# hyperbola

## Definitions

• WordNet 3.6
• n hyperbola an open curve formed by a plane that cuts the base of a right circular cone
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Webster's Revised Unabridged Dictionary
• n Hyperbola (Geom) A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.
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Century Dictionary and Cyclopedia
• n hyperbola A curve formed by the intersection of a plane with a double cone—that is, with two similar cones placed vertex to vertex, so that one is the continuation of the other. If the plane cuts only one of the cones, the section is a circle, an ellipse, or a parabola; but if both cones are cut, the section is a hyperbola. A hyperbola may be formed by throwing upon a table the shadow of a ball the top of which is higher than the source of light. It has two asymptotes. If through any point of the curve lines be drawn parallel to the asymptotes, the parallelogram so formed will be of constant area for any given hyperbola. The point of intersection of the asymptotes is the center of the hyperbola, and is equidistant from the two intersections of any line through it with the hyperbola. The two lines through the Center bisecting the angles of the asymptotes are the linesof the axes of the hyperbola, and the curve is symmetrical with respect to each of these. One of these lines cuts the curve, and the points of intersection are called the vertices of the hyperbola. The line between the vertices is the major or transverse axis of the hyperbola. If from the vertices lines be drawn parallel to the two asymptotes, the two points at which these lines will meet will be the extremities of the minor or conjugate axis. Although the axes bear these names, the minor may be longer than the major axis. The equation of the hyperbola, referred to its center and axes, is The foci'of the hyperbola are two points on the line of the transverse axis distant from the center as far as the vertices are from the extremities of the conjugate axis. If from any point of the curve lines be drawn to the two foci, the difference of the lengths of these lines is constant for any given hyperbola, and the angle between them is bisected by the tangent at that point. The eccentricity of the hyperbola is the secant of half the angle between the asymptotes. The parameter or latus rectum of a hyperbola is a chord through the focus perpendicular to the transverse axis.
• n hyperbola An algebraic curve having asymptotes greater in number by one than its order. This meaning was introduced by Newton.
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Chambers's Twentieth Century Dictionary
• n Hyperbola hī-per′bo-la (geom.) one of the conic sections or curves formed when the intersecting plane makes a greater angle with the base than the side of the cone makes
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## Etymology

Webster's Revised Unabridged Dictionary
Gr. , prop., an overshooting, excess, i. e., of the angle which the cutting plane makes with the base. See Hyperbole
Chambers's Twentieth Century Dictionary
L.,—Gr. hyperbolē, from hyperballeinhyper, beyond, ballein, to throw.

## Usage

### In literature:

The Lord Brouncker employed this series to square the hyperbola.
"Letters on England" by Voltaire
An oval is never mistaken for a circle, nor an hyperbola for an ellipsis.
"An Enquiry Concerning Human Understanding" by David Hume
The following analysis shows that with the aid of an hyperbola any arc, and therefore any angle, may be trisected.
"Scientific American Supplement, No. 787, January 31, 1891" by Various
It is easily seen that the curves traced by the shadow of the point P are hyperbolas whose convexity is turned toward A B.
"Scientific American Supplement, No. 810, July 11, 1891" by Various
There's nothing more to fear from your hyperbolas or parabolas or any other of your open curves!
"All Around the Moon" by Jules Verne
In the hyperbola we have the mathematical demonstration of the error of an axiom.
"Fables of Infidelity and Facts of Faith" by Robert Patterson
Griffin (R. W.) on Parabola, Ellipse, and Hyperbola.
"France and the Republic" by William Henry Hurlbert
It is said that he was the first to introduce the words ellipse and hyperbola.
"History of the Intellectual Development of Europe, Volume I (of 2)" by John William Draper
In such cases the orbit might be changed to a hyperbola, and then the comet would never return.
"Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 7" by Various
Just a plain hyperbola is bad enough.
"The Romantic Analogue" by W.W. Skupeldyckle
The planet would then have moved in a parabola, or an hyperbola, curves not returning into themselves.
"A System of Logic: Ratiocinative and Inductive" by John Stuart Mill
Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines.
"The Teaching of Geometry" by David Eugene Smith
The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.
"Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8" by Various
The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.
"Encyclopaedia Britannica, 11th Edition, Volume 14, Slice 2" by Various
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### In science:

On the other hand, the curves of parabolas in the middle (resp. right) ﬁgure are the osculating parabolic congruence of a hyperbola (resp. an ellipse ).
Curvature Functionals for Curves in the Equi-Affine Plane
Region in the (ρ, η) plane allowed by constraints on |Vub /Vcb | (dotted semicircles), B 0 − ¯B 0 mixing (dashed semicircles), and CP-violating K − ¯K mixing (solid hyperbolae).
Present and Future Aspects of CP Violation
Equation (19) speciﬁes a hyperbola in the ( ¯̺, ¯η) plane.
Theoretical review of K physics
Vub/Vcb | which produces a circular ring and from a hyperbola deﬁned from the theoretical formula for ε.
Analysis of \epsilon'/\epsilon in the 1/N_c Expansion
The position of the hyperbola depends on mt , |Vcb |, and ˆBK .
Analysis of \epsilon'/\epsilon in the 1/N_c Expansion
This is valid since geodesics on the Taub-NUT space are hyperbolae , so in a scattering process r asymptotically approaches inﬁnity and there is only one turning point.
The curve for N = 6 dominates that for N = 4, which in turn dominates the hyperbola for N = 2.
Increased Efficiency of Quantum State Estimation Using Non-Separable Measurements
The eq.(21) represents a hyperbola and shows that to an external observer a radially in-falling time like or null particle approaches the radius r = q(D)asymptotically but can never reach it.
Wormhole and C-field
Here we also see the s - r relationship represents a hyperbola.
Wormhole and C-field
For example, the constraint obtained from the C P -violating parameter ǫK in the neutral K system corresponds to the vertex A of the unitarity triangle lying on a hyperbola for ﬁxed values of the (imprecisely known) hadronic matrix elements , .
Unitarity Triangle from CP invariant quantities
Note that this has the required form (two hyperbolae with the same asymptotes, and possibly different intercepts).
Analogue spacetime based on 2-component Bose-Einstein condensates
It is known that for any point (p, q) ∈ R2 below the critical hyperbola the Hamiltonian system has a nontrivial solution (see [CdFM], [dFF], [HvdV], [FM] and [dFR]), whereas for points (p, q) on the critical hyperbola one ﬁnds the typical problems of non-compactness and non-existence of solutions (see [vdV] and [M]).
Perturbation from symmetry and multiplicity of solutions for strongly indefinite elliptic systems
We remark here that the pair (p, q) lies below the critical hyperbola; for any ﬁxed (p, q), the value of r, which identiﬁes the space E r , is not ﬁxed, but can be chosen in the range deﬁned by (60) (see Theorem 2.1).
Perturbation from symmetry and multiplicity of solutions for strongly indefinite elliptic systems
Let us ﬁx (p, q) below the critical hyperbola 1 N .
Perturbation from symmetry and multiplicity of solutions for strongly indefinite elliptic systems
Condition (81) deﬁnes a new region in the (p, q) plane which is contained in the subcritical region delimited by the critical hyperbola.
Perturbation from symmetry and multiplicity of solutions for strongly indefinite elliptic systems
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