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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $P_0, \\dots, P_n$ be points of $\\R^d$ (or real numbers), and let $t \\in \\R$.\n*De Casteljau's algorithm* computes points $P_i^{(j)}$ for $0 \\le j \\le n$ and $0 \\le i \\le n - j$ by\n\n1. $P_i^{(0)} = P_i$ for $i = 0, \\dots, n$;\n2. $P_i^{(j)} = (1 - t)\\, P_i^{(j-1)} + t\\, P_{i+1}^{(j-1)}$ for $j = 1, \\dots, n$ and $i = 0, \\dots, n - j$;\n\nand returns $P_0^{(n)}$.\n\n## Remarks\n\nThe output equals $B(t) = \\sum_{k=0}^{n} \\binom{n}{k} (1-t)^{n-k} t^k P_k$, the point of the [[Bézier curve|NEXBTGYYRG]] with these control points; by induction on $j$, $P_i^{(j)} = \\sum_{k=0}^{j} \\binom{j}{k} (1-t)^{j-k} t^k P_{i+k}$.\nFor $t \\in [0, 1]$ each step is a [[convex combination|21SXYJYC59]].\nExample: $n = 2$, $P_0 = (0, 0)$, $P_1 = (1, 2)$, $P_2 = (2, 0)$, $t = 1/2$: then $P_0^{(1)} = (1/2, 1)$, $P_1^{(1)} = (3/2, 1)$ and $P_0^{(2)} = (1, 1)$, which is $B(1/2)$.\nNon-example: replacing step 2 by the average $\\tfrac{1}{2}(P_i^{(j-1)} + P_{i+1}^{(j-1)})$ for all $t$ returns $B(1/2)$ no matter what $t$ is, so it does not compute $B(t)$ for $t \\ne 1/2$.\nThe points $P_0^{(0)}, P_0^{(1)}, \\dots, P_0^{(n)}$ and $P_0^{(n)}, P_1^{(n-1)}, \\dots, P_n^{(0)}$ are the control points of the curve restricted to $[0, t]$ and to $[t, 1]$, reparametrized to $[0, 1]$.\nThe algorithm costs $O(n^2)$ operations, more than Horner's scheme, but it is more stable.\n",
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