Hyperbola - Relation To Other Conic Sections

Relation To Other Conic Sections

There are three major types of conic sections: hyperbolas, ellipses and parabolas. Since the parabola may be seen as a limiting case poised exactly between an ellipse and a hyperbola, there are effectively only two major types, ellipses and hyperbolas. These two types are related in that formulae for one type can often be applied to the other.

The canonical equation for a hyperbola is

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$

Any hyperbola can be rotated so that it is east-west opening and positioned with its center at the origin, so that the equation describing it is this canonical equation.

The canonical equation for the hyperbola may be seen as a version of the corresponding ellipse equation

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

in which the semi-minor axis length b is imaginary. That is, if in the ellipse equation b is replaced by ib where b is real, one obtains the hyperbola equation.

Similarly, the parametric equations for a hyperbola and an ellipse are expressed in terms of hyperbolic and trigonometric functions, respectively, which are again related by an imaginary number, e.g.,

$\cosh \mu = \cos i\mu$

Hence, many formulae for the ellipse can be extended to hyperbolas by adding the imaginary unit i in front of the semi-minor axis b and the angle. For example, the arc length of a segment of an ellipse can be determined using an incomplete elliptic integral of the second kind. The corresponding arclength of a hyperbola is given by the same function with imaginary parameters b and μ, namely, ib E(iμ, c).

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