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This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).
Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian ...
The unit hyperbola is blue, its conjugate is green, and the asymptotes are red. In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation = In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
The blue path in this image is an example of a hyperbolic trajectory. A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases ...
In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. [ citation needed ] More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface .
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form =, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates , the equation is represented by a hyperbola ; solutions occur wherever the curve passes through a point whose x and y ...
A ray through the unit hyperbola = in the point (,), where is twice the area between the ray, the hyperbola, and the -axis. The earliest and most widely adopted symbols use the prefix arc-(that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).