Cartesian coordinates[edit]An east-west opening hyperbola centered at (h,k) has the equation

The major axis runs through the center of the hyperbola and intersects both arms of the hyperbola at the vertices (bend points) of the arms. The foci lie on the extension of the major axis of the hyperbola.The minor axis runs through the center of the hyperbola and is perpendicular to the major axis.In both formulasais thesemi-major axis(half the distance between the two arms of the hyperbola measured along the major axis),[2]andbis thesemi-minor axis(half the distance between the asymptotes along a line tangent to the hyperbola at a vertex).If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are2bin length while the sides that run parallel to the line between the foci (the major axis) are2ain length. Note thatbmay be larger thanadespite the namesminorandmajor.If one calculates the distance from any point on the hyperbola to each focus, the absolute value of the difference of those two distances is always2a.Theeccentricityis given by

Ifcequals the distance from the center to either focus, then

where.The distancecis known as thelinear eccentricityof the hyperbola. The distance between the foci is 2cor 2a.The foci for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.The directrices for an east-west opening hyperbola are given by

and for a north-south opening hyperbola are given by.

HyperbolaA hyperbola (plural "hyperbolas"; Gray 1997, p.45) is aconic sectiondefined as thelocusof all pointsin theplanethe difference of whose distancesandfrom two fixed points (thefociand) separated by a distanceis a givenpositiveconstant,(1)

(Hilbert and Cohn-Vossen 1999, p.3). Lettingfall on the left-intercept requires that(2)

so the constant is given by, i.e., the distance between the-intercepts (left figure above). The hyperbola has the important property that a ray originating at afocusreflects in such a way that the outgoing path lies along the line from the otherfocusthrough the point of intersection (right figure above).The special case of therectangular hyperbola, corresponding to a hyperbola with eccentricity, was first studied by Menaechmus. Euclid and Aristaeus wrote about the general hyperbola, but only studied one branch of it. The hyperbola was given its present name by Apollonius, who was the first to study both branches. Thefocusandconic section directrixwere considered by Pappus (MacTutor Archive). The hyperbola is the shape of an orbit of a body on an escape trajectory (i.e., a body with positive energy), such as some comets, about a fixed mass, such as the sun.

The hyperbola can be constructed by connecting the free endof a rigid bar, whereis afocus, and the otherfocuswith a string. As the baris rotated aboutandis kept taut against the bar (i.e., lies on the bar), thelocusofis one branch of a hyperbola (left figure above; Wells 1991). A theorem of Apollonius states that for a line segment tangent to the hyperbola at a pointandintersectingthe asymptotes at pointsand, thenis constant, and(right figure above; Wells 1991).

Let the pointon the hyperbola have Cartesian coordinates, then the definition of the hyperbolagives(3)

Rearranging and completing the square gives(4)

and dividing both sides byresults in(5)

By analogy with the definition of theellipse, define(6)

so the equation for a hyperbola withsemimajor axisparallel to thex-axisandsemiminor axisparallel to they-axisis given by(7)

or, for a center at the pointinstead of,(8)

Unlike theellipse, no points of the hyperbola actually lie on thesemiminor axis, but rather the ratiodetermines the vertical scaling of the hyperbola. Theeccentricityof the hyperbola (which always satisfies) is then defined as(9)

In the standard equation of the hyperbola, the center is located at, thefociare at, and the vertices are at. The so-calledasymptotes(shown as the dashed lines in the above figures) can be found by substituting 0 for the 1 on the right side of the general equation (8),(10)

and therefore haveslopes.The special case(the left diagram above) is known as arectangular hyperbolabecause theasymptotesareperpendicular.

The hyperbola can also be defined as thelocusof points whose distance from thefocusis proportional to the horizontal distance from a vertical lineknown as theconic section directrix, where the ratio is. Lettingbe the ratio andthe distance from the center at which the directrix lies, then(11)

(12)

whereis therefore simply theeccentricity.Like noncircularellipses, hyperbolas havetwodistinctfociand two associatedconic section directrices, eachconic section directrixbeingperpendicularto the line joining the two foci (Eves 1965, p.275).Thefocal parameterof the hyperbola is(13)

(14)

(15)

Inpolar coordinates, the equation of a hyperbola centered at theorigin(i.e., with) is(16)

Inpolar coordinatescentered at afocus,(17)

as illustrated above.The two-centerbipolar coordinatesequation with origin at afocusis(18)

Parametric equationsfor the right branch of a hyperbola are given by(19)

(20)

whereis thehyperbolic cosineandis thehyperbolic sine, which ranges over the right branch of the hyperbola.A parametric representation which ranges over both branches of the hyperbola is(21)

(22)

withand discontinuities at. Thearc length,curvature, andtangential anglefor the above parametrization are(23)

(24)

(25)

whereis anelliptic integral of the second kind.Thespecial affine curvatureof the hyperbola is(26)

Thelocusof the apex of a variableconecontaining anellipsefixed in three-space is a hyperbola through thefociof theellipse. In addition, thelocusof the apex of aconecontaining that hyperbola is the originalellipse. Furthermore, theeccentricitiesof theellipseand hyperbola are reciprocals