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Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of coefficients. The center ( x c , y c ) {\displaystyle (x_{c},y_{c})} of the hyperbola may be determined from the formulae
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes.A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
Similarly, a given mass M of gas with changing volume will have variable density δ = M / V, and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the quadrant of absolute temperature and gas density. For hyperbolic coordinates in the theory of relativity see the History section.
The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis (b), the same value b as in the standard equation below. By analogy, for a hyperbola the parameter b in the standard equation is also called the semi-minor axis. The following relations hold: [6] = =
Through a change of coordinates (a rotation of axes and a translation of axes), equation can be put into a standard form, which is usually easier to work with. It is always possible to rotate the coordinates at a specific angle so as to eliminate the x′y′ term. Substituting equations and into equation , we obtain
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.
Conservation form or Eulerian form refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term is usually used in the context of continuum mechanics.