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  "body": "\n## Example\n\nLet $n \\ge 1$ and give $\\R^{n+1}$, with coordinates $x = (x_0, x_1, \\ldots, x_n)$, the symmetric [[bilinear form|0SQZMXB6GP]]\n$$B(x, y) = x_0 y_0 - x_1 y_1 - \\cdots - x_n y_n.$$\nThe *hyperboloid model* of hyperbolic $n$-space is the upper sheet of a hyperboloid of two sheets,\n$$H^n = \\set{x \\in \\R^{n+1} : B(x, x) = 1, \\ x_0 \u003e 0},$$\nwith the distance $d(x, y) = \\operatorname{arcosh} B(x, y)$.\nOn each tangent space of $H^n$ the form $-B$ is positive definite, and the resulting [[Riemannian manifold|7CBT4SM83Y]] has constant sectional curvature $-1$.\nIts geodesics, the lines of [[hyperbolic geometry|GN4N94GQ0Z]], are the nonempty intersections of $H^n$ with 2-dimensional linear subspaces of $\\R^{n+1}$.\n\nCheck for $n = 1$: the curve $\\gamma(t) = (\\cosh t, \\sinh t)$ lies on $H^1$, since $\\cosh^2 t - \\sinh^2 t = 1$ and $\\cosh t \u003e 0$.\nIts velocity is $\\gamma'(t) = (\\sinh t, \\cosh t)$, and $-B(\\gamma', \\gamma') = \\cosh^2 t - \\sinh^2 t = 1$, so $t$ is arc length.\nThe formula gives $d(\\gamma(0), \\gamma(t)) = \\operatorname{arcosh}(\\cosh t) = \\abs{t}$, in agreement.\n\n## Remarks\n\nThe isometries of $H^n$ are the linear maps of $\\R^{n+1}$ that preserve $B$ and map the upper sheet to itself, the subgroup $O^+(1, n)$ of the [[indefinite orthogonal group|M2900DMEW6]].\nSo isometries act by matrices, which makes computations in this model linear algebra.\nThe central projection $x \\mapsto (x_1/x_0, \\ldots, x_n/x_0)$ maps $H^n$ onto the [[Beltrami–Klein model|1WN9B18H86]].\nThe projection $x \\mapsto (x_1, \\ldots, x_n)/(1 + x_0)$ from the point $(-1, 0, \\ldots, 0)$ maps $H^n$ onto the [[Poincaré disk model|85MM2EA0TK]] when $n = 2$, and onto the Poincaré ball model in general.\nFor $n = 3$, $\\R^4$ with the form $B$ is Minkowski spacetime, and $H^3$ is the set of unit future-pointing timelike vectors.\n\n## Explanations\n\n### L1 human\n\nTake the bowl-shaped sheet $x_0 = \\sqrt{1 + x_1^2 + x_2^2}$ in space, and measure lengths on it with a modified rule in which the vertical direction counts negatively.\nWith this rule the bowl becomes the hyperbolic plane.\nHyperbolic lines are the curves cut out of the bowl by planes through the origin, and rotations and Lorentz boosts move the bowl onto itself.\nThis is the hyperbolic analogue of the round sphere, which is the set of points at distance $1$ from the origin for the usual form $x_0^2 + x_1^2 + x_2^2$.\n",
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