Hyperbola

In mathematics, a hyperbola (/haɪˈpɜːrbələ/ ; pl. hyperbolas or hyperbolae /-liː/ ; adj. hyperbolic /ˌhaɪpərˈbɒlɪk/ ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

This article is about a geometric curve. For the term used in rhetoric, see Hyperbole.

Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship $xy=1.$ ^[1] In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.

Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve $y(x)=1/x$ the asymptotes are the two coordinate axes.^[2]

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

Etymology and history[edit]

The word "hyperbola" derives from the Greek ὑπερβολή, meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones.^[3] The term hyperbola is believed to have been coined by Apollonius of Perga (c. 262 – c. 190 BC) in his definitive work on the conic sections, the Conics.^[4] The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.^[5]

the coordinate axes as asymptotes,

the line $y=x$ as major axis ,

the center $(0,0)$ and the semi-axis $a=b={\sqrt {2A}}\;,$

the vertices $\left({\sqrt {A}},{\sqrt {A}}\right),\left(-{\sqrt {A}},-{\sqrt {A}}\right)\;,$

the semi-latus rectum and radius of curvature at the vertices $p=a={\sqrt {2A}}\;,$

the linear eccentricity $c=2{\sqrt {A}}$ and the eccentricity $e={\sqrt {2}}\;,$

the tangent $y=-{\tfrac {A}{x_{0}^{2}}}x+2{\tfrac {A}{x_{0}}}$ at point $(x_{0},A/x_{0})\;.$

In Cartesian coordinates[edit]

Equation[edit]

If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the x-axis is the major axis, then the hyperbola is called east-west-opening and

For a point (pole) on the hyperbola the polar is the tangent at this point (see diagram: $P_{1},\ p_{1}$ ).

For a pole $P$ outside the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing $P$ (see diagram: $P_{2},\ p_{2},\ P_{3},\ p_{3}$ ).

For a point within the hyperbola the polar has no point with the hyperbola in common. (see diagram: $P_{4},\ p_{4}$ ).

Arc length[edit]

The arc length of a hyperbola does not have an elementary expression. The upper half of a hyperbola can be parameterized as

$y=b{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}.$

Then the integral giving the arc length $s$ from $x_{1}$ to $x_{2}$ can be computed as:

$s=b\int _{\operatorname {arcosh} {\frac {x_{1}}{a}}}^{\operatorname {arcosh} {\frac {x_{2}}{a}}}{\sqrt {1+\left(1+{\frac {a^{2}}{b^{2}}}\right)\sinh ^{2}v}}\,\mathrm {d} v.$

After using the substitution $z=iv$ , this can also be represented using the incomplete elliptic integral of the second kind $E$ with parameter $m=k^{2}$ :

$s=ib{\Biggr [}E\left(iv\,{\Biggr |}\,1+{\frac {a^{2}}{b^{2}}}\right){\Biggr ]}_{\operatorname {arcosh} {\frac {x_{2}}{a}}}^{\operatorname {arcosh} {\frac {x_{1}}{a}}}.$

Using only real numbers, this becomes^[26]

$s=b\left[F\left(\operatorname {gd} v\,{\Biggr |}-{\frac {a^{2}}{b^{2}}}\right)-E\left(\operatorname {gd} v\,{\Biggr |}-{\frac {a^{2}}{b^{2}}}\right)+{\sqrt {1+{\frac {a^{2}}{b^{2}}}\tanh ^{2}v}}\,\sinh v\right]_{\operatorname {arcosh} {\tfrac {x_{1}}{a}}}^{\operatorname {arcosh} {\tfrac {x_{2}}{a}}}$

where $F$ is the incomplete elliptic integral of the first kind with parameter $m=k^{2}$ and $\operatorname {gd} v=\arctan \sinh v$ is the Gudermannian function.

Elliptic coordinates[edit]

A family of confocal hyperbolas is the basis of the system of elliptic coordinates in two dimensions. These hyperbolas are described by the equation

$\left({\frac {x}{c\cos \theta }}\right)^{2}-\left({\frac {y}{c\sin \theta }}\right)^{2}=1$

where the foci are located at a distance c from the origin on the x-axis, and where θ is the angle of the asymptotes with the x-axis. Every hyperbola in this family is orthogonal to every ellipse that shares the same foci. This orthogonality may be shown by a conformal map of the Cartesian coordinate system w = z + 1/z, where z= x + iy are the original Cartesian coordinates, and w=u + iv are those after the transformation.

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping w = z² transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.

Elliptic

cone

Hyperbolic

cylinder

Hyperbolic paraboloid

Hyperboloid of one sheet

Hyperboloid of two sheets

Hyperbolas appear as plane sections of the following quadrics:

(1970), Ruler and the Round, Boston: Prindle, Weber & Schmidt, ISBN 0-87150-113-9

Kazarinoff, Nicholas D.

Oakley, C. O., Ph.D. (1944), An Outline of the Calculus, New York: {{citation}}: CS1 maint: multiple names: authors list (link)

Barnes & Noble

; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042