Hyperbola - Nomenclature and Features

Nomenclature and Features

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 2b, 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

A hyperbola aligned in this way is called an "East-West opening hyperbola". Likewise, a hyperbola with its transverse axis aligned with the y-axis is called a "North-South opening hyperbola" and has equation

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε (its shape, or degree of "spread"), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated (rigidly moved in the plane) so that it is centered at the origin. For convenience, hyperbolas are usually analyzed in terms of their centered East-West opening form.

The shape of a hyperbola is defined entirely by its eccentricity ε, which is a dimensionless number always greater than one. The distance c from the center to the foci equals aε. The eccentricity can also be defined as the ratio of the distances to either focus and to a corresponding line known as the directrix; hence, the distance from the center to the directrices equals a/ε. In terms of the parameters a, b, c and the angle θ, the eccentricity equals

For example, the eccentricity of a rectangular hyperbola (θ = 45°, a = b) equals the square root of two: ε = .

Every hyperbola has a conjugate hyperbola, in which the transverse and conjugate axes are exchanged without changing the asymptotes. The equation of the conjugate hyperbola of is . If the graph of the conjugate hyperbola is rotated 90° to restore the east-west opening orientation (so that x becomes y and vice versa), the equation of the resulting rotated conjugate hyperbola is the same as the equation of the original hyperbola except with a and b exchanged. For example, the angle θ of the conjugate hyperbola equals 90° minus the angle of the original hyperbola. Thus, the angles in the original and conjugate hyperbolas are complementary angles, which implies that they have different eccentricities unless θ = 45° (a rectangular hyperbola). Hence, the conjugate hyperbola does not in general correspond to a 90° rotation of the original hyperbola; the two hyperbolas are generally different in shape.

A few other lengths are used to describe hyperbolas. Consider a line perpendicular to the transverse axis (i.e., parallel to the conjugate axis) that passes through one of the hyperbola's foci. The line segment connecting the two intersection points of this line with the hyperbola is known as the latus rectum and has a length . The semi-latus rectum l is half of this length, i.e., . The focal parameter p is the distance from a focus to its corresponding directrix, and equals .