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  "body": "\n## Definition\n\nA *Benz plane* is an incidence structure of points and circles that is a [[Möbius plane|DED008EJVH]], a [[Laguerre plane|F0JQ1TGKB0]] or a [[Minkowski plane|VB2D03PD9Y]].\n\n## Remarks\n\nThe three types differ in their parallelism of points.\nA Möbius plane has no parallel points, a Laguerre plane has one parallelism, and a Minkowski plane has two.\nIn each type, three pairwise non-parallel points lie on exactly one circle, and a touching axiom holds.\n\nExample: the real plane with a point $\\infty$ added, with circles and lines through $\\infty$ as circles, is a Möbius plane, hence a Benz plane.\n\nNon-example: a projective plane, with its lines as circles, is not a Benz plane.\nThree non-collinear points lie on no line, so the axiom on three points fails for each of the three types.\n\nThe classical Benz planes are the geometries of plane sections of a sphere, a cylinder and a hyperboloid of one sheet.\n",
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