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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
s 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 A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
, formed by the intersection of a plane and a double
cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
. (The other conic sections are the
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
and the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. A
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
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. 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
at that point, or as the solution of certain bivariate
quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
s such as the reciprocal relationship xy = 1. In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a
sundial A sundial is a horology, horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the position of the Sun, apparent position of the Sun in the sky. In the narrowest sense of the ...
's
gnomon A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields, typically to measure directions, position, or time. History A painted stick dating from 2300 BC that was ...
, the shape of an open orbit such as that of a celestial object exceeding the
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming: * Ballistic trajectory – no other forces are acting on the object, such as ...
of the nearest gravitational body, or the scattering trajectory of a
subatomic particle In physics, a subatomic particle is a particle smaller than an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a baryon, lik ...
, among others. Each
branch A branch, also called a ramus in botany, is a stem that grows off from another stem, or when structures like veins in leaves are divided into smaller veins. History and etymology In Old English, there are numerous words for branch, includ ...
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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
of those two arms. So there are two asymptotes, whose intersection is at the center of
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
. Hyperbolas share many of the ellipses' analytical properties such as
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
,
focus Focus (: foci or focuses) may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in East Australia Film *Focus (2001 film), ''Focus'' (2001 film), a 2001 film based on the Arthur Miller novel *Focus (2015 ...
, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s have their origin in the hyperbola, such as
hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...
s (saddle surfaces),
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
s ("wastebaskets"),
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
( Lobachevsky's celebrated
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
),
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
s (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
which is not Euclidean).


Etymology and history

The word "hyperbola" derives from the
Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...
, meaning "over-thrown" or "excessive", from which the English term
hyperbole Hyperbole (; adj. hyperbolic ) is the use of exaggeration as a rhetorical device or figure of speech. In rhetoric, it is also sometimes known as auxesis (literally 'growth'). In poetry and oratory, it emphasizes, evokes strong feelings, and cre ...
also derives. Hyperbolae were discovered by
Menaechmus Menaechmus (, c. 380 – c. 320 BC) was an ancient Greek mathematician, list of geometers, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher P ...
in his investigations of the problem of
doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...
, but were then called sections of obtuse cones. The term hyperbola is believed to have been coined by
Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
() in his definitive work on the
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
s, the ''Conics''. The names of the other two general conic sections, the
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
and the
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
, 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.


Definitions


As locus of points

A hyperbola can be defined geometrically as a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of points ( locus of points) in the Euclidean plane: The midpoint M of the line segment joining the foci is called the ''center'' of the hyperbola. The line through the foci is called the ''major axis''. It contains the ''vertices'' V_1, V_2, which have distance a to the center. The distance c of the foci to the center is called the ''focal distance'' or ''linear eccentricity''. The quotient \tfrac c a is the ''eccentricity'' e. The equation \left, \left, PF_2\ - \left, PF_1\\ = 2a can be viewed in a different way (see diagram):
If c_2 is the circle with midpoint F_2 and radius 2a, then the distance of a point P of the right branch to the circle c_2 equals the distance to the focus F_1: , PF_1, =, Pc_2, . c_2 is called the ''circular directrix'' (related to focus F_2) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the circular directrix related to F_1. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.


Hyperbola with equation

If the ''xy''-coordinate system is rotated about the origin by the angle +45^\circ and new coordinates \xi,\eta are assigned, then x = \tfrac,\; y = \tfrac .
The rectangular hyperbola \tfrac = 1 (whose semi-axes are equal) has the new equation \tfrac = 1. Solving for \eta yields \eta = \tfrac \ . Thus, in an ''xy''-coordinate system the graph of a function f: x \mapsto \tfrac,\; A>0\; , with equation y = \frac\;, A>0\; , is a ''rectangular hyperbola'' entirely in the first and third quadrants with *the coordinate axes as ''asymptotes'', *the line y = x as ''major axis'' , *the ''center'' (0,0) and the ''semi-axis'' a = b = \sqrt \; , *the ''vertices'' \left(\sqrt,\sqrt\right), \left(-\sqrt,-\sqrt\right) \; , *the ''semi-latus rectum'' and ''radius of curvature '' at the vertices p=a=\sqrt \; , *the ''linear eccentricity'' c=2\sqrt and the eccentricity e=\sqrt \; , *the ''tangent'' y=-\tfracx+2\tfrac at point (x_0,A/x_0)\; . A rotation of the original hyperbola by -45^\circ results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for the case of +45^\circ rotation, with equation y = -\frac \; , ~~ A>0\; , *the ''semi-axes'' a = b = \sqrt \; , *the line y = -x as major axis, *the ''vertices'' \left(-\sqrt,\sqrt\right), \left(\sqrt,-\sqrt\right) \; . Shifting the hyperbola with equation y=\frac, \ A\ne 0\ , so that the new center is yields the new equation y=\frac+d_0\; , and the new asymptotes are x=c_0 and y=d_0. The shape parameters a,b,p,c,e remain unchanged.


By the directrix property

The two lines at distance d = \fracc from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram). For an arbitrary point P of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: \frac = \frac = e= \frac \, . The proof for the pair F_1, l_1 follows from the fact that , PF_1, ^2 = (x-c)^2+y^2,\ , Pl_1, ^2 = \left(x-\tfrac\right)^2 and y^2 = \tfracx^2-b^2 satisfy the equation , PF_1, ^2-\frac, Pl_1, ^2 = 0\ . The second case is proven analogously. The ''inverse statement'' is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola): For any point F (focus), any line l (directrix) not through F and any
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
e with e > 1 the set of points (locus of points), for which the quotient of the distances to the point and to the line is e H = \left\ is a hyperbola. (The choice e = 1 yields a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
and if e < 1 an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
.)


Proof

Let F=(f,0) ,\ e >0 and assume (0,0) is a point on the curve. The directrix l has equation x=-\tfrac. With P=(x,y), the relation , PF, ^2 = e^2, Pl, ^2 produces the equations :(x-f)^2+y^2 = e^2\left(x+\tfrac\right)^2 = (e x+f)^2 and x^2(e^2-1)+2xf(1+e)-y^2 = 0. The substitution p=f(1+e) yields x^2(e^2-1)+2px-y^2 = 0. This is the equation of an ''ellipse'' (e<1) or a ''parabola'' (e=1) or a ''hyperbola'' (e>1). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). If e > 1, introduce new parameters a,b so that e^2-1 = \tfrac, \text \ p = \tfrac, and then the equation above becomes \frac - \frac = 1 \, , which is the equation of a hyperbola with center (-a,0), the ''x''-axis as major axis and the major/minor semi axis a,b.


Construction of a directrix

Because of c \cdot \tfrac=a^2 point L_1 of directrix l_1 (see diagram) and focus F_1 are inverse with respect to the
circle inversion In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...
at circle x^2+y^2=a^2 (in diagram green). Hence point E_1 can be constructed using the theorem of Thales (not shown in the diagram). The directrix l_1 is the perpendicular to line \overline through point E_1. ''Alternative construction of E_1'': Calculation shows, that point E_1 is the intersection of the asymptote with its perpendicular through F_1 (see diagram).


As plane section of a cone

The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two
Dandelin spheres In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the pl ...
d_1, d_2, which are spheres that touch the cone along circles c_2 and the intersecting (hyperbola) plane at points F_1 and It turns out: F_1, F_2 are the ''foci'' of the hyperbola. # Let P be an arbitrary point of the intersection curve. # The generatrix of the cone containing P intersects circle c_1 at point A and circle c_2 at a point B. # The line segments \overline and \overline are tangential to the sphere d_1 and, hence, are of equal length. # The line segments \overline and \overline are tangential to the sphere d_2 and, hence, are of equal length. # The result is: , PF_1, - , PF_2, = , PA, - , PB, = , AB, is independent of the hyperbola point because no matter where point P is, A, B have to be on circles and line segment AB has to cross the apex. Therefore, as point P moves along the red curve (hyperbola), line segment \overline simply rotates about apex without changing its length.


Pin and string construction

The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler: #
  • Choose the ''foci'' F_1,F_2 and one of the ''circular directrices'', for example c_2 (circle with radius 2a)
  • # A ''ruler'' is fixed at point F_2 free to rotate around F_2. Point B is marked at distance 2a. # A ''string'' gets its one end pinned at point A on the ruler and its length is made , AB, . # The free end of the string is pinned to point F_1. # Take a ''pen'' and hold the string tight to the edge of the ruler. # ''Rotating'' the ruler around F_2 prompts the pen to draw an arc of the right branch of the hyperbola, because of , PF_1, = , PB, (see the definition of a hyperbola by ''circular directrices'').


    Steiner generation of a hyperbola

    The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section: For the generation of points of the hyperbola \tfrac-\tfrac = 1 one uses the pencils at the vertices V_1,V_2. Let P = (x_0,y_0) be a point of the hyperbola and A = (a,y_0), B = (x_0,0). The line segment \overline is divided into n equally-spaced segments and this division is projected parallel with the diagonal AB as direction onto the line segment \overline (see diagram). The parallel projection is part of the projective mapping between the pencils at V_1 and V_2 needed. The intersection points of any two related lines S_1 A_i and S_2 B_i are points of the uniquely defined hyperbola. ''Remarks:'' * The subdivision could be extended beyond the points A and B in order to get more points, but the determination of the intersection points would become more inaccurate. A better idea is extending the points already constructed by symmetry (see animation). * The Steiner generation exists for ellipses and parabolas, too. * The Steiner generation is sometimes called a ''parallelogram method'' because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.


    Inscribed angles for hyperbolas and the 3-point-form

    A hyperbola with equation y=\tfrac+c,\ a \ne 0 is uniquely determined by three points (x_1,y_1),\;(x_2,y_2),\; (x_3,y_3) with different ''x''- and ''y''-coordinates. A simple way to determine the shape parameters a,b,c uses the ''inscribed angle theorem'' for hyperbolas: Analogous to the
    inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an ...
    theorem for circles one gets the A consequence of the inscribed angle theorem for hyperbolas is the


    As an affine image of the unit hyperbola

    Another definition of a hyperbola uses
    affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
    s:


    Parametric representation

    An affine transformation of the Euclidean plane has the form \vec x \to \vec f_0+A\vec x, where A is a regular
    matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
    (its
    determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
    is not 0) and \vec f_0 is an arbitrary vector. If \vec f_1, \vec f_2 are the column vectors of the matrix A, the unit hyperbola (\pm\cosh(t),\sinh(t)), t \in \R, is mapped onto the hyperbola \vec x = \vec p(t)=\vec f_0 \pm\vec f_1 \cosh t +\vec f_2 \sinh t \ . \vec f_0 is the center, \vec f_0+ \vec f_1 a point of the hyperbola and \vec f_2 a tangent vector at this point.


    Vertices

    In general the vectors \vec f_1, \vec f_2 are not perpendicular. That means, in general \vec f_0\pm \vec f_1 are ''not'' the vertices of the hyperbola. But \vec f_1\pm \vec f_2 point into the directions of the asymptotes. The tangent vector at point \vec p(t) is \vec p'(t) = \vec f_1\sinh t + \vec f_2\cosh t \ . Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter t_0 of a vertex from the equation \vec p'(t)\cdot \left(\vec p(t) -\vec f_0\right) = \left(\vec f_1\sinh t + \vec f_2\cosh t\right) \cdot \left(\vec f_1 \cosh t +\vec f_2 \sinh t\right) = 0 and hence from \coth (2t_0)= -\tfrac \ , which yields t_0=\tfrac\ln\tfrac. The formulae and \operatorname x = \tfrac\ln\tfrac were used. The two ''vertices'' of the hyperbola are \vec f_0\pm\left(\vec f_1\cosh t_0 +\vec f_2 \sinh t_0\right).


    Implicit representation

    Solving the parametric representation for \cosh t, \sinh t by
    Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
    and using \;\cosh^2t-\sinh^2t -1 = 0\; , one gets the implicit representation \det\left(\vec x\!-\!\vec f\!_0,\vec f\!_2\right)^2 - \det\left(\vec f\!_1,\vec x\!-\!\vec f\!_0\right)^2 - \det\left(\vec f\!_1,\vec f\!_2\right)^2 = 0 .


    Hyperbola in space

    The definition of a hyperbola in this section gives a parametric representation of an arbitrary hyperbola, even in space, if one allows \vec f\!_0, \vec f\!_1, \vec f\!_2 to be vectors in space.


    As an affine image of the hyperbola

    Because the unit hyperbola x^2-y^2=1 is affinely equivalent to the hyperbola y=1/x, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola \vec x = \vec p(t) = \vec f_0 + \vec f_1 t + \vec f_2 \tfrac, \quad t\ne 0\, . M: \vec f_0 is the center of the hyperbola, the vectors \vec f_1 , \vec f_2 have the directions of the asymptotes and \vec f_1 + \vec f_2 is a point of the hyperbola. The tangent vector is \vec p'(t)=\vec f_1 - \vec f_2 \tfrac. At a vertex the tangent is perpendicular to the major axis. Hence \vec p'(t)\cdot \left(\vec p(t) -\vec f_0\right) = \left(\vec f_1 - \vec f_2 \tfrac\right)\cdot\left(\vec f_1 t+ \vec f_2 \tfrac\right) = \vec f_1^2t-\vec f_2^2 \tfrac = 0 and the parameter of a vertex is t_0= \pm \sqrt \left, \vec f\!_1\ = \left, \vec f\!_2\ is equivalent to t_0 = \pm 1 and \vec f_0 \pm (\vec f_1+\vec f_2) are the vertices of the hyperbola. The following properties of a hyperbola are easily proven using the representation of a hyperbola introduced in this section.


    Tangent construction

    The tangent vector can be rewritten by factorization: \vec p'(t)=\tfrac\left(\vec f_1t - \vec f_2 \tfrac\right) \ . This means that This property provides a way to construct the tangent at a point on the hyperbola. This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem. ;Area of the grey parallelogram: The area of the grey parallelogram MAPB in the above diagram is \text = \left, \det\left( t\vec f_1, \tfrac\vec f_2\right)\ = \left, \det\left(\vec f_1,\vec f_2\right)\ = \cdots = \frac and hence independent of point P. The last equation follows from a calculation for the case, where P is a vertex and the hyperbola in its canonical form \tfrac-\tfrac=1 \, .


    Point construction

    For a hyperbola with parametric representation \vec x = \vec p(t) = \vec f_1 t + \vec f_2 \tfrac (for simplicity the center is the origin) the following is true: The simple proof is a consequence of the equation \tfrac\vec a = \tfrac\vec b. This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given. This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem.


    Tangent–asymptotes triangle

    For simplicity the center of the hyperbola may be the origin and the vectors \vec f_1,\vec f_2 have equal length. If the last assumption is not fulfilled one can first apply a parameter transformation (see above) in order to make the assumption true. Hence \pm (\vec f_1 + \vec f_2) are the vertices, \pm(\vec f_1-\vec f_2) span the minor axis and one gets , \vec f_1 + \vec f_2, = a and , \vec f_1 - \vec f_2, = b. For the intersection points of the tangent at point \vec p(t_0) = \vec f_1 t_0 + \vec f_2 \tfrac with the asymptotes one gets the points C = 2t_0\vec f_1,\ D = \tfrac\vec f_2. The ''
    area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
    '' of the triangle M,C,D can be calculated by a 2 × 2 determinant: A = \tfrac\Big, \det\left( 2t_0\vec f_1, \tfrac\vec f_2\right)\Big, = 2\Big, \det\left(\vec f_1,\vec f_2\right)\Big, (see rules for
    determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
    s). \left, \det(\vec f_1,\vec f_2)\ is the area of the rhombus generated by \vec f_1,\vec f_2. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes a,b of the hyperbola. Hence:


    Reciprocation of a circle

    The reciprocation of a
    circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
    ''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 In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into i ...
    , 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 e = \frac. Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle ''C''. This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle ''B'', as well as the
    envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...
    of the polar lines of the points on ''B''. Conversely, the circle ''B'' is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to ''B'' have no (finite) poles because they pass through the center C of the reciprocation circle ''C''; the polars of the corresponding tangent points on ''B'' are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle ''B'' that are separated by these tangent points.


    Quadratic equation

    A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (x, y) in the plane, A_ x^2 + 2 A_ xy + A_ y^2 + 2 B_x x + 2 B_y y + C = 0, provided that the constants A_, A_, A_, B_x, B_y, and C satisfy the determinant condition D := \begin A_ & A_ \\ A_ & A_ \end < 0. This determinant is conventionally called the
    discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
    of the conic section. A special case of a hyperbola—the '' degenerate hyperbola'' consisting of two intersecting lines—occurs when another determinant is zero: \Delta := \begin A_ & A_ & B_x \\ A_ & A_ & B_y \\ B_x & B_y & C \end = 0. This determinant \Delta is sometimes called the discriminant of the conic section. The general equation's coefficients can be obtained from known semi-major axis a, semi-minor axis b, center coordinates (x_\circ, y_\circ), and rotation angle \theta (the angle from the positive horizontal axis to the hyperbola's major axis) using the formulae: \begin A_ &= -a^2 \sin^2\theta + b^2 \cos^2\theta, & B_ &= -A_ x_\circ - A_ y_\circ, \\ ex A_ &= -a^2 \cos^2\theta + b^2 \sin^2\theta, & B_ &= - A_ x_\circ - A_ y_\circ, \\ ex A_ &= \left(a^2 + b^2\right) \sin\theta \cos\theta, & C &= A_ x_\circ^2 + 2A_ x_\circ y_\circ + A_ y_\circ^2 - a^2 b^2. \end These expressions can be derived from the canonical equation \frac - \frac = 1 by a translation and rotation of the coordinates \begin X &= \phantom+\left(x - x_\circ\right) \cos\theta &&+ \left(y - y_\circ\right) \sin\theta, \\ Y &= -\left(x - x_\circ\right) \sin\theta &&+ \left(y - y_\circ\right) \cos\theta. \end Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of coefficients. The center (x_c, y_c) of the hyperbola may be determined from the formulae \begin x_c &= -\frac \, \begin B_x & A_ \\ B_y & A_ \end \,, \\ exy_c &= -\frac \, \begin A_ & B_x \\ A_ & B_y \end \,. \end In terms of new coordinates, \xi = x - x_c and \eta = y - y_c, the defining equation of the hyperbola can be written A_ \xi^2 + 2A_ \xi\eta + A_ \eta^2 + \frac \Delta D = 0. The principal axes of the hyperbola make an angle \varphi with the positive x-axis that is given by \tan (2\varphi) = \frac. Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical form \frac - \frac = 1. The major and minor semiaxes a and b are defined by the equations \begin a^2 &= -\frac = -\frac, \\ exb^2 &= -\frac = -\frac, \end where \lambda_1 and \lambda_2 are the
    roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
    of the
    quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
    \lambda^2 - \left( A_ + A_ \right)\lambda + D = 0. For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is \frac - \frac = 0. The tangent line to a given point (x_0, y_0) on the hyperbola is defined by the equation E x + F y + G = 0 where E, F, and G are defined by \begin E &= A_ x_0 + A_ y_0 + B_x, \\ exF &= A_ x_0 + A_ y_0 + B_y, \\ exG &= B_x x_0 + B_y y_0 + C. \end The normal line to the hyperbola at the same point is given by the equation F(x - x_0) - E(y - y_0) = 0. The normal line is perpendicular to the tangent line, and both pass through the same point (x_0, y_0). From the equation \frac - \frac = 1, \qquad 0 < b \leq a, the left focus is (-ae,0) and the right focus is (ae,0), where e is the eccentricity. Denote the distances from a point (x, y) to the left and right foci as r_1 and r_2. For a point on the right branch, r_1 - r_2 = 2 a, and for a point on the left branch, r_2 - r_1 = 2 a. This can be proved as follows: If (x, y) is a point on the hyperbola the distance to the left focal point is r_1^2 = (x+a e)^2 + y^2 = x^2 + 2 x a e + a^2 e^2 + \left(x^2-a^2\right) \left(e^2-1\right) = (e x + a)^2. To the right focal point the distance is r_2^2 = (x-a e)^2 + y^2 = x^2 - 2 x a e + a^2 e^2 + \left(x^2-a^2\right) \left(e^2-1\right) = (e x - a)^2. If (x, y) is a point on the right branch of the hyperbola then ex > a and \begin r_1 &= e x + a, \\ r_2 &= e x - a. \end Subtracting these equations one gets r_1 - r_2 = 2a. If (x, y) is a point on the left branch of the hyperbola then ex < -a and \begin r_1 &= - e x - a, \\ r_2 &= - e x + a. \end Subtracting these equations one gets r_2 - r_1 = 2a.


    In Cartesian coordinates


    Equation

    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 :the ''foci'' are the points F_1=(c,0),\ F_2=(-c,0), :the ''vertices'' are V_1=(a, 0),\ V_2=(-a,0). For an arbitrary point (x,y) the distance to the focus (c,0) is \sqrt and to the second focus \sqrt. Hence the point (x,y) is on the hyperbola if the following condition is fulfilled \sqrt - \sqrt = \pm 2a \ . Remove the square roots by suitable squarings and use the relation b^2 = c^2-a^2 to obtain the equation of the hyperbola: \frac - \frac = 1 \ . This equation is called the
    canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
    of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is
    congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
    to the original (see
    below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fred Belo ...
    ). The axes of
    symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
    or ''principal axes'' are the ''transverse axis'' (containing the segment of length 2''a'' with endpoints at the vertices) and the ''conjugate axis'' (containing the segment of length 2''b'' perpendicular to the transverse axis and with midpoint at the hyperbola's center). As opposed to an ellipse, a hyperbola has only two vertices: (a,0),\; (-a,0). The two points (0,b),\; (0,-b) on the conjugate axes are ''not'' on the hyperbola. It follows from the equation that the hyperbola is ''symmetric'' with respect to both of the coordinate axes and hence symmetric with respect to the origin.


    Eccentricity

    For a hyperbola in the above canonical form, the
    eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
    is given by e=\sqrt. Two hyperbolas are geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by rigid left and right movements,
    rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
    , taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.


    Asymptotes

    Solving the equation (above) of the hyperbola for y yields y=\pm\frac \sqrt. It follows from this that the hyperbola approaches the two lines y=\pm \fracx for large values of , x, . These two lines intersect at the center (origin) and are called ''asymptotes'' of the hyperbola \tfrac-\tfrac= 1 \ . With the help of the second figure one can see that : The ''perpendicular distance from a focus to either asymptote'' is b (the semi-minor axis). From the
    Hesse normal form In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane \mathbb^2, a plane in Euclidean space \mathbb^3, or a hyperplane in higher dimensions.John Vince: ''Geometry for C ...
    \tfrac=0 of the asymptotes and the equation of the hyperbola one gets:Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", ''Mathematical Gazette'' 96, July 2012, 299–301. : The ''product of the distances from a point on the hyperbola to both the asymptotes'' is the constant \tfrac\ , which can also be written in terms of the eccentricity ''e'' as \left( \tfrac\right) ^2. From the equation y=\pm\frac\sqrt of the hyperbola (above) one can derive: : The ''product of the slopes of lines from a point P to the two vertices'' is the constant b^2/a^2\ . In addition, from (2) above it can be shown that : ''The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes'' is the constant \tfrac.


    Semi-latus rectum

    The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' p. A calculation shows p = \fraca. The semi-latus rectum p may also be viewed as the ''
    radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius ...
    '' at the vertices.


    Tangent

    The simplest way to determine the equation of the tangent at a point (x_0,y_0) is to implicitly differentiate the equation \tfrac-\tfrac= 1 of the hyperbola. Denoting ''dy/dx'' as ''y′'', this produces \frac-\frac= 0 \ \Rightarrow \ y'=\frac\frac\ \Rightarrow \ y=\frac\frac(x-x_0) +y_0. With respect to \tfrac-\tfrac= 1, the equation of the tangent at point (x_0,y_0) is \fracx-\fracy = 1. A particular tangent line distinguishes the hyperbola from the other conic sections. Let ''f'' be the distance from the vertex ''V'' (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2''f''. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.


    Rectangular hyperbola

    In the case a = b the hyperbola is called ''rectangular'' (or ''equilateral''), because its asymptotes intersect at right angles. For this case, the linear eccentricity is c=\sqrta, the eccentricity e=\sqrt and the semi-latus rectum p=a. The graph of the equation y=1/x is a rectangular hyperbola.


    Parametric representation with hyperbolic sine/cosine

    Using the hyperbolic sine and cosine functions \cosh,\sinh, a parametric representation of the hyperbola \tfrac-\tfrac= 1 can be obtained, which is similar to the parametric representation of an ellipse: (\pm a \cosh t, b \sinh t),\, t \in \R \ , which satisfies the Cartesian equation because \cosh^2 t -\sinh^2 t =1 . Further parametric representations are given in the section
    Parametric equations In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. In the case of a single parameter, parametric equations are commonly used to ...
    below.


    Conjugate hyperbola

    For the hyperbola \frac - \frac = 1, change the sign on the right to obtain the equation of the conjugate hyperbola: :\frac-\frac = -1 (which can also be written as \frac-\frac = 1). A hyperbola and its conjugate may have diameters which are conjugate. In the theory of
    special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
    , such diameters may represent axes of time and space, where one hyperbola represents events at a given spatial distance from the center, and the other represents events at a corresponding temporal distance from the center. :xy = c^2 and xy = -c^2 also specify conjugate hyperbolas.


    In polar coordinates


    Origin at the focus

    The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its ''origin in a focus'' and its x-axis pointing toward the origin of the "canonical coordinate system" as illustrated in the first diagram. In this case the angle \varphi is called true anomaly. Relative to this coordinate system one has that r = \frac, \quad p = \frac and -\arccos \left(-\frac 1 e\right) < \varphi < \arccos \left(-\frac 1 e\right).


    Origin at the center

    With polar coordinates relative to the "canonical coordinate system" (see second diagram) one has that r =\frac .\, For the right branch of the hyperbola the range of \varphi is -\arccos \left(\frac 1 e\right) < \varphi < \arccos \left(\frac 1 e\right).


    Eccentricity

    When using polar coordinates, the eccentricity of the hyperbola can be expressed as \sec\varphi_\text where \varphi_\text is the limit of the angular coordinate. As \varphi approaches this limit, ''r'' approaches infinity and the denominator in either of the equations noted above approaches zero, hence:Casey, John, (1885
    "A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples"
    /ref> e^2 \cos^2 \varphi_\text - 1 = 0 1 \pm e \cos \varphi_\text = 0 \implies e = \sec\varphi_\text


    Parametric equations

    A hyperbola with equation \tfrac - \tfrac = 1 can be described by several parametric equations: # Through hyperbolic trigonometric functions \begin x = \pm a \cosh t, \\ y = b \sinh t, \end \qquad t \in \R. # As a ''rational'' representation \begin x = \pm a \dfrac, \\ ex y = b \dfrac, \end \qquad t > 0 # Through circular trigonometric functions \begin x = \frac = a \sec t, \\ y = \pm b \tan t, \end \qquad 0 \le t < 2\pi,\ t \ne \frac,\ t \ne \frac \pi. # With the tangent slope as parameter: A parametric representation, which uses the slope m of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case b^2 by -b^2 and use formulae for the
    hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
    s. One gets \vec c_\pm(m) = \left(-\frac, \frac\right),\quad , m, > b/a. Here, \vec c_- is the upper, and \vec c_+ the lower half of the hyperbola. The points with vertical tangents (vertices (\pm a, 0)) are not covered by the representation. The equation of the tangent at point \vec c_\pm(m) is y = m x \pm\sqrt. This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.


    Hyperbolic functions

    Just as the
    trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
    s are defined in terms of the
    unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
    , so also the
    hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
    s are defined in terms of the
    unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...
    , as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the
    circular sector A circular sector, also known as circle sector or disk sector or simply a sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, with the smaller area being known as the ''minor ...
    which that angle subtends. The analogous
    hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functio ...
    is likewise defined as twice the area of a
    hyperbolic sector A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbol ...
    . Let a be twice the area between the x axis and a ray through the origin intersecting the unit hyperbola, and define (x,y) = (\cosh a,\sinh a) = (x, \sqrt) as the coordinates of the intersection point. Then the area of the hyperbolic sector is the area of the triangle minus the curved region past the vertex at (1,0): \begin \frac &= \frac - \int_1^x \sqrt \, dt \\ ex &= \frac \left(x\sqrt\right) - \frac \left(x\sqrt - \ln \left(x+\sqrt\right)\right), \end which simplifies to the area hyperbolic cosine a=\operatornamex=\ln \left(x+\sqrt\right). Solving for x yields the exponential form of the hyperbolic cosine: x=\cosh a=\frac. From x^2-y^2=1 one gets y=\sinh a=\sqrt=\frac, and its inverse the area hyperbolic sine: a=\operatornamey=\ln \left(y+\sqrt\right). Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example \operatornamea=\frac=\frac.


    Properties


    Reflection property

    The tangent at a point P bisects the angle between the lines \overline, \overline. This is called the ''optical property'' or ''reflection property'' of a hyperbola. ;Proof: Let L be the point on the line \overline with the distance 2a to the focus F_2 (see diagram, a is the semi major axis of the hyperbola). Line w is the bisector of the angle between the lines \overline, \overline. In order to prove that w is the tangent line at point P, one checks that any point Q on line w which is different from P cannot be on the hyperbola. Hence w has only point P in common with the hyperbola and is, therefore, the tangent at point P.
    From the diagram and the
    triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...
    one recognizes that , QF_2, <, LF_2, +, QL, =2a+, QF_1, holds, which means: , QF_2, -, QF_1, <2a. But if Q is a point of the hyperbola, the difference should be 2a.


    Midpoints of parallel chords

    The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola. The proof of the property on midpoints is best done for the hyperbola y=1/x. Because any hyperbola is an affine image of the hyperbola y=1/x (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:
    For two points P=\left(x_1,\tfrac \right), \ Q=\left(x_2,\tfrac \right) of the hyperbola y=1/x :the midpoint of the chord is M=\left(\tfrac,\cdots\right)=\cdots =\tfrac\; \left(1,\tfrac\right) \ ; :the slope of the chord is \frac=\cdots =-\tfrac \ . For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line y=\tfrac \; x \ . Consequence: for any pair of points P,Q of a chord there exists a ''skew reflection'' with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points P,Q and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line m, where all point-image pairs are on a line perpendicular to m. Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint M of a chord P Q divides the related line segment \overline P \, \overline Q between the asymptotes into halves, too. This means that , P\overline P, =, Q\overline Q, . This property can be used for the construction of further points Q of the hyperbola if a point P and the asymptotes are given. If the chord degenerates into a ''tangent'', then the touching point divides the line segment between the asymptotes in two halves.


    Orthogonal tangents – orthoptic

    For a hyperbola \frac-\frac=1, \, a>b the intersection points of ''orthogonal'' tangents lie on the circle x^2+y^2=a^2-b^2.
    This circle is called the ''orthoptic'' of the given hyperbola. The tangents may belong to points on different branches of the hyperbola. In case of a\le b there are no pairs of orthogonal tangents.


    Pole-polar relation for a hyperbola

    Any hyperbola can be described in a suitable coordinate system by an equation \tfrac-\tfrac= 1. The equation of the tangent at a point P_0=(x_0,y_0) of the hyperbola is \tfrac-\tfrac=1. If one allows point P_0=(x_0,y_0) to be an arbitrary point different from the origin, then :point P_0=(x_0,y_0)\ne(0,0) is mapped onto the line \frac-\frac=1 , not through the center of the hyperbola. This relation between points and lines is a
    bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
    . The
    inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
    maps :line y=mx+d,\ d\ne 0 onto the point \left(-\frac,-\frac\right) and :line x=c,\ c\ne 0 onto the point \left(\frac,0\right)\ . Such a relation between points and lines generated by a conic is called pole-polar relation or just ''polarity''. The pole is the point, the polar the line. See
    Pole and polar In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into i ...
    . By calculation one checks the following properties of the pole-polar relation of the hyperbola: * 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). ''Remarks:'' # The intersection point of two polars (for example: p_2,p_3) is the pole of the line through their poles (here: P_2,P_3). # The foci (c,0), and (-c,0) respectively and the directrices x=\tfrac and x=-\tfrac respectively belong to pairs of pole and polar. Pole-polar relations exist for ellipses and parabolas, too.


    Other properties

    * The following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola. * The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes. * Since both the transverse axis and the conjugate axis are axes of symmetry, the
    symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
    of a hyperbola is the
    Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
    . * The rectangular hyperbolas ''xy'' = constant admit
    group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...
    s by
    squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation (mathematics), rotation or shear mapping. For a fixed p ...
    s which have the hyperbolas as invariant sets.


    Arc length

    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. Then the integral giving the arc length s from x_ to x_ can be computed as: s = b\int_^ \sqrt \, \mathrm dv. 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 \, 1 + \frac\right)\Biggr_. Using only real numbers, this becomes s=b\left -\frac\right) - E\left(\operatornamev\,\Biggr, -\frac\right) + \sqrt\,\sinh v\right^ where F is the incomplete elliptic integral of the first kind with parameter m = k^2 and \operatornamev=\arctan\sinh v is the
    Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...
    .


    Derived curves

    Several other curves can be derived from the hyperbola by inversion, the so-called inverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the
    lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
    ; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a
    limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...
    or a strophoid, respectively.


    Elliptic coordinates

    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 \right)^2 - \left(\frac y \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 In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
    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'' = ''z''2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.


    Conic section analysis of the hyperbolic appearance of circles

    Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of circles, or more generally an ellipse. The viewer is typically a camera or the human eye and the image of the scene a
    central projection In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspa ...
    onto an image plane, that is, all projection rays pass a fixed point ''O'', the center. The lens plane is a plane parallel to the image plane at the lens ''O''. The image of a circle c is (Special positions where the circle plane contains point ''O'' are omitted.) These results can be understood if one recognizes that the projection process can be seen in two steps: 1) circle c and point ''O'' generate a cone which is 2) cut by the image plane, in order to generate the image. One sees a hyperbola whenever catching sight of a portion of a circle cut by one's lens plane. The inability to see very much of the arms of the visible branch, combined with the complete absence of the second branch, makes it virtually impossible for the human visual system to recognize the connection with hyperbolas.


    Applications


    Sundials

    Hyperbolas may be seen in many
    sundial A sundial is a horology, horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the position of the Sun, apparent position of the Sun in the sky. In the narrowest sense of the ...
    s. On any given day, the sun revolves in a circle on the
    celestial sphere In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
    , and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the ''declination line''). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a ''pelekinon'' by the Greeks, since it resembles a double-bladed axe.


    Multilateration

    A hyperbola is the basis for solving
    multilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth ( geopositioning). When more than three distances are involved, it may be called multilateration, f ...
    problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a
    LORAN LORAN (Long Range Navigation) was a hyperbolic navigation, hyperbolic radio navigation system developed in the United States during World War II. It was similar to the UK's Gee (navigation), Gee system but operated at lower frequencies in order ...
    or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2''a'' from two given points is a hyperbola of vertex separation 2''a'' whose foci are the two given points.


    Path followed by a particle

    The path followed by any particle in the classical
    Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force that varies in strength as the inverse square of the distance between them. The force may be either attra ...
    is a
    conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
    . In particular, if the total energy ''E'' of the particle is greater than zero (that is, if the particle is unbound), the path of such a particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering high-energy particles; for example, the Rutherford experiment demonstrated the existence of an
    atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
    by examining the scattering of
    alpha particle Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay but may also be produce ...
    s from
    gold Gold is a chemical element; it has chemical symbol Au (from Latin ) and atomic number 79. In its pure form, it is a brightness, bright, slightly orange-yellow, dense, soft, malleable, and ductile metal. Chemically, gold is a transition metal ...
    atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive
    Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the ''electrostatic ...
    , which satisfies the
    inverse square law In science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cau ...
    requirement for a Kepler problem.


    Korteweg–de Vries equation

    The hyperbolic trig function \operatorname\, x appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.


    Angle trisection

    As shown first by
    Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
    , a hyperbola can be used to trisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertex O, which intersects the sides of the angle at points A and B. Next draw the line segment with endpoints A and B and its perpendicular bisector \ell. Construct a hyperbola of
    eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
    ''e''=2 with \ell as directrix and B as a focus. Let P be the intersection (upper) of the hyperbola with the circle. Angle POB trisects angle AOB. To prove this, reflect the line segment OP about the line \ell obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.This construction is due to
    Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
    (circa 300 A.D.) and the proof comes from .


    Efficient portfolio frontier

    In
    portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversificatio ...
    , the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.


    Biochemistry

    In
    biochemistry Biochemistry, or biological chemistry, is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology, a ...
    and
    pharmacology Pharmacology is the science of drugs and medications, including a substance's origin, composition, pharmacokinetics, pharmacodynamics, therapeutic use, and toxicology. More specifically, it is the study of the interactions that occur betwee ...
    , the Hill equation and Hill-Langmuir equation respectively describe biological responses and the formation of
    protein–ligand complex A protein–ligand complex is a complex of a protein bound with a ligand (biochemistry), ligand that is formed following molecular recognition between proteins that interact with each other or with other molecules. Formation of a protein-ligand co ...
    es as functions of ligand concentration. They are both rectangular hyperbolae.


    Hyperbolas as plane sections of quadrics

    Hyperbolas appear as plane sections of the following
    quadric In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hype ...
    s: * Elliptic
    cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...
    * Hyperbolic
    cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
    *
    Hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...
    *
    Hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Equation, principal axes. A hyperboloid is the surface obtained ...
    * Hyperboloid of two sheets File:Quadric Cone.jpg, Elliptic cone File:Hyperbolic Cylinder Quadric.png, Hyperbolic cylinder File:Hyperbol Paraboloid.pov.png, Hyperbolic paraboloid File:Hyperboloid1.png, Hyperboloid of one sheet File:Hyperboloid2.png, Hyperboloid of two sheets


    See also


    Other conic sections

    *
    Circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
    *
    Ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
    *
    Parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
    *
    Degenerate conic In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more ...


    Other related topics

    * Elliptic coordinates, an orthogonal coordinate system based on families of
    ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
    s and hyperbolas. * Hyperbolic growth *
    Hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
    *
    Hyperbolic sector A hyperbolic sector is a region (mathematics), region of the Cartesian plane bounded by a hyperbola and two ray (geometry), rays from the origin to it. For example, the two points and on the Hyperbola#Rectangular hyperbola, rectangular hyperbol ...
    * Hyperboloid structure *
    Hyperbolic trajectory In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the ...
    *
    Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
    *
    Multilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth ( geopositioning). When more than three distances are involved, it may be called multilateration, f ...
    *
    Rotation of axes In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes ar ...
    *
    Translation of axes In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y' ...
    *
    Unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...


    Notes


    References

    * * *


    External links

    *
    Apollonius' Derivation of the Hyperbola
    a
    Convergence

    ''Mathematische Oeffeningen''
    Frans van Schooten, 1659 * {{Authority control Algebraic curves Analytic geometry Conic sections