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In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
For example, two points uniquely define a line, and line segments can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting ...
The free end of the string is ... an angle between two lines with equations = ... according to the hyperbolic cosine and hyperbolic sine, so for example ...
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
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.
In the Beltrami-Klein model of the hyperbolic geometry: two ultraparallel lines correspond to two non-intersecting chords. The poles of these two lines are the respective intersections of the tangent lines to the boundary circle at the endpoints of the chords. Lines perpendicular to line l are modeled by chords whose extension passes through ...
Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers , which are built around the unit circle, can be ...
A straight line in hyperbolic n-space is modeled by a geodesic on the hyperboloid. A geodesic on the hyperboloid is the (non-empty) intersection of the hyperboloid with a two-dimensional linear subspace (including the origin) of the n+1-dimensional Minkowski space. If we take u and v to be basis vectors of that linear subspace with