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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.
Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value u, and all points on the same red hyperbola share the same coordinate value v. In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane
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 curve represents xy = 1. A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if a = 1. In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane.
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.
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.
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L , one can construct a hypercycle by taking all points Q on the same side of L as P , with perpendicular distance to L equal to that ...
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane.