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  "body": "\n## Definition\n\nLet $k$ be a field and let $n \\ge 0$.\nA point $p$ of the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^n(k)$ is an [[equivalence class|0W95W33CMP]] of nonzero vectors of $k^{n+1}$.\nTwo vectors are equivalent when one is a nonzero scalar multiple of the other.\nA tuple $(x_0, \\dots, x_n) \\in k^{n+1} \\setminus \\set{0}$ in this class is a tuple of *homogeneous coordinates* of $p$, and one writes $p = [x_0 : \\dots : x_n]$.\n\n## Remarks\n\nTwo tuples are homogeneous coordinates of the same point exactly when they differ by a nonzero factor $\\lambda \\in k$.\n\nExample: in $\\mathbb{P}^2(\\R)$, $[1 : 2 : 3] = [-2 : -4 : -6]$.\nCheck: $(-2, -4, -6) = -2 \\cdot (1, 2, 3)$.\n\nNon-example: $(0, 0, 0)$ is not a tuple of homogeneous coordinates of any point, since the zero vector is excluded.\n\nThe affine point $(x_1, \\dots, x_n)$ of $k^n$ corresponds to $[x_1 : \\dots : x_n : 1]$; the points with last coordinate $0$ form the hyperplane at infinity.\nFor a homogeneous polynomial $F$, the condition $F(x_0, \\dots, x_n) = 0$ does not depend on the chosen coordinates of a point, since $F(\\lambda x) = \\lambda^d F(x)$ for $F$ of degree $d$.\n",
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