{{short description|Plane curve: conic section}}{{About|a geometric curve|the term used in rhetoric|Hyperbole}}File:Hyperbola (PSF).svg|right|thumb|210px|A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double conedouble cone(File:Hyperbel-def-ass-e.svg|300px|thumb|Hyperbola (red): features)In mathematics, a hyperbola ({{IPAc-en|h|aɪ|ˈ|p|ɜr|b|ə|l|ə|audio=En-hyperbola (spoken word).ogg}}; pl. hyperbolas or hyperbolae {{IPAc-en|-|l|iː|audio=En-hyperbole (spoken word).ogg}}; adj. hyperbolic {{IPAc-en|ˌ|h|aɪ|p|ər|ˈ|b|ɒ|l|ɪ|k|audio=En-hyperbolic (spoken word).ogg}}) 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.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.{{harvtxt|Oakley|1944|p=17}} 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.{{harvtxt|Oakley|1944|p=17}}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

The word "hyperbola" derives from the Greek , 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.{{citation |title=Apollonius of Perga: Treatise on Conic Sections with Introductions Including an Essay on Earlier History on the Subject |first=Sir Thomas Little |last=Heath |publisher=Cambridge University Press |year=1896 |contribution=Chapter I. The discovery of conic sections. Menaechmus |pages=xvii–xxx |url=https://books.google.com/books?id=B0k0AQAAMAAJ&pg=PR17}}. The term hyperbola is believed to have been coined by Apollonius of Perga ({{circa|262|190 BC}}) in his definitive work on the conic sections, the Conics.{{citation |title=A History of Mathematics |first1=Carl B. |last1=Boyer |first2=Uta C. |last2=Merzbach |author2-link=Uta Merzbach|publisher=Wiley |year=2011 |isbn=9780470630563 |url=https://books.google.com/books?id=bR9HAAAAQBAJ&pg=RA2-PT73 |page=73 |quote=It was Apollonius (possibly following up a suggestion of Archimedes) who introduced the names "ellipse" and "hyperbola" in connection with these curves.}}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.{{citation|pages=30–31|first=Howard|last=Eves|title=A Survey of Geometry (Vol. One)|year=1963|publisher=Allyn and Bacon}}

Definitions

As locus of points

(File:Hyperbel-def-e.svg|thumb|Hyperbola: definition by the distances of points to two fixed points (foci))(File:Hyperbel-def-dc.svg|thumb|Hyperbola: definition with circular directrix)A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:{{block indent | em = 1.5 | text =A hyperbola is a set of points, such that for any point P of the set, the absolute difference of the distances |PF_1|,, |PF_2| to two fixed points F_1, F_2 (the foci) is constant, usually denoted by {{nowrap|2a,, a>0:}}{{harvtxt|Protter|Morrey|1970|pp=308–310}}H = left{P : left|left|PF_2right| - left|PF_1right|right| = 2a right} .}}The midpoint M of the line segment joining the foci is called the center of the hyperbola.{{harvtxt|Protter|Morrey|1970|p=310}} 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_2right| - left|PF_1right|right| = 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:

=

.c_2 is called the circular directrix (related to focus F_2) of the hyperbola.{{citation | first1=Tom M.|last1=Apostol|first2=Mamikon A.|last2=Mnatsakanian|title=New Horizons in Geometry|year=2012|publisher=The Mathematical Association of America | series=The Dolciani Mathematical Expositions #47 | isbn=978-0-88385-354-2 | page=251}}The German term for this circle is Leitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle). 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 {{math|1y A/x}} (File:Hyperbel-gs-hl.svg|thumb|Rotating the coordinate system in order to describe a rectangular hyperbola as graph of a function)(File:Hyperbeln-gs-3.svg|thumb|Three rectangular hyperbolas y = A / x with the coordinate axes as asymptotesred: A = 1; magenta: A = 4; blue: A = 9)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{xi+eta}{sqrt{2}},; y = tfrac{-xi+eta}{sqrt{2}} .The rectangular hyperbola tfrac{x^2-y^2}{a^2} = 1 (whose semi-axes are equal) has the new equation tfrac{2xieta}{a^2} = 1.Solving for eta yields eta = tfrac{a^2/2}{xi} . Thus, in an xy-coordinate system the graph of a function f: x mapsto tfrac{A}{x},; A>0; , with equationy = frac{A}{x};, 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{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=2sqrt{A} and the eccentricity e=sqrt{2} ; , the tangent y=-tfrac{A}{x_0^2}x+2tfrac{A}{x_0} 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 equationy = -frac{A}{x} ; , ~~ A>0; , the semi-axes a = b = sqrt{2A} ; , the line y = -x as major axis, the vertices left(-sqrt{A},sqrt{A}right), left(sqrt{A},-sqrt{A}right) ; . Shifting the hyperbola with equation y=frac{A}{x}, Ane 0 , so that the new center is {{nowrap|(c_0,d_0),}} yields the new equationy=frac{A}{x-c_0}+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 (File:Hyperbel-ll-e.svg|300px|thumb|Hyperbola: directrix property)(File:Hyperbel-ll-def.svg|300px|thumb|Hyperbola: definition with directrix property)The two lines at distance d = frac{a^2}c 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{|PF_1|}{|Pl_1|} = frac{|PF_2|}{|Pl_2|} = e= frac{c}{a} , .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{a^2}{c}right)^2 and y^2 = tfrac{b^2}{a^2}x^2-b^2 satisfy the equation

< 1 an ellipse.)

Proof

Let F=(f,0) , e >0 and assume (0,0) is a point on the curve.The directrix l has equation x=-tfrac{f}{e}. With P=(x,y), the relation |PF|^2 = e^2|Pl|^2 produces the equations The substitution p=f(1+e) yieldsx^2(e^2-1)+2px-y^2 = 0.This is the equation of an ellipse (e1). 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{b^2}{a^2}, text { and } p = tfrac{b^2}{a}, and then the equation above becomesfrac{(x+a)^2}{a^2} - frac{y^2}{b^2} = 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.(File:Hyperbel-leitl-e.svg|thumb|upright=1.4|Hyperbola: construction of a directrix)

Construction of a directrix

Because of c cdot tfrac{a^2}{c}=a^2 point L_1 of directrix l_1 (see diagram) and focus F_1 are inverse with respect to the circle inversion 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{F_1F_2} 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

(File:Dandelin-hyperbel.svg|thumb|upright=2|Hyperbola (red): two views of a cone and two Dandelin spheres d1, d2)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 d_1, d_2, which are spheres that touch the cone along circles {{nowrap|c_1,}} c_2 and the intersecting (hyperbola) plane at points F_1 and {{nowrap|F_2.}} It turns out: F_1, F_2 are the foci of the hyperbola.

1. Let P be an arbitrary point of the intersection curve .
2. The generatrix of the cone containing P intersects circle c_1 at point A and circle c_2 at a point B.
3. The line segments overline{PF_1} and overline{PA} are tangential to the sphere d_1 and, hence, are of equal length.
4. The line segments overline{PF_2} and overline{PB} are tangential to the sphere d_2 and, hence, are of equal length.
5. The result is: |PF_1| - |PF_2| = |PA| - |PB| = |AB| is independent of the hyperbola point {{nowrap|P,}} because no matter where point P is, A, B have to be on circles {{nowrap|c_1,}} {{nowrap|c_2 ,}} and line segment AB has to cross the apex. Therefore, as point P moves along the red curve (hyperbola), line segment overline{AB} simply rotates about apex without changing its length.

Pin and string construction

(File:Hyperbola-pin-string.svg|300px|thumb|Hyperbola: 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: Frans van Schooten: Mathematische Oeffeningen, Leyden, 1659, p. 327

1. Choose the foci F_1,F_2, the vertices V_1,V_2 and one of the circular directrices , for example c_2 (circle with radius 2a)
2. A ruler is fixed at point F_2 free to rotate around F_2. Point B is marked at distance 2a.
3. A string with length |AB| is prepared.
4. One end of the string is pinned at point A on the ruler, the other end is pinned to point F_1.
5. Take a pen and hold the string tight to the edge of the ruler.
6. 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

(File:Hyperbel-steiner-e.svg|250px|thumb|Hyperbola: Steiner generation)(File:Hyperbola construction - parallelogram method.gif|200px|thumb|Hyperbola y = 1/x: Steiner generation)The following method to construct single points of a hyperbola relies on the Steiner generation of a non degenerate conic section:{{block indent | em = 1.5 | text = Given two pencils B(U),B(V) of lines at two points U,V (all lines containing U and V, respectively) and a projective but not perspective mapping pi of B(U) onto B(V), then the intersection points of corresponding lines form a non-degenerate projective conic section.}}For the generation of points of the hyperbola tfrac{x^2}{a^2}-tfrac{y^2}{b^2} = 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{BP} is divided into n equally-spaced segments and this division is projected parallel with the diagonal AB as direction onto the line segment overline{AP} (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 {{math|1y a/(x − b) + c}} and the 3-point-form

(File:Hyperbel-pws-s.svg|250px|thumb|Hyperbola: inscribed angle theorem)A hyperbola with equation y=tfrac{a}{x-b}+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:{{block indent | em = 1.5 | text = In order to measure an angle between two lines with equations y=m_1x+d_1, y=m_2x + d_2 ,m_1,m_2 ne 0 in this context one uses the quotientfrac{m_1}{m_2} .}}Analogous to the inscribed angle theorem for circles one gets the{{math theorem, quad tne 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 isvec p'(t)=vec f_1 - vec f_2 tfrac{1}{t^2}.At a vertex the tangent is perpendicular to the major axis. Hencevec p'(t)cdot left(vec p(t) -vec f_0right) = left(vec f_1 - vec f_2 tfrac{1}{t^2}right)cdotleft(vec f_1 t+ vec f_2 tfrac{1}right) = vec f_1^2t-vec f_2^2 tfrac{1}{t^3} = 0and the parameter of a vertex ist_0= pm sqrt[4]{frac{vec f_2^2}{vec f_1^2}}.left|vec f!_1right| = left|vec f!_2right| 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

(File:Hyperbel-tang-s.svg|thumb|Tangent construction: asymptotes and P given → tangent)The tangent vector can be rewritten by factorization:vec p'(t)=tfrac{1}left(vec f_1t - vec f_2 tfrac{1}right) .This means that{{block indent | em = 1.5 | text = the diagonal AB of the parallelogram M: vec f_0, A=vec f_0+vec f_1t, B: vec f_0+ vec f_2 tfrac{1}, P: vec f_0+vec f_1t+vec f_2 tfrac{1} is parallel to the tangent at the hyperbola point P (see diagram).}}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.Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, S. 33, (PDF; 757 kB)

Area of the grey parallelogram:

The area of the grey parallelogram MAPB in the above diagram istext{Area} = left|detleft( tvec f_1, tfrac{1}vec f_2right)right| = left|detleft(vec f_1,vec f_2right)right| = cdots = frac{a^2+b^2}{4} 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{x^2}{a^2}-tfrac{y^2}{b^2}=1 , .

Point construction

(File:Hyperbel-pasc4-s.svg|thumb|Point construction: asymptotes and P1 are given → P2)For a hyperbola with parametric representation vec x = vec p(t) = vec f_1 t + vec f_2 tfrac{1} (for simplicity the center is the origin) the following is true:{{block indent | em = 1.5 | text = For any two points P_1: vec f_1 t_1+ vec f_2 tfrac{1}{t_1}, P_2: vec f_1 t_2+ vec f_2 tfrac{1}{t_2} the pointsA: vec a =vec f_1 t_1+ vec f_2 tfrac{1}{t_2}, B: vec b=vec f_1 t_2+ vec f_2 tfrac{1}{t_1}are collinear with the center of the hyperbola (see diagram).}}The simple proof is a consequence of the equation tfrac{1}{t_1}vec a = tfrac{1}{t_2}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.Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, S. 32, (PDF; 757 kB)

Tangent–asymptotes triangle

(File:Hyperbel-tad-s.svg|thumb|Hyperbola: 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{1}{t_0} with the asymptotes one gets the pointsC = 2t_0vec f_1, D = tfrac{2}{t_0}vec f_2.The area of the triangle M,C,D can be calculated by a 2 × 2 determinant:A = tfrac{1}{2}Big|detleft( 2t_0vec f_1, tfrac{2}{t_0}vec f_2right)Big| = 2Big|detleft(vec f_1,vec f_2right)Big|(see rules for determinants).left|det(vec f_1,vec f_2)right| 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:{{block indent | em = 1.5 | text = The area of the triangle MCD is independent of the point of the hyperbola: A = ab.}}

Reciprocation of a circle

The reciprocation of a circleB in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, thene = frac{overline{BC}}{r}.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 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_{xx} x^2 + 2 A_{xy} xy + A_{yy} y^2 + 2 B_x x + 2 B_y y + C = 0,provided that the constants A_{xx}, A_{xy}, A_{yy}, B_x, B_y, and C satisfy the determinant conditionD := begin{vmatrix} This determinant is conventionally called the discriminant of the conic section.BOOK, Math refresher for scientists and engineers, John R., Fanchi, John Wiley and Sons, 2006, 0-471-75715-2,weblinkSection 3.2, pages 44–45, A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:Delta := begin{vmatrix} end{vmatrix} = 0.This determinant Delta is sometimes called the discriminant of the conic section.BOOK, Korn, Granino A, Theresa M. Korn, Korn, Theresa M., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover Publ., second, 2000, 40, 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{align} end{align}These expressions can be derived from the canonical equationfrac{X^2}{a^2} - frac{Y^2}{b^2} = 1by a translation and rotation of the coordinates {{nobr|(x, y):}}begin{alignat}{2}X &= phantom+left(x - x_circright) costheta &&+ left(y - y_circright) sintheta, Y &= -left(x - x_circright) sintheta &&+ left(y - y_circright) costheta.end{alignat}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 formulaebegin{align}x_c &= -frac{1}{D} , begin{vmatrix} B_x & A_{xy} B_y & A_{yy} end{vmatrix} ,, [1ex]y_c &= -frac{1}{D} , begin{vmatrix} A_{xx} & B_x A_{xy} & B_y end{vmatrix} ,.end{align}In terms of new coordinates, xi = x - x_c and eta = y - y_c, the defining equation of the hyperbola can be writtenA_{xx} xi^2 + 2A_{xy} xieta + A_{yy} eta^2 + frac Delta D = 0.The principal axes of the hyperbola make an angle varphi with the positive x-axis that is given bytan (2varphi) = frac{2A_{xy}}{A_{xx} - A_{yy}}.Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical formfrac{x^2}{a^2} - frac{y^2}{b^2} = 1.The major and minor semiaxes a and b are defined by the equationsbegin{align}a^2 &= -frac{Delta}{lambda_1 D} = -frac{Delta}{lambda_1^2 lambda_2}, [1ex]b^2 &= -frac{Delta}{lambda_2 D} = -frac{Delta}{lambda_1 lambda_2^2},end{align}where lambda_1 and lambda_2 are the roots of the quadratic equationlambda^2 - left( A_{xx} + A_{yy} right)lambda + D = 0.For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) isfrac{x^2}{a^2} - frac{y^2}{b^2} = 0.The tangent line to a given point (x_0, y_0) on the hyperbola is defined by the equationE x + F y + G = 0where E, F, and G are defined bybegin{align}E &= A_{xx} x_0 + A_{xy} y_0 + B_x, [1ex]F &= A_{xy} x_0 + A_{yy} y_0 + B_y, [1ex]G &= B_x x_0 + B_y y_0 + C.end{align}The normal line to the hyperbola at the same point is given by the equationF(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 equationfrac{x^2}{a^2} - frac{y^2}{b^2} = 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, and for a point on the left branch, This can be proved as follows:If (x, y) is a point on the hyperbola the distance to the left focal point isr_1^2

(x+a e)^2 + y^2

x^2 + 2 x a e + a^2 e^2 + left(x^2-a^2right) left(e^2-1right)

(e x + a)^2.

To the right focal point the distance isr_2^2

(x-a e)^2 + y^2

x^2 - 2 x a e + a^2 e^2 + left(x^2-a^2right) left(e^2-1right)

(e x - a)^2.

If (x, y) is a point on the right branch of the hyperbola then ex > a andbegin{align}r_1 &= e x + a, r_2 &= e x - a.end{align}Subtracting these equations one getsr_1 - r_2 = 2a.If (x, y) is a point on the left branch of the hyperbola then ex < -a andbegin{align}r_1 &= - e x - a, r_2 &= - e x + a.end{align}Subtracting these equations one getsr_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 For an arbitrary point (x,y) the distance to the focus (c,0) is sqrt{ (x-c)^2 + y^2 } and to the second focus sqrt{ (x+c)^2 + y^2 }. Hence the point (x,y) is on the hyperbola if the following condition is fulfilledsqrt{(x-c)^2 + y^2} - sqrt{(x+c)^2 + y^2} = 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{x^2}{a^2} - frac{y^2}{b^2} = 1 .This equation is called the canonical form 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 to the original (see below).The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center).{{harvtxt|Protter|Morrey|1970|p=310}} 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 is given bye=sqrt{1+frac{b^2}{a^2}}.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, taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.

Asymptotes

(File:Hyperbel-param-e.svg|250px|thumb|Hyperbola: semi-axes a,b, linear eccentricity c, semi latus rectum p)(File:Hyperbola-3prop.svg|300px|thumb|Hyperbola: 3 properties)Solving the equation (above) of the hyperbola for y yieldsy=pmfrac{b}{a} sqrt{x^2-a^2}.It follows from this that the hyperbola approaches the two linesy=pm frac{b}{a}x for large values of |x|. These two lines intersect at the center (origin) and are called asymptotes of the hyperbola tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1 .{{harvtxt|Protter|Morrey|1970|pp=APP-29–APP-30}}With the help of the second figure one can see that From the Hesse normal form tfrac{bxpm ay}{sqrt{a^2+b^2}}=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. From the equation y=pmfrac{b}{a}sqrt{x^2-a^2} of the hyperbola (above) one can derive: In addition, from (2) above it can be shown that

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 showsp = frac{b^2}a.The semi-latus rectum p may also be viewed as the radius of curvature 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{x^2}{a^2}-tfrac{y^2}{b^2}= 1 of the hyperbola. Denoting dy/dx as y′, this producesfrac{2x}{a^2}-frac{2yy'}{b^2}= 0 Rightarrow y'=frac{x}{y}frac{b^2}{a^2} Rightarrow y=frac{x_0}{y_0}frac{b^2}{a^2}(x-x_0) +y_0.With respect to tfrac{x_0^2}{a^2}-tfrac{y_0^2}{b^2}= 1, the equation of the tangent at point (x_0,y_0) isfrac{x_0}{a^2}x-frac{y_0}{b^2}y = 1.A particular tangent line distinguishes the hyperbola from the other conic sections.J. W. Downs, Practical Conic Sections, Dover Publ., 2003 (orig. 1993): p. 26. 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 2f. 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=sqrt{2}a, the eccentricity e=sqrt{2} 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{x^2}{a^2}-tfrac{y^2}{b^2}= 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 below.File:Conjugate-unit-hyperbolas.svg|thumb|Here {{nowrap|a {{=}} b {{=}} 1}} giving the unit hyperbolaunit hyperbola

Conjugate hyperbola

Exchange frac{x^2}{a^2} and frac{y^2}{b^2} to obtain the equation of the conjugate hyperbola (see diagram):frac{y^2}{b^2}-frac{x^2}{a^2}= 1 , also written asfrac{x^2}{a^2}-frac{y^2}{b^2}= -1 .A hyperbola and its conjugate may have diameters which are conjugate. In the theory of special relativity, 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.

In polar coordinates

(File:Hyperbel-pold-f-s.svg|thumb|Hyperbola: Polar coordinates with pole = focus)(File:Hyperbel-pold-m-s.svg|thumb|Hyperbola: Polar coordinates with pole = center)(File:Hyperbola_polar_animation.gif|thumb|Animated plot of Hyperbola by using r = frac{p}{1 - e cos theta})

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 towards 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 thatr = frac{p}{1 mp e cos varphi}, quad p = frac{b^2}{a}and-arccos left(-frac 1 eright) < varphi < arccos left(-frac 1 eright).

Origin at the center

With polar coordinates relative to the "canonical coordinate system" (see second diagram)one has thatr =frac{b}{sqrt{e^2 cos^2 varphi -1}} .,For the right branch of the hyperbola the range of varphi is-arccos left(frac 1 eright) < varphi < arccos left(frac 1 eright).

Parametric equations

A hyperbola with equation tfrac{x^2}{a^2} - tfrac{y^2}{b^2} = 1 can be described by several parametric equations:

1. Through hyperbolic trigonometric functions

1. As a rational representation

1. Through circular trigonometric functions

1. With the tangent slope as parameter: {{pb}} 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 functions. One gets vec c_pm(m) = left(-frac{ma^2}{pmsqrt{m^2a^2 - b^2}}, frac{-b^2}{pmsqrt{m^2a^2 - b^2}}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. {{pb}} The equation of the tangent at point vec c_pm(m) is y = m x pmsqrt{m^2a^2 - b^2}. This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.

Hyperbolic functions

Image:Hyperbolic functions-2.svg|thumb|296px|right|A ray through the unit hyperbolaunit hyperbolaJust as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector which that angle subtends. The analogous hyperbolic angle is likewise defined as twice the area of a hyperbolic sector.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{x^2-1}) 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{align}frac{a}{2} &= frac{xy}{2} - int_1^x sqrt{t^{2}-1} , dt [1ex] end{align}which simplifies to the area hyperbolic cosinea=operatorname{arcosh}x=ln left(x+sqrt{x^2-1}right).Solving for x yields the exponential form of the hyperbolic cosine:x=cosh a=frac{e^a+e^{-a}}{2}.From x^2-y^2=1 one getsy=sinh a=sqrt{cosh^2 a - 1}=frac{e^a-e^{-a}}{2},and its inverse the area hyperbolic sine:a=operatorname{arsinh}y=ln left(y+sqrt{y^2+1}right).Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for exampleoperatorname{tanh}a=frac{sinh a}{cosh a}=frac{e^{2a}-1}{e^{2a}+1}.

Properties

Reflection property

(File:Hyperbel-wh-s.svg|300px|thumb|Hyperbola: the tangent bisects the lines through the foci)The tangent at a point P bisects the angle between the lines overline{PF_1}, overline{PF_2}. This is called the optical property or reflection property of a hyperbola. {{citation |last1=Coffman |first1=R. T. |last2=Ogilvy |first2=C. S. |year=1963 |title=The 'Reflection Property' of the Conics |journal=Mathematics Magazine |volume=36 |number=1 |pages=11–12 |doi=10.2307/2688124 }} {{pb}} {{citation |last=Flanders |first=Harley |year=1968 |title=The Optical Property of the Conics |journal=American Mathematical Monthly |volume=75 |number=4 |page=399 |doi=10.2307/2313439 }} {{pb}}{{citation |last=Brozinsky |first=Michael K. |year=1984 |title=Reflection Property of the Ellipse and the Hyperbola |journal=College Mathematics Journal |volume=15 |number=2 |pages=140–42 |doi=10.2307/2686519 }}

Proof:

Let L be the point on the line overline{PF_2} 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{PF_1}, overline{PF_2}. 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 one recognizes that |QF_2|

^2-frac{c^2}{a^2}

^2 = 0 .The second case is proven analogously.(File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex and common semi latus rectum)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 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 eH = left{P , Biggr| , frac{|PF|}{|Pl|} = eright} is a hyperbola.(The choice e = 1 yields a parabola and if e

E. Hartmann: Lecture Note 'Planar Circle Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 93W. Benz: Vorlesungen über Geomerie der Algebren, Springer Science+Business Media>Springer (1973)| math_statement = For four points P_i = (x_i,y_i), i=1,2,3,4, x_ine x_k, y_ine y_k, ine k (see diagram) the following statement is true:The four points are on a hyperbola with equation y = tfrac{a}{x-b} + c if and only if the angles at P_3 and P_4 are equal in the sense of the measurement above. That means if frac{(y_4-y_1)}{(x_4-x_1)}frac{(x_4-x_2)}{(y_4-y_2)}=frac{(y_3-y_1)}{(x_3-x_1)}frac{(x_3-x_2)}{(y_3-y_2)}The proof can be derived by straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is {{nowrap|y = a/x.}}}}A consequence of the inscribed angle theorem for hyperbolas is the{{math theorem| name = 3-point-form of a hyperbola's equation| math_statement = The equation of the hyperbola determined by 3 points P_i=(x_i,y_i), i=1,2,3, x_ine x_k, y_ine y_k, ine k is the solution of the equation frac{({color{red}y}-y_1)}{({color{green}x}-x_1)}frac{({color{green}x}-x_2)}{({color{red}y}-y_2)}=frac{(y_3-y_1)}{(x_3-x_1)}frac{(x_3-x_2)}{(y_3-y_2)} for {color{red}y}.}} As an affine image of the unit hyperbola {{math|1x2 − y2 1}} (File:Hyperbel-aff-s.svg|300px|thumb|Hyperbola as an affine image of the unit hyperbola)Another definition of a hyperbola uses affine transformations:{{block indent | em = 1.5 | text = Any hyperbola is the affine image of the unit hyperbola with equation x^2 - y^2 = 1.}} Parametric representation An affine transformation of the Euclidean plane has the form vec x to vec f_0+Avec x, where A is a regular matrix (its determinant 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 (pmcosh(t),sinh(t)), t in R, is mapped onto the hyperbolavec x = vec p(t)=vec f_0 pmvec 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_0pm vec f_1 are not the vertices of the hyperbola. But vec f_1pm vec f_2 point into the directions of the asymptotes. The tangent vector at point vec p(t) isvec p'(t) = vec f_1sinh t + vec f_2cosh 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 equationvec p'(t)cdot left(vec p(t) -vec f_0right) = left(vec f_1sinh t + vec f_2cosh tright) cdot left(vec f_1 cosh t +vec f_2 sinh tright) = 0and hence fromcoth (2t_0)= -tfrac{vec f_1^{, 2}+vec f_2^{, 2}}{2vec f_1 cdot vec f_2} ,which yieldst_0=tfrac{1}{4}lntfrac{left(vec f_1-vec f_2right)^2}{left(vec f_1+vec f_2right)^2}.The formulae {{nowrap|cosh^2 x + sinh^2 x = cosh 2x,}} {{nowrap|2sinh x cosh x = sinh 2x,}} and operatorname{arcoth} x = tfrac{1}{2}lntfrac{x+1}{x-1} were used.The two vertices of the hyperbola are vec f_0pmleft(vec f_1cosh t_0 +vec f_2 sinh t_0right). Implicit representation Solving the parametric representation for cosh t, sinh t by Cramer's rule and using ;cosh^2t-sinh^2t -1 = 0; , one gets the implicit representationdetleft(vec x!-!vec f!_0,vec f!_2right)^2 - detleft(vec f!_1,vec x!-!vec f!_0right)^2 - detleft(vec f!_1,vec f!_2right)^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 {{math|1y 1/x}} (File:Hyperbel-aff2.svg|thumb|300px|Hyperbola as affine image of y = 1/x)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 {{nowrap|y = 1/x , :}}vec x = vec p(t) = vec f_0 + vec f_1 t + vec f_2 tfrac{1}