Mathematical definitions

A hyperbola can be defined mathematically in several equivalent ways.

Conic section

Three major types of conic sections.

A hyperbola may be defined as the curve of

intersection between a right circular conical surface and a plane that cuts through both halves of the cone. The other major types of conic sections are the ellipse and the parabola ; in these cases, the plane cuts through only one half of the double cone. If the plane passes through the central apex of the double cone a

degenerate hyperbola results — two straight lines that cross at the apex point.

Difference of distances to foci

A hyperbola may be defined equivalently as the

locus of points where the absolute value of the

difference of the distances to the two foci is a constant equal to 2 a , the distance between its two vertices. This definition accounts for many of the hyperbola's applications, such as

multilateration; this is the problem of determining position from the difference in arrival times of synchronized signals, as in GPS .

This definition may be expressed also in terms of tangent circles. The center of any circles externally tangent to two given circles lies on a hyperbola, whose foci are the centers of the given circles and where the vertex distance 2 a equals the difference in radii of the two circles. As a special case, one given circle may be a point located at one focus; since a point may be considered as a circle of zero radius, the other given circle—which is centered on the other focus—must have radius 2 a . This provides a simple technique for constructing a hyperbola, as shown below . It follows from this definition that a tangent line to the hyperbola at a point P bisects the angle formed with the two foci, i.e., the angle F 1 P F 2 . Consequently, the feet of perpendiculars drawn from each focus to such a tangent line lies on a circle of radius

a that is centered on the hyperbola's own center.

A proof that this characterization of the hyperbola is equivalent to the conic-section characterization can be done without coordinate geometry by means of Dandelin spheres .

Directrix and focus

A hyperbola can be defined as the locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant e that is larger than 1. This constant is the eccentricity of the hyperbola. The eccentricity equals the secant of half the angle between the asymptotes of the hyperbola, so the eccentricity of the hyperbola xy = 1 equals the square root of 2.

By symmetry a hyperbola has two directrices, which are parallel to the conjugate axis and are between it and the tangent to the hyperbola at a vertex. One directrix and its focus is enough to produce both arms of the hyperbola.

Reciprocation of a circle

The reciprocation of a circle B in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C " consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The

pole of a line is the inversion of its closest point to the circle C , whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.

The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then