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  "body": "\n## Definition\n\nLet $(\\mathcal{P}, \\mathcal{Z}, \\in)$ be an [[incidence structure|FGKNAFNHA3]] of points and circles, where circles are subsets of $\\mathcal{P}$.\nLet $\\parallel$ be an equivalence relation on $\\mathcal{P}$, called parallelism.\nThe structure is a *Laguerre plane* if:\n\n1. any three pairwise non-parallel points lie on exactly one circle;\n2. for every circle $z$, every point $P \\in z$ and every point $Q \\notin z$ with $P \\not\\parallel Q$, there is exactly one circle $z'$ with $P, Q \\in z'$ and $z \\cap z' = \\set{P}$;\n3. for every circle $z$ and every point $P$, there is exactly one point $P' \\in z$ with $P \\parallel P'$;\n4. every circle has at least three points, and there is a circle $z$ and a point not on $z$.\n\n## Remarks\n\nExample: the real parabola model has points $\\R^2$, parallel classes the vertical lines $x = \\text{const}$, and circles the graphs $y = ax^2 + bx + c$ with $a, b, c \\in \\R$.\nClause 3 holds because each circle is the graph of a function of $x$.\nClause 1 holds because through three points with distinct $x$-coordinates there is exactly one polynomial of degree at most $2$.\nFor clause 2, two such polynomials whose graphs touch at $P = (x_0, y_0)$ differ by $\\lambda (x - x_0)^2$, and exactly one value of $\\lambda$ makes the graph pass through $Q$.\n\nNon-example: keep the same circles but let parallelism be equality of points.\nThen three points on a vertical line are pairwise non-parallel and lie on no circle, so clause 1 fails.\n\nThe classical model is the set of plane sections of a circular cylinder not parallel to its axis, with the generators of the cylinder as parallel classes.\nMore generally, an oval cone in three-dimensional projective space with its vertex removed gives an ovoidal Laguerre plane.\n",
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