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  "body": "\n## Definition\n\nLet $(\\mathcal{P}, \\mathcal{Z}, \\in)$ be an [[incidence structure|FGKNAFNHA3]] whose elements of $\\mathcal{P}$ are called points.\nThe elements of $\\mathcal{Z}$ are subsets of $\\mathcal{P}$, called circles.\nThe structure is a *Möbius plane* if:\n\n1. any three distinct points lie on exactly one circle;\n2. for every circle $z$, every point $P \\in z$ and every point $Q \\notin z$, there is exactly one circle $z'$ with $P, Q \\in z'$ and $z \\cap z' = \\set{P}$;\n3. every circle has at least three points, there is at least one circle, and there are four points not on a common circle.\n\n## Remarks\n\nClause 2 is the touching axiom: $z'$ touches $z$ at $P$.\n\nExample: the classical real Möbius plane has as points $\\R^2 \\cup \\set{\\infty}$, the [[Riemann sphere|0J78KBPFZB]].\nIts circles are the Euclidean [[circles|060W5CBZP6]] and the sets $g \\cup \\set{\\infty}$ for lines $g$.\nThree distinct points lie on exactly one such circle: on a line with $\\infty$ if they are collinear or one is $\\infty$, and on the circumcircle otherwise.\nGiven $z$, $P \\in z$ and $Q \\notin z$, the circles through $P$ tangent to $z$ at $P$ form a pencil, and exactly one of them passes through $Q$.\n\nNon-example: the Euclidean plane with only its circles as blocks fails clause 1, since three collinear points lie on no circle.\n\nThe plane sections of an [[ovoid|HNTE5FPS14]] in a three-dimensional projective space form a Möbius plane, called ovoidal; the classical plane comes from a sphere.\nA finite Möbius plane has an order $n$: every circle has $n + 1$ points and there are $n^2 + 1$ points.\n",
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