Nomenclature and features

The hyperbola consists of the red curves. The asymptotes of the hyperbola are shown as blue dashed lines and intersect at the center of the hyperbola, C . The two focal points are labeled F 1 and F 2 , and the thin black line joining them is the transverse axis. The perpendicular thin black line through the center is the conjugate axis. The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D 1 and D 2 . The eccentricity e equals the ratio of the distances from a point

P on the hyperbola to one focus and its corresponding directrix line (shown in green). The two vertices are located on the transverse axis at ± a relative to the center. So the parameters are: a — distance from center C to either vertex

b — length of a segment perpendicular to the transverse axis drawn from each vertex to the asymptotes

c — distance from center C to either Focus point, F 1 and F 2 , and

θ — angle formed by each asymptote with the transverse axis.

Similar to a parabola , a hyperbola is an open curve, meaning that it continues indefinitely to infinity, rather than closing on itself as an

ellipse does. A hyperbola consists of two disconnected curves called its arms or

branches .

The points on the two branches that are closest to each other are called the vertices ; they are the points where the curve has its smallest radius of curvature . The line segment connecting the vertices is called the transverse axis or major axis , corresponding to the major diameter of an ellipse. The midpoint of the transverse axis is known as the hyperbola's

center . The distance a from the center to each vertex is called the semi-major axis . Outside of the transverse axis but on the same line are the two focal points (foci) of the hyperbola. The line through these five points is one of the two principal axes of the hyperbola, the other being the perpendicular bisector of the transverse axis. The hyperbola has mirror symmetry about its principal axes, and is also symmetric under a 180° turn about its center.

At large distances from the center, the hyperbola approaches two lines, its asymptotes , which intersect at the hyperbola's center. A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them; however, a degenerate hyperbola consists only of its asymptotes. Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x -axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±

b

/

a , where b = a × tan(θ) and where θ is the angle between the transverse axis and either asymptote. The distance b (not shown) is the length of the perpendicular segment from either vertex to the asymptotes.

A conjugate axis of length 2 b , corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ± b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices. Because of the minus sign in some of the formulas below, it is also called the

imaginary axis of the hyperbola.

If b = a , the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be
rectangular or equilateral. In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b .
If the transverse axis of any hyperbola is aligned with the x -axis of a Cartesian coordinate system and is centered on the origin, the equation of the hyperbola can be written as