hyperbola: Quick Facts

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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.

Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.

A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.

Figure 6: The hyperbola (see text).

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hyperbolic orbit Fermat’s hyperbola Landen’s theorem hyperbolic navigation system hyperbolic space hyperbolic geometry hyperbolic functions conic section ellipse parabola circle