$$s(x) = a_0 + a^T x + \\sum_{i=1}^{N} w_i\\, \\varphi(\\norm{x - x_i}), \\qquad x \\in \\R^2,$$\nwith $a_0 \\in \\R$, $a \\in \\R^2$ and $w \\in \\R^N$ determined by:\n\n1. $s(x_i) = y_i$ for $i = 1, \\dots, N$;\n2. $\\sum_{i=1}^{N} w_i = 0$;\n3. $\\sum_{i=1}^{N} w_i x_i = 0$ in $\\R^2$.\n\n## Remarks\n\nThe conditions form a square linear system, and it has exactly one solution when the points are not collinear.\nThis is the [[polyharmonic spline|XJ0JGPBC8R]] with $d = 2$ and $m = 2$, built from the [[radial basis function|ZBSXSTQM2J]] $\\varphi$.\nAmong all functions $f$ with square integrable second derivatives and $f(x_i) = y_i$, the spline $s$ minimizes the bending energy\n$$\\iint_{\\R^2} \\big( f_{x_1 x_1}^2 + 2 f_{x_1 x_2}^2 + f_{x_2 x_2}^2 \\big)\\, dx.$$\nAway from the points, $s$ satisfies the [[biharmonic equation|AS8A9F5SKE]] $\\Delta^2 s = 0$.\nExample: if $y_i = c_0 + c^T x_i$ for an affine function, then $w = 0$, $a_0 = c_0$, $a = c$ satisfy all three clauses, so $s$ is that affine function.\nNon-example: $f(x) = \\varphi(\\norm{x})$ with the single centre $0$ fails clause 2, since its only weight is $1$.\nThin plate splines are also used with one map per coordinate to define smooth deformations of the plane from matched landmark points.\n",
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