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Hyperbola: definition by the distances of points to two fixed points (foci) ... The graph of the equation = / is a rectangular hyperbola. Parametric representation ...
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).
Hyperbolic may refer to: . of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics . Hyperbolic geometry, a non-Euclidean geometry; Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.
A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.
The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion ...
In all these formulae (h, k) are the center coordinates of the hyperbola, a is the length of the semi-major axis, and b is the length of the semi-minor axis. Note that in the rational forms of these formulae, the points ( −a , 0) and (0 , −a ) , respectively, are not represented by a real value of t , but are the limit of x and y as t tends ...