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  "body": "\n## Definition\n\nLet $n \\ge 1$ and give $\\R^{n+3}$, with coordinates $(x_1, \\ldots, x_{n+3})$, the symmetric [[bilinear form|0SQZMXB6GP]] of signature $(n+1, 2)$\n$$\\ip{x, y} = -x_1 y_1 + x_2 y_2 + \\cdots + x_{n+2} y_{n+2} - x_{n+3} y_{n+3}.$$\nThe *Lie quadric* is the [[quadric|Q3Q4FF23VG]] $Q = \\set{[x] \\in \\mathbb{RP}^{n+2} : \\ip{x, x} = 0}$ in the [[projective space|ZT9A6JK8QF]] $\\mathbb{RP}^{n+2}$.\nThe points of $Q$ correspond bijectively to the following objects of $\\R^n$:\n\n1. the oriented [[sphere|P9CM4GQ2T0]] with centre $p \\in \\R^n$ and signed radius $r \\ne 0$, and the point $p$ for $r = 0$, to $\\bigl[ \\tfrac{1 + \\abs{p}^2 - r^2}{2}, \\tfrac{1 - \\abs{p}^2 + r^2}{2}, p, r \\bigr]$;\n2. the oriented hyperplane $\\set{x : x \\cdot N = h}$ with unit normal $N$ and $h \\in \\R$, to $[h, -h, N, 1]$;\n3. the point at infinity, to $[1, -1, 0, 0]$.\n\nTwo such objects are in *oriented contact* when their points $[x], [y] \\in Q$ satisfy $\\ip{x, y} = 0$.\nA *Lie sphere transformation* is a projective transformation of $\\mathbb{RP}^{n+2}$ that maps $Q$ onto itself.\n*Lie sphere geometry* is the study of the properties of these objects that are invariant under Lie sphere transformations.\n\n## Remarks\n\nFor two oriented spheres with centres $p_1, p_2$ and signed radii $r_1, r_2$, let $k_1, k_2$ be their vectors as in clause 1; a computation gives\n$$\\ip{k_1, k_2} = -\\tfrac{1}{2} \\bigl( \\abs{p_1 - p_2}^2 - (r_1 - r_2)^2 \\bigr).$$\nSo they are in oriented contact exactly when $\\abs{p_1 - p_2} = \\abs{r_1 - r_2}$, that is when they are tangent with matching orientations at the point of contact.\nExample: in $\\R^2$, take the circle with centre $(0, 0)$ and radius $2$ and the circle with centre $(1, 0)$ and radius $1$.\nThey have $\\abs{p_1 - p_2} = 1 = \\abs{2 - 1}$, so they are in oriented contact; indeed they touch at $(2, 0)$.\nNon-example: the circles of radius $1$ with centres $(0, 0)$ and $(1, 0)$ have $\\abs{p_1 - p_2} = 1 \\ne 0 = \\abs{r_1 - r_2}$; they cross at two points and are not in oriented contact.\nA point ($r = 0$) is in oriented contact with a sphere exactly when it lies on it, and every hyperplane is in oriented contact with the point at infinity.\nThe group of Lie sphere transformations is $O(n+1, 2) / \\set{\\pm 1}$.\nThe Möbius transformations of $\\R^n \\cup \\set{\\infty}$ are the Lie sphere transformations that map points to points.\n",
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