hyperbola (idea)

One of the conic section functions. Others include the ellipse, the parabola and the circle. A major feature of any hyperbola is the asymptotes, unlike the parabola which has none. The generic equation of an hyperbola centered at the origin is:

x²/_a2-y²/_b2=1

Just as the sine and cosine functions relate to the circle and ellipse, the hyperbolic cosine and hyperbolic sine relate to the hyperbola. Consider the following identities and equalities:

cosh²(x)-sinh²(x)=1
cos²(x)+sin²(x)=1

          (ix)    (-ix)
         e     + e
cos(x) = -------------
               2

            x    -x
           e  + e
cosh(x) = ----------
              2

Another form of the hyperbola can be constructed with the equation xy=c for c a constant. This is not the "standard" form, and it is not so easily relatable to the ellipse, but it is nonetheless an hyperbola by virtue of its asymtotes. A further note on the relationship between hyperbolas and ellipses: the ellipse is defined to be the set of points whose distances from a pair of points sum to a constant, while the hyperbola is defined as the set of points whose distances from a pair of points are different by a constant.