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The distance between two points in the half-plane model can be computed in terms of Euclidean distances in an isosceles trapezoid formed by the points and their reflection across the x-axis: a "side length" s, a "diagonal" d, and two "heights" h 1 and h 2.
The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers. The model generalizes directly to an additional dimension: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.
Fixing a point yields a natural distance on : two points represented by rays , originating at are at distance ( ()). When X {\displaystyle X} is the unit disk, i.e. the Poincaré disk model for the hyperbolic plane, the hyperbolic metric on the disk is
Choose a line (the x-axis) in the hyperbolic plane (with a standardized curvature of -1) and label the points on it by their distance from an origin (x=0) point on the x-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x-axis.
Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the scalar square. If the signature (+, −, −) is chosen, scalar square between distinct points on the hyperboloid will be negative, so various definitions of basic terms must be adjusted, which can be inconvenient.
The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point. The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of ...
Specializing to the case where one of the points is the origin and the Euclidean distance between the points is r, the hyperbolic distance is: (+) = where is the inverse hyperbolic function of the hyperbolic tangent. If the two points lie on the same radius and point ′ = (′,) lies between the origin and point = (,), their hyperbolic ...
In the second case (−1 in the right-hand side of the equation): a two-sheet hyperboloid, also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point. The surface is convex in the sense that the tangent plane at every point intersects the surface only in this point.