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A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
It was introduced by David Hilbert as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.
By measuring the orbit distance between the reference point and the point calculated with low precision, it can be detected that it is not possible to calculate the point correctly, and the calculation can be stopped. These incorrect points can later be re-calculated e.g. from another closer reference point.
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
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.
It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane.
The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, y) : y > 0 } with logarithmic measure. Let the hyperbolic distance from (0, y) to (0, 1) be a, so: log y − log 1 = a, so y = e a where e is the base of the natural logarithm.
In fact the quantity (A,B) C is just the hyperbolic distance p from C to either of the points of contact of the incircle with the adjacent sides: for from the diagram c = (a – p) + (b – p), so that p = (a + b – c)/2 = (A,B) C. [7] The Euclidean plane is not hyperbolic, for example because of the existence of homotheties.