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hyperbola
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Eccentricity of conic sectionsThe eccentricity of a conic section completely characterizes its shape. For example, all circles have zero eccentricity, and all parabolas have unit eccentricity; hence, all circles (and all parabolas) have the same shape, only varying in size. (Under appropriate magnification they are indistinguishable.) In contrast, ellipses and hyperbolas vary greatly in shape. conic sections Projective conic sectionsThe conic sections (ellipse, parabola, and hyperbola) can be generated by projecting the circle formed by the intersection of a cone with a plane (the reality plane, or RP) perpendicular to the cone's central axis. The image of the circle is projected onto a plane (the projective plane, or PP) that is oriented at the same angle as the cutting plane (Ω) passing through the apex (“eye”) of the double cone. In this example, the orientation of Ω produces an ellipse in PP. Figure 6: The hyperbola (see text).
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hyperbola

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hyperbola, two-branched open curve, a conic section, produced by the intersection of a circular cone and a plane that cuts both nappes (see cone) of the cone. As a plane curve it may be defined as the path (locus) of a point moving so that the ratio of the distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant greater than one. The hyperbola, however, because of its symmetry, has two foci. Another definition is that of a point moving so that the difference of its distances from two fixed points, or foci, is a constant. A degenerate hyperbola (two intersecting lines) is formed by the intersection of a circular cone and a plane that cuts both nappes of the cone through the apex.

A line drawn through the foci and prolonged beyond is the transverse axis of the hyperbola; perpendicular to that axis, and intersecting it at the geometric centre of the hyperbola, a point midway between the two foci, lies the conjugate axis. The hyperbola is symmetrical with respect to both axes.

Two straight lines, the asymptotes of the curve, pass through the geometric centre. The hyperbola does not intersect the asymptotes, but its distance from them becomes arbitrarily small at great distances from the centre. The hyperbola when revolved about either axis forms a hyperboloid (q.v.).

For a hyperbola that has its centre at the origin of a Cartesian coordinate system and has its transverse axis lying on the x axis, the coordinates of its points satisfy the equation x2/a2 - y2/b2 = 1, in which a and b are constants.