hyperbola

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.

The general equation for a hyperbola is

(x-h)2 - (y-k)2 = +/-1
  a2       b2

In this form (h,k) will represent the center of the hyperbola.
If the 1 on the right side of the equation is positive, the equation will represent a hyperbola whose transverse axis is horizontal, looking something like:

\      /
 \    /
  |  |
 /    \
/      \

In this case, the vertices of the hyperbola (i.e the points on the transverse axis, represented in the diagram above by the | symbol) will be (h + a, k) and (h - a, k).

If, on the other hand, the 1 is negative, the hyperbola will have a vertical transverse axis, looking like:

\   /
 \_/
  _
 / \
/   \

In that case, the vertices of the hyperbola will be (h, k + b) and (h, k - b).
The asymptotes of a hyperbola are the lines that the hyperbola approaches; for the hyperbola depicted above, they will look like this:

\   /
 \ /
  X
 / \
/   \

These asymptotes can be found by setting the right side of the equation above to 0, and isolating y:

(x-h)2 - (y-k)2 = 0
  a2       b2

giving:

y = (b/a)x + (k - bh/a)
y = -(b/a)x + (k + bh/a)

By drawing the vertices and the asymptotes of a hyperbola, an approximate sketch of it may be done. Other points of interest include the focii of the hyperbola. The hyperbola itself is the locus of all points such that the difference in their distances from the two focii is constant.
The focii for a hyperbola with the centre at the origin are (c,0) and (-c,0), or (0,c) and (0,-c) for a vertical hyperbola, where

c = a² + b²

Other forms of the equation for a hyperbola exist, as noted in the writeup above. These are all actually transformations of a single, universal form, given by:

ax² + 2hxy + by² + fx + gy + c = 0.

This is a standard form of the equation for all conic sections; when the curve is a hyperbola,

ab - h² > 0

This equation represents hyperbolae rotated to any angle, while the form described above and the form xy=c only denote hyperbolas that are vertical, horizontal, or oriented at 45 degrees to the axes.

Other than annoying students through the vast amounts of work required to graph and rotate these muthas, hyperbolas seem to have no useful application.

Useful information about hyperbolas:
(note: c is the distance from the center to the focus, a is the distance from the vertex to the center, and a²+b²=c²)

			Horizontal transverse axis	Horizontal conjugate axis
Standard Form:		(x-h)²/a²-(y-k)²/b²=1		(y-k)²/a²-(x-h)²/b²=1
Eccentricity:		e=c/a (e>1)			e=c/a (e>1)
Length of latera recta:	l=2b²/a 			l=2b²/a 
Directrices:		x=±a²/c+h			y=±a²/c+k
Vertices:		(h±a,k±b)			(h±b,k±a)
Foci:			(h±c,k)				(h,k±c)
Asymptotes:		y=k±(b/a)(x-h)			y=k±(a/b)(x-h)
Parametric Form:	x′=asec(t)			x′=btan(t)
			y′=btan(t)			y′=asec(t)

See also:

• Conic Section
• Parabola
• Ellipse
• Circle

Hy*per"bo*la (?), n. [Gr. , prop., an overshooting, excess, i. e., of the angle which the cutting plane makes with the base. See Hyperbole.] Geom.

A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.

 

© Webster 1913.