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  "body": "\n## Definition\n\nLet $x_0 \u003c x_1 \u003c \\cdots \u003c x_n$ be real nodes, and let $p_0, \\dots, p_n$ and $m_0, \\dots, m_n$ be real numbers.\nLet $h_i = x_{i+1} - x_i$.\nThe *cubic Hermite spline* for these data is the function $s \\colon [x_0, x_n] \\to \\R$ given on each $[x_i, x_{i+1}]$ by\n$$s(x) = h_{00}(u)\\, p_i + h_{10}(u)\\, h_i m_i + h_{01}(u)\\, p_{i+1} + h_{11}(u)\\, h_i m_{i+1},$$\nwhere $u = (x - x_i)/h_i$ and\n$$h_{00}(u) = 2u^3 - 3u^2 + 1, \\quad h_{10}(u) = u^3 - 2u^2 + u, \\quad h_{01}(u) = -2u^3 + 3u^2, \\quad h_{11}(u) = u^3 - u^2.$$\n\n## Remarks\n\nThe piece on $[x_i, x_{i+1}]$ is the unique cubic with $s(x_i) = p_i$, $s'(x_i) = m_i$, $s(x_{i+1}) = p_{i+1}$ and $s'(x_{i+1}) = m_{i+1}$, that is, a [[Hermite interpolant|9Z50M2WPTP]].\nCheck: $h_{00}(0) = 1$, $h_{00}(1) = 0$, $h_{01}(1) = 1$, $h_{10}'(0) = 1$, $h_{11}'(1) = 1$, and the other values and slopes at $0$ and $1$ vanish.\nSo $s$ is a [[spline|84TJ8KQ926]] of class [[$C^1$|0VW90BBSJ4]], but in general not of class $C^2$.\nExample: nodes $0, 1$ with $p = (0, 1)$ and $m = (0, 0)$ give $s(x) = 3x^2 - 2x^3$.\nNon-example: the piecewise linear interpolant of the same values is not a cubic Hermite spline for these slopes, since its slope on $[0, 1]$ is $1$, not $0$.\nWhen the slopes are not given, they are chosen from the values, for instance by finite differences (Catmull–Rom spline) or by monotonicity-preserving rules.\n",
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