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The inverse statement is ... The tangent vector ... Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example ...
Most hyperbolic surfaces have a non-trivial fundamental group π 1 = Γ; the groups that arise this way are known as Fuchsian groups. The quotient space H 2 / Γ of the upper half-plane modulo the fundamental group is known as the Fuchsian model of the hyperbolic surface. The Poincaré half plane is also hyperbolic, but is simply ...
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola , as can be seen when graphed on a Minkowski diagram whose coordinates represent a suitable inertial (non-accelerated) frame.
Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry. [ 1 ] This article exhibits these examples of the use of hyperbolic motions: the extension of the metric d ( a , b ) = | log ( b / a ) | {\displaystyle d(a,b)=\vert \log(b/a)\vert } to the half-plane and the unit disk .
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.
In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. Example hyperbolic flow, illustrating stable and unstable manifolds. The vector field equation is (+ (),). The stable manifold is the x-axis, and the unstable manifold is the other asymptotic curve crossing the x-axis.
The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.
where α is a basis vector orthogonal to the hyperboloid axis. For example, he obtained the hyperbolic law of cosines through use of his Algebra of Physics. [1] H. Jansen made the hyperboloid model the explicit focus of his 1909 paper "Representation of hyperbolic geometry on a two sheeted hyperboloid". [15]