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  "body": "\n## Definition\n\nWork in the Euclidean plane $\\R^2$ with the [[dot product|0S52TNNTH9]].\n\n1. A *cycle* is a pair $(c, r)$ with $c \\in \\R^2$ and $r \\in \\R$. For $r \\ne 0$ it is the circle with centre $c$ and radius $\\abs{r}$, oriented by the sign of $r$. For $r = 0$ it is the point $c$.\n2. A *spear* is an oriented [[line|0MP9BCF6DP]], given by a unit vector $N \\in \\R^2$ and $h \\in \\R$ as the line $\\set{x : x \\cdot N = h}$ with the orientation determined by $N$.\n3. A cycle $(c, r)$ and a spear $(N, h)$ are in *oriented contact* when $c \\cdot N - h = r$.\n\nA *Laguerre transformation* is a pair of bijections, one of the set of spears and one of the set of cycles.\nIt is required that a cycle and a spear are in oriented contact if and only if their images are.\n*Laguerre geometry* is the study of the properties of spears and cycles that are invariant under all Laguerre transformations.\n\n## Remarks\n\nTwo cycles $(c_1, r_1)$ and $(c_2, r_2)$ touch with matching orientations when $\\abs{c_1 - c_2} = \\abs{r_1 - r_2}$.\nIn the cyclographic model, the cycle $(c, r)$ is the point $(c, r) \\in \\R^3$, and this condition says that the difference of the two points is a null vector of the form $x^2 + y^2 - z^2$.\nExample: for $t \\in \\R$, the map sending the cycle $(c, r)$ to $(c, r + t)$ and the spear $(N, h)$ to $(N, h - t)$ is a Laguerre transformation.\nCheck: $c \\cdot N - (h - t) = r + t$ holds exactly when $c \\cdot N - h = r$, and both maps are bijections.\nIt sends each point $(c, 0)$ to a circle of radius $\\abs{t}$, so it is not a map of points.\nNon-example: inversion in the unit circle is not a Laguerre transformation in this sense.\nIt sends the lines not through the origin to circles, so it does not map spears to spears.\nEuclidean isometries and dilations $x \\mapsto \\lambda x$, $\\lambda \u003e 0$, acting on centres and radii, are also Laguerre transformations.\n",
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