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The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, = (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity .
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
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 approaches a straight line.
The mean eccentricity of an object is the average eccentricity as a result of perturbations over a given time period. Neptune currently has an instant (current epoch ) eccentricity of 0.011 3 , [ 13 ] but from 1800 to 2050 has a mean eccentricity of 0.008 59 .
Conic sections of varying eccentricity sharing 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 pair of lines.
In a hyperbola, a conjugate axis or minor axis of length , corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola.
A pencil of confocal ellipses and hyperbolas is specified by choice of linear eccentricity c (the x-coordinate of one focus) and can be parametrized by the semi-major axis a (the x-coordinate of the intersection of a specific conic in the pencil and the x-axis). When 0 < a < c the conic is a hyperbola; when c < a the conic is an ellipse.
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals . [1] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry.