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  "body": "\n## Definition\n\nLet $n \\ge 1$, and let $\\R^{n+1,1} = \\R^n \\oplus \\R e_+ \\oplus \\R e_-$, where $\\R^n$ has the dot product, $e_+ \\cdot e_+ = 1$, $e_- \\cdot e_- = -1$, and the three summands are pairwise orthogonal.\nPut $n_0 = \\tfrac{1}{2}(e_- - e_+)$ and $n_\\infty = e_- + e_+$.\nThe *conformal geometric algebra* of $\\R^n$ is the [[geometric algebra|N8H460W3E2]] $\\mathcal{G}(\\R^{n+1,1})$, together with the embedding\n$$x \\mapsto X = x + \\tfrac{1}{2} \\abs{x}^2 n_\\infty + n_0, \\qquad x \\in \\R^n.$$\n\n## Remarks\n\nFrom $e_\\pm \\cdot e_\\pm = \\pm 1$ one gets $n_0 \\cdot n_0 = n_\\infty \\cdot n_\\infty = 0$ and $n_0 \\cdot n_\\infty = -1$.\nHence $X \\cdot X = \\abs{x}^2 - \\abs{x}^2 = 0$, so points become null vectors, and $X \\cdot Y = x \\cdot y - \\tfrac{1}{2}(\\abs{x}^2 + \\abs{y}^2) = -\\tfrac{1}{2} \\abs{x - y}^2$.\nExample: for $c \\in \\R^n$ and $r \u003e 0$, put $S = C - \\tfrac{1}{2} r^2 n_\\infty$ with $C$ the image of $c$.\nSince $X \\cdot n_\\infty = -1$, one gets $X \\cdot S = \\tfrac{1}{2}(r^2 - \\abs{x - c}^2)$, which vanishes exactly when $x$ lies on the sphere with centre $c$ and radius $r$.\nLikewise, for a unit vector $m \\in \\R^n$ and $\\delta \\in \\R$, the vector $m + \\delta n_\\infty$ satisfies $X \\cdot (m + \\delta n_\\infty) = x \\cdot m - \\delta$, and it represents the plane $x \\cdot m = \\delta$.\nNon-example: the vector $e_- \\in \\R^{n+1,1}$ is not the image of a point, since $e_- \\cdot e_- = -1 \\ne 0$.\nThe versors of the algebra, the products of invertible vectors, act on $\\R^{n+1,1}$ by conjugation, up to sign.\nThey induce exactly the Möbius transformations of $\\R^n \\cup \\set{\\infty}$, the maps generated by translations, rotations, dilations and inversions in spheres.\n",
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