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  "body": "\n## Definition\n\nA *Minkowski plane* is an [[incidence structure|FGKNAFNHA3]] $(\\mathcal{P}, \\mathcal{Z}, \\in)$ of points and circles, where circles are subsets of $\\mathcal{P}$.\nIt comes with two equivalence relations $\\parallel_+$ and $\\parallel_-$ on $\\mathcal{P}$.\nTwo points are *parallel* if they are $(+)$-parallel or $(-)$-parallel.\nThe following conditions hold:\n\n1. for any points $P, Q$ there is exactly one point $R$ with $P \\parallel_+ R$ and $R \\parallel_- Q$;\n2. for every circle $z$, every point $P$ and each sign $\\pm$, exactly one point of $z$ is $(\\pm)$-parallel to $P$;\n3. any three pairwise non-parallel points lie on exactly one circle;\n4. for every circle $z$, every point $P \\in z$ and every point $Q \\notin z$ not parallel to $P$, there is exactly one circle $z'$ with $P, Q \\in z'$ and $z \\cap z' = \\set{P}$;\n5. every circle has at least three points, and there is a circle $z$ and a point not on $z$.\n\n## Remarks\n\nExample: let $L = \\R \\cup \\set{\\infty}$ be the real projective line.\nThe classical real Minkowski plane has points $L \\times L$, with $(x_1, y_1) \\parallel_+ (x_2, y_2)$ when $x_1 = x_2$ and $(x_1, y_1) \\parallel_- (x_2, y_2)$ when $y_1 = y_2$.\nIts circles are the graphs of the maps $x \\mapsto (ax + b)/(cx + d)$ of $L$ with $ad - bc \\ne 0$.\nClause 1 holds with $R = (x_1, y_2)$, and clause 2 holds because each circle is the graph of a bijection of $L$.\nClause 3 holds because for distinct $x_1, x_2, x_3$ and distinct $y_1, y_2, y_3$ there is exactly one such map with $x_i \\mapsto y_i$.\n\nNon-example: the same points and parallelisms with the graphs of all bijections of $L$ as circles fail clause 3, since many bijections take three given values at three given points.\n\nEquivalently, the classical plane comes from a hyperboloid of one sheet.\nIts circles are the plane sections that contain no line of the hyperboloid, and its two families of lines give the parallel classes.\n",
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