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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $S^n \\subset \\R^{n+1}$ be the unit [[sphere|T62369RFH6]], and let $x \\cdot y$ be the [[dot product|0S52TNNTH9]].\n*Elliptic $n$-space* is the [[real projective space|1DDYB6VVSR]] $\\mathbb{RP}^n = S^n / \\set{x \\sim -x}$ with these data:\n\n1. A point is a pair $\\set{x, -x}$ of antipodal points of $S^n$, written $[x]$.\n2. A line is the image of a great circle of $S^n$, the intersection of $S^n$ with a 2-dimensional linear subspace.\n3. The distance is $d([x], [y]) = \\arccos \\abs{x \\cdot y} \\in [0, \\pi/2]$.\n\n*Elliptic geometry* is the geometry of these spaces; for $n = 2$ it is the geometry of the elliptic plane.\n\n## Remarks\n\nThe distance is well defined, since replacing $x$ by $-x$ does not change $\\abs{x \\cdot y}$.\nExample: two distinct great circles of $S^2$ meet in exactly one antipodal pair $\\set{x, -x}$, so two distinct lines of the elliptic plane meet in exactly one point.\nTwo distinct points $[x] \\ne [y]$ span a 2-dimensional subspace, so they lie on exactly one line.\nNon-example: the sphere $S^2$ with great circles as lines, that is [[spherical geometry|SQ9RQVANMG]], does not satisfy the fact that two distinct points lie on exactly one line.\nThe antipodal points $(0, 0, 1)$ and $(0, 0, -1)$ lie on infinitely many great circles.\nFor this reason spherical geometry is sometimes called double elliptic geometry, and elliptic geometry single elliptic geometry.\nThe elliptic plane has constant curvature $1$, and the double cover $S^2 \\to \\mathbb{RP}^2$ is a local isometry.\n",
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