Editing Hyperbola

{{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 (geometry)|plane]] with both halves of a [[double cone (geometry)|double cone]]. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of the line.]]
[[File:Hyperbel-def-ass-e.svg|300px|thumb|Hyperbola (red): features]]

In [[mathematics]], a '''hyperbola''' is a type of [[smooth function|smooth]] [[plane curve|curve lying in a plane]], defined by its geometric properties or by [[equation]]s for which it is the solution set. A hyperbola has two pieces, called [[connected component (topology)|connected components]] or branches, that are mirror images of each other and resemble two infinite [[bow (weapon)|bows]]. The hyperbola is one of the three kinds of [[conic section]], formed by the intersection of a [[plane (mathematics)|plane]] and a double [[cone (geometry)|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 (mathematics)|locus]] of points whose difference of distances to two fixed [[focus (geometry)|foci]] is constant, as a curve for each point of which the rays to two fixed foci are [[reflection (mathematics)|reflection]]s across the [[tangent line]] at that point, or as the solution of certain bivariate [[quadratic function|quadratic equation]]s such as the [[multiplicative inverse|reciprocal]] relationship <math>xy = 1.</math>{{sfn|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 [[Rutherford scattering|scattering trajectory]] of a [[subatomic particle]], among others.

Each [[branch (algebraic geometry)|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 <math>y(x) = 1/x</math> the asymptotes are the two [[coordinate axes]].{{sfn|Oakley|1944|p=17}}

Hyperbolas share many of the ellipses' analytical properties such as [[eccentricity (mathematics)|eccentricity]], [[focus (geometry)|focus]], and [[directrix (conic section)|directrix]]. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other [[mathematical object]]s have their origin in the hyperbola, such as [[hyperbolic paraboloid]]s (saddle surfaces), [[hyperboloid]]s ("wastebaskets"), [[hyperbolic geometry]] ([[Nikolai Lobachevsky|Lobachevsky]]'s celebrated [[non-Euclidean geometry]]), [[hyperbolic function]]s (sinh, cosh, tanh, etc.), and [[gyrovector space]]s (a geometry proposed for use in both [[theory of relativity|relativity]] and [[quantum mechanics]] which is not [[Euclidean geometry|Euclidean]]).

==Etymology and history==
The word "hyperbola" derives from the [[Greek language|Greek]] {{lang|grc|ὑπερβολή}}, meaning "over-thrown" or "excessive", from which the English term [[hyperbole]] also derives. Hyperbolae were discovered by [[Menaechmus]] in his investigations of the problem of [[doubling the cube]], but were then called sections of obtuse cones.<ref>{{citation |title=Apollonius of Perga: Treatise on Conic Sections with Introductions Including an Essay on Earlier History on the Subject |last=Heath |first=Sir Thomas Little |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}}.</ref> The term hyperbola is believed to have been coined by [[Apollonius of Perga]] ({{circa|262|190 BC}}) in his definitive work on the [[conic section]]s, the ''Conics''.<ref>{{citation |title=A History of Mathematics |last1=Boyer |first1=Carl B. |last2=Merzbach |first2=Uta C. |author2-link=Uta Merzbach |publisher=Wiley |year=2011 |isbn=9780470630563 |page=73 |url=https://books.google.com/books?id=bR9HAAAAQBAJ&pg=RA2-PT73 |quote=It was Apollonius (possibly following up a suggestion of Archimedes) who introduced the names "ellipse" and "hyperbola" in connection with these curves.}}</ref>
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.<ref>{{citation |pages=30–31 |last=Eves |first=Howard |title=A Survey of Geometry (Vol. One) |year=1963 |publisher=Allyn and Bacon}}</ref>

==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 (mathematics)|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 <math>P</math> of the set, the absolute difference of the distances <math>|PF_1|,\, |PF_2|</math> to two fixed points <math>F_1, F_2</math> (the ''foci'') is constant, usually denoted by {{nowrap|<math>2a,\, a>0</math>:}}{{sfn|Protter|Morrey|1970|pp=308–310}}

<math display="block">H = \left\{P : \left|\left|PF_2\right| - \left|PF_1\right|\right| = 2a \right\} .</math>
}}
The midpoint <math>M</math> of the line segment joining the foci is called the ''center'' of the hyperbola.{{sfn|Protter|Morrey|1970|p=310}} The line through the foci is called the ''major axis''. It contains the ''vertices'' <math>V_1, V_2</math>, which have distance <math>a</math> to the center. The distance <math>c</math> of the foci to the center is called the ''focal distance'' or ''linear eccentricity''. The quotient <math>\tfrac c a</math> is the ''eccentricity'' <math>e</math>.

The equation <math>\left|\left|PF_2\right| - \left|PF_1\right|\right| = 2a</math> can be viewed in a different way (see diagram):<br/>
If <math>c_2</math> is the circle with midpoint <math>F_2</math> and radius <math>2a</math>, then the distance of a point <math>P</math> of the right branch to the circle <math>c_2</math> equals the distance to the focus <math>F_1</math>:
<math display="block">|PF_1|=|Pc_2|.</math>
<math>c_2</math> is called the ''circular directrix'' (related to focus <math>F_2</math>) of the hyperbola.<ref>{{citation |last1=Apostol |first1=Tom M. |last2=Mnatsakanian |first2=Mamikon A. |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}}</ref><ref>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]]).</ref> In order to get the left branch of the hyperbola, one has to use the circular directrix related to <math>F_1</math>. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.

===Hyperbola with equation {{math|1=''y'' = ''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 <math>y = A / x</math> with the coordinate axes as asymptotes<br/>
red: ''A'' = 1; magenta: ''A'' = 4; blue: ''A'' = 9]]
If the ''xy''-coordinate system is [[rotation matrix|rotated]] about the origin by the angle <math>+45^\circ</math> and new coordinates <math>\xi,\eta</math> are assigned, then <math>x = \tfrac{\xi+\eta}{\sqrt{2}},\; y = \tfrac{-\xi+\eta}{\sqrt{2}} </math>.<br/>
The rectangular hyperbola <math>\tfrac{x^2-y^2}{a^2} = 1</math> (whose [[semi-major and semi-minor axes|semi-axes]] are equal) has the new equation <math>\tfrac{2\xi\eta}{a^2} = 1</math>.
Solving for <math>\eta</math> yields <math>\eta = \tfrac{a^2/2}{\xi} \ . </math>

Thus, in an ''xy''-coordinate system the graph of a function <math>f: x \mapsto \tfrac{A}{x},\; A>0\; , </math> with equation
<math display="block">y = \frac{A}{x}\;, A>0\; ,</math> is a ''rectangular hyperbola'' entirely in the first and third [[quadrant (plane geometry)|quadrants]] with
*the coordinate axes as ''asymptotes'',
*the line <math>y = x</math> as ''major axis'' ,
*the ''center'' <math>(0,0)</math> and the ''semi-axis'' <math> a = b = \sqrt{2A} \; ,</math>
*the ''vertices'' <math>\left(\sqrt{A},\sqrt{A}\right), \left(-\sqrt{A},-\sqrt{A}\right) \; ,</math>
*the ''semi-latus rectum'' and ''radius of curvature '' at the vertices <math> p=a=\sqrt{2A} \; ,</math>
*the ''linear eccentricity'' <math>c=2\sqrt{A}</math> and the eccentricity <math>e=\sqrt{2} \; ,</math>
*the ''tangent'' <math>y=-\tfrac{A}{x_0^2}x+2\tfrac{A}{x_0}</math> at point <math>(x_0,A/x_0)\; .</math>

A rotation of the original hyperbola by <math>-45^\circ</math> 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 <math>+45^\circ</math> rotation, with equation
<math display="block">y = -\frac{A}{x} \; , ~~ A>0\; ,</math>
*the ''semi-axes'' <math> a = b = \sqrt{2A} \; ,</math>
*the line <math> y = -x</math> as major axis,
*the ''vertices'' <math>\left(-\sqrt{A},\sqrt{A}\right), \left(\sqrt{A},-\sqrt{A}\right) \; .</math>

Shifting the hyperbola with equation <math>y=\frac{A}{x}, \ A\ne 0\ ,</math> so that the new center is {{nowrap|<math>(c_0,d_0)</math>,}} yields the new equation
<math display="block">y=\frac{A}{x-c_0}+d_0\; ,</math>
and the new asymptotes are <math>x=c_0 </math> and <math>y=d_0</math>. The shape parameters <math>a,b,p,c,e</math> 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 <math display="inline">d = \frac{a^2}c</math> from the center and parallel to the minor axis are called '''directrices''' of the hyperbola (see diagram).

For an arbitrary point <math>P</math> of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
<math display="block">\frac{|PF_1|}{|Pl_1|} = \frac{|PF_2|}{|Pl_2|} = e= \frac{c}{a} \, .</math>
The proof for the pair <math>F_1, l_1</math> follows from the fact that <math>|PF_1|^2 = (x-c)^2+y^2,\ |Pl_1|^2 = \left(x-\tfrac{a^2}{c}\right)^2</math> and <math>y^2 = \tfrac{b^2}{a^2}x^2-b^2</math> satisfy the equation
<math display="block">|PF_1|^2-\frac{c^2}{a^2}|Pl_1|^2 = 0\ .</math>
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 <math>F</math> (focus), any line <math>l</math> (directrix) not through <math>F</math> and any [[real number]] <math>e</math> with <math>e > 1</math> the set of points (locus of points), for which the quotient of the distances to the point and to the line is <math>e</math>
<math display="block">H = \left\{P \, \Biggr| \, \frac{|PF|}{|Pl|} = e\right\} </math>
is a hyperbola.

(The choice <math>e = 1</math> yields a [[parabola]] and if <math>e < 1</math> an [[ellipse]].)

====Proof====
Let <math>F=(f,0) ,\ e >0</math> and assume <math>(0,0)</math> is a point on the curve.
The directrix <math>l</math> has equation <math>x=-\tfrac{f}{e}</math>. With <math>P=(x,y)</math>, the relation <math>|PF|^2 = e^2|Pl|^2</math> produces the equations
:<math>(x-f)^2+y^2 = e^2\left(x+\tfrac{f}{e}\right)^2 = (e x+f)^2</math> and <math>x^2(e^2-1)+2xf(1+e)-y^2 = 0.</math>
The substitution <math>p=f(1+e)</math> yields
<math display="block">x^2(e^2-1)+2px-y^2 = 0.</math>
This is the equation of an ''ellipse'' (<math>e<1</math>) or a ''parabola'' (<math>e=1</math>) or a ''hyperbola'' (<math>e>1</math>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).

If <math>e > 1</math>, introduce new parameters <math>a,b</math> so that <math>e^2-1 = \tfrac{b^2}{a^2}, \text { and }\ p = \tfrac{b^2}{a}</math>, and then the equation above becomes
<math display="block">\frac{(x+a)^2}{a^2} - \frac{y^2}{b^2} = 1 \, ,</math>
which is the equation of a hyperbola with center <math>(-a,0)</math>, the ''x''-axis as major axis and the major/minor semi axis <math>a,b</math>.

[[File:Hyperbel-leitl-e.svg|thumb|upright=1.4|Hyperbola: construction of a directrix]]
====Construction of a directrix====
Because of <math>c \cdot \tfrac{a^2}{c}=a^2</math> point <math>L_1</math> of directrix <math>l_1</math> (see diagram) and focus <math>F_1</math> are inverse with respect to the [[circle inversion]] at circle <math>x^2+y^2=a^2</math> (in diagram green). Hence point <math>E_1</math> can be constructed using the [[theorem of Thales]] (not shown in the diagram). The directrix <math>l_1</math> is the perpendicular to line <math>\overline{F_1F_2}</math> through point <math>E_1</math>.

''Alternative construction of <math>E_1</math>'': Calculation shows, that point <math>E_1</math> is the intersection of the asymptote with its perpendicular through <math>F_1</math> (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 ''d''<sub>1</sub>, ''d''<sub>2</sub>]]
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]] <math>d_1, d_2</math>, which are spheres that touch the cone along circles {{nowrap|<math>c_1</math>,}} <math>c_2 </math> and the intersecting (hyperbola) plane at points <math>F_1</math> and {{nowrap|<math>F_2</math>.}} It turns out: <math>F_1, F_2</math> are the ''foci'' of the hyperbola.
# Let <math>P</math> be an arbitrary point of the intersection curve.
# The [[generatrix]] of the cone containing <math>P</math> intersects circle <math>c_1</math> at point <math>A</math> and circle <math>c_2</math> at a point <math>B</math>.
# The line segments <math>\overline{PF_1}</math> and <math>\overline{PA}</math> are tangential to the sphere <math>d_1</math> and, hence, are of equal length.
# The line segments <math>\overline{PF_2}</math> and <math>\overline{PB}</math> are tangential to the sphere <math>d_2</math> and, hence, are of equal length.
# The result is: <math>|PF_1| - |PF_2| = |PA| - |PB| = |AB|</math> is independent of the hyperbola point {{nowrap|<math>P</math>,}} because no matter where point <math>P</math> is, <math>A, B</math> have to be on circles {{nowrap|<math>c_1</math>,}} {{nowrap|<math>c_2 </math>,}} and line segment <math>AB</math> has to cross the apex. Therefore, as point <math>P</math> moves along the red curve (hyperbola), line segment <math>\overline{AB}</math> 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:<ref> [[Frans van Schooten]]: ''Mathematische Oeffeningen'', Leyden, 1659, p. 327</ref>

#<li value="0"> Choose the ''foci'' <math>F_1,F_2</math> and one of the ''circular directrices'', for example <math>c_2</math> (circle with radius <math>2a</math>)</li>
# A ''ruler'' is fixed at point <math>F_2</math> free to rotate around <math>F_2</math>. Point <math>B</math> is marked at distance <math>2a</math>.
# A ''string'' gets its one end pinned at point <math>A</math> on the ruler and its length is made <math>|AB|</math>.
# The free end of the string is pinned to point <math>F_1</math>.
# Take a ''pen'' and hold the string tight to the edge of the ruler.
# ''Rotating'' the ruler around <math>F_2</math> prompts the pen to draw an arc of the right branch of the hyperbola, because of <math>|PF_1| = |PB|</math> (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 conic|Steiner generation of a non degenerate conic section]]:

{{block indent |em=1.5 |text=Given two [[pencil (mathematics)|pencils]] <math>B(U),B(V)</math> of lines at two points <math>U,V</math> (all lines containing <math>U</math> and <math>V</math>, respectively) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>, then the intersection points of corresponding lines form a non-degenerate projective conic section.}}

For the generation of points of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2} = 1</math> one uses the pencils at the vertices <math>V_1,V_2</math>. Let <math>P = (x_0,y_0)</math> be a point of the hyperbola and <math>A = (a,y_0), B = (x_0,0)</math>. The line segment <math>\overline{BP}</math> is divided into n equally-spaced segments and this division is projected parallel with the diagonal <math>AB</math> as direction onto the line segment <math>\overline{AP}</math> (see diagram). The parallel projection is part of the projective mapping between the pencils at <math>V_1</math> and <math>V_2</math> needed. The intersection points of any two related lines <math>S_1 A_i</math> and <math>S_2 B_i</math> are points of the uniquely defined hyperbola.

''Remarks:''
* The subdivision could be extended beyond the points <math>A</math> and <math>B</math> 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|1=''y'' = ''a''/(''x'' − ''b'') + ''c''}} and the 3-point-form===
[[File:Hyperbel-pws-s.svg|250px|thumb|Hyperbola: inscribed angle theorem]]
A hyperbola with equation <math>y=\tfrac{a}{x-b}+c,\ a \ne 0 </math> is uniquely determined by three points <math>(x_1,y_1),\;(x_2,y_2),\; (x_3,y_3)</math> with different ''x''- and ''y''-coordinates. A simple way to determine the shape parameters <math>a,b,c</math> uses the ''inscribed angle theorem'' for hyperbolas:

{{block indent |em=1.5 |text=In order to '''measure an angle''' between two lines with equations <math>y=m_1x+d_1, \ y=m_2x + d_2\ ,m_1,m_2 \ne 0</math> in this context one uses the quotient

<math display="block">\frac{m_1}{m_2}\ .</math>}}

Analogous to the [[inscribed angle]] theorem for circles one gets the
{{math theorem
|name= Inscribed angle theorem for hyperbolas<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf E. Hartmann: Lecture Note ''Planar Circle Geometries'', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 93]</ref><ref>W. Benz: ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973)</ref>
|math_statement= For four points <math>P_i = (x_i,y_i),\ i=1,2,3,4,\ x_i\ne x_k, y_i\ne y_k, i\ne k</math> (see diagram) the following statement is true:

The four points are on a hyperbola with equation <math>y = \tfrac{a}{x-b} + c</math> if and only if the angles at <math>P_3</math> and <math>P_4</math> are equal in the sense of the measurement above. That means if <math display="block">\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)}</math>

The proof can be derived by straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is {{nowrap|<math>y = a/x</math>.}}
}}

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 <math>P_i=(x_i,y_i),\ i=1,2,3,\ x_i\ne x_k, y_i\ne y_k, i\ne k</math> is the solution of the equation <math display="block">\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)}</math> for <math>{\color{red}y}</math>.}}

===As an affine image of the unit hyperbola {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}}===
[[File:Hyperbel-aff-s.svg|300px|thumb|Hyperbola as an affine image of the unit hyperbola]]
Another definition of a hyperbola uses [[affine transformation]]s:

{{block indent |em=1.5 |text=Any ''hyperbola'' is the affine image of the unit hyperbola with equation <math>x^2 - y^2 = 1</math>.}}

====Parametric representation====
An affine transformation of the Euclidean plane has the form <math>\vec x \to \vec f_0+A\vec x</math>, where <math>A</math> is a regular [[matrix (mathematics)|matrix]] (its [[determinant]] is not 0) and <math>\vec f_0</math> is an arbitrary vector. If <math>\vec f_1, \vec f_2</math> are the column vectors of the matrix <math>A</math>, the unit hyperbola <math>(\pm\cosh(t),\sinh(t)), t \in \R,</math> is mapped onto the hyperbola

<math display="block">\vec x = \vec p(t)=\vec f_0 \pm\vec f_1 \cosh t +\vec f_2 \sinh t \ .</math>

<math>\vec f_0</math> is the center, <math>\vec f_0+ \vec f_1</math> a point of the hyperbola and <math>\vec f_2</math> a tangent vector at this point.

====Vertices====
In general the vectors <math>\vec f_1, \vec f_2</math> are not perpendicular. That means, in general <math>\vec f_0\pm \vec f_1</math> are ''not'' the vertices of the hyperbola. But <math>\vec f_1\pm \vec f_2</math> point into the directions of the asymptotes. The tangent vector at point <math>\vec p(t)</math> is
<math display="block">\vec p'(t) = \vec f_1\sinh t + \vec f_2\cosh t \ .</math>
Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter <math>t_0</math> of a vertex from the equation
<math display="block">\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</math>
and hence from
<math display="block">\coth (2t_0)= -\tfrac{\vec f_1^{\, 2}+\vec f_2^{\, 2}}{2\vec f_1 \cdot \vec f_2} \ ,</math>
which yields

<math display="block">t_0=\tfrac{1}{4}\ln\tfrac{\left(\vec f_1-\vec f_2\right)^2}{\left(\vec f_1+\vec f_2\right)^2}.</math>

The formulae {{nowrap|<math>\cosh^2 x + \sinh^2 x = \cosh 2x</math>,}} {{nowrap|<math>2\sinh x \cosh x = \sinh 2x</math>,}} and <math>\operatorname{arcoth} x = \tfrac{1}{2}\ln\tfrac{x+1}{x-1}</math> were used.

The two ''vertices'' of the hyperbola are <math>\vec f_0\pm\left(\vec f_1\cosh t_0 +\vec f_2 \sinh t_0\right).</math>

====Implicit representation====
Solving the parametric representation for <math> \cosh t, \sinh t</math> by [[Cramer's rule]] and using <math>\;\cosh^2t-\sinh^2t -1 = 0\; </math>, one gets the implicit representation
<math display="block">\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 .</math>

====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 <math>\vec f\!_0, \vec f\!_1, \vec f\!_2</math> to be vectors in space.

===As an affine image of the hyperbola {{math|1=''y'' = 1/''x''}}===
[[File:Hyperbel-aff2.svg|thumb|300px|Hyperbola as affine image of ''y'' = 1/''x'']]
Because the unit hyperbola <math>x^2-y^2=1</math> is affinely equivalent to the hyperbola <math>y=1/x</math>, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola {{nowrap|<math>y = 1/x \, </math>:}}

<math display="block">\vec x = \vec p(t) = \vec f_0 + \vec f_1 t + \vec f_2 \tfrac{1}{t}, \quad t\ne 0\, .</math>

<math>M: \vec f_0 </math> is the center of the hyperbola, the vectors <math>\vec f_1 , \vec f_2 </math> have the directions of the asymptotes and <math>\vec f_1 + \vec f_2 </math> is a point of the hyperbola. The tangent vector is
<math display="block">\vec p'(t)=\vec f_1 - \vec f_2 \tfrac{1}{t^2}.</math>
At a vertex the tangent is perpendicular to the major axis. Hence
<math display="block">\vec p'(t)\cdot \left(\vec p(t) -\vec f_0\right) = \left(\vec f_1 - \vec f_2 \tfrac{1}{t^2}\right)\cdot\left(\vec f_1 t+ \vec f_2 \tfrac{1}{t}\right) = \vec f_1^2t-\vec f_2^2 \tfrac{1}{t^3} = 0</math>
and the parameter of a vertex is

<math display="block">t_0= \pm \sqrt[4]{\frac{\vec f_2^2}{\vec f_1^2}}.</math>

<math>\left|\vec f\!_1\right| = \left|\vec f\!_2\right|</math> is equivalent to <math>t_0 = \pm 1</math> and <math>\vec f_0 \pm (\vec f_1+\vec f_2)</math> 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:
<math display="block">\vec p'(t)=\tfrac{1}{t}\left(\vec f_1t - \vec f_2 \tfrac{1}{t}\right) \ .</math>
This means that

{{block indent |em=1.5 |text=the diagonal <math>AB</math> of the parallelogram <math>M: \ \vec f_0, \ A=\vec f_0+\vec f_1t,\ B:\ \vec f_0+ \vec f_2 \tfrac{1}{t},\ P:\ \vec f_0+\vec f_1t+\vec f_2 \tfrac{1}{t}</math> is parallel to the tangent at the hyperbola point <math>P</math> (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]].<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius-, Laguerre- and Minkowski Planes], S.&nbsp;33, (PDF; 757&nbsp;kB)</ref>

;Area of the grey parallelogram:
The area of the grey parallelogram <math>MAPB</math> in the above diagram is
<math display="block">\text{Area} = \left|\det\left( t\vec f_1, \tfrac{1}{t}\vec f_2\right)\right| = \left|\det\left(\vec f_1,\vec f_2\right)\right| = \cdots = \frac{a^2+b^2}{4} </math>
and hence independent of point <math>P</math>. The last equation follows from a calculation for the case, where <math>P</math> is a vertex and the hyperbola in its canonical form <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}=1 \, .</math>

====Point construction====
[[File:Hyperbel-pasc4-s.svg|thumb|Point construction: asymptotes and ''P''<sub>1</sub> are given → ''P''<sub>2</sub>]]

For a hyperbola with parametric representation <math>\vec x = \vec p(t) = \vec f_1 t + \vec f_2 \tfrac{1}{t}</math> (for simplicity the center is the origin) the following is true:

{{block indent |em=1.5 |text=For any two points <math>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}</math> the points
<math display="block">A:\ \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}</math>
are collinear with the center of the hyperbola (see diagram).}}
The simple proof is a consequence of the equation <math>\tfrac{1}{t_1}\vec a = \tfrac{1}{t_2}\vec b</math>.

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]].<ref>[https://www2.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius-, Laguerre- and Minkowski Planes], S.&nbsp;32, (PDF; 757&nbsp;kB)</ref>

====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 <math>\vec f_1,\vec f_2</math> 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 <math>\pm (\vec f_1 + \vec f_2)</math> are the vertices, <math>\pm(\vec f_1-\vec f_2)</math> span the minor axis and one gets <math>|\vec f_1 + \vec f_2| = a</math> and <math>|\vec f_1 - \vec f_2| = b</math>.

For the intersection points of the tangent at point <math>\vec p(t_0) = \vec f_1 t_0 + \vec f_2 \tfrac{1}{t_0}</math> with the asymptotes one gets the points
<math display="block">C = 2t_0\vec f_1,\ D = \tfrac{2}{t_0}\vec f_2.</math>
The ''[[area]]'' of the triangle <math>M,C,D</math> can be calculated by a 2&thinsp;×&thinsp;2 determinant:
<math display="block">A = \tfrac{1}{2}\Big|\det\left( 2t_0\vec f_1, \tfrac{2}{t_0}\vec f_2\right)\Big| = 2\Big|\det\left(\vec f_1,\vec f_2\right)\Big|</math>
(see rules for [[determinant]]s).
<math>\left|\det(\vec f_1,\vec f_2)\right|</math> is the area of the rhombus generated by <math>\vec f_1,\vec f_2</math>. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes <math>a,b</math> of the hyperbola. Hence:

{{block indent |em=1.5 |text=The ''area'' of the triangle <math>MCD</math> is independent of the point of the hyperbola: <math>A = ab.</math>}}

===Reciprocation of a circle===
The [[reciprocation (geometry)|reciprocation]] of a [[circle]] ''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]], respectively. The ''pole'' of a line is the [[inversive geometry#Circle inversion|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

<math display="block">e = \frac{\overline{BC}}{r}.</math>

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 (mathematics)|locus]] of the poles of the tangent lines to the circle ''B'', as well as the [[envelope (mathematics)|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 <math>(x, y)</math> in the [[plane (geometry)|plane]],

<math display=block>
A_{xx} x^2 + 2 A_{xy} xy + A_{yy} y^2 + 2 B_x x + 2 B_y y + C = 0,
</math>

provided that the constants <math>A_{xx},</math> <math>A_{xy},</math> <math>A_{yy},</math> <math>B_x,</math> <math>B_y,</math> and <math>C</math> satisfy the determinant condition

<math display=block>
D := \begin{vmatrix}
  A_{xx} & A_{xy} \\
  A_{xy} & A_{yy} \end{vmatrix} < 0.
</math>

This determinant is conventionally called the [[discriminant#Discriminant of a conic section|discriminant]] of the conic section.<ref>{{cite book
|title=Math refresher for scientists and engineers
|last1=Fanchi
|first1=John R.
|publisher=John Wiley and Sons
|year=2006
|isbn=0-471-75715-2
|url=https://books.google.com/books?id=75mAJPcAWT8C
|at=[https://books.google.com/books?id=75mAJPcAWT8C&pg=PA44 Section 3.2, pages 44–45]
}}</ref>

A special case of a hyperbola—the ''[[degenerate conic|degenerate hyperbola]]'' consisting of two intersecting lines—occurs when another determinant is zero:

<math display=block>
\Delta := \begin{vmatrix}
  A_{xx} & A_{xy} & B_x \\
  A_{xy} & A_{yy} & B_y \\
  B_x & B_y & C
\end{vmatrix} = 0.
</math>

This determinant <math>\Delta</math> is sometimes called the discriminant of the conic section.<ref>{{cite book |last1=Korn |first1=Granino A |author2-link=Theresa M. Korn |last2=Korn |first2=Theresa M. |title=Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review |publisher=Dover Publ. |edition=second |year=2000 |page=40}}</ref>

The general equation's coefficients can be obtained from known semi-major axis <math>a,</math> semi-minor axis <math>b,</math> center coordinates <math>(x_\circ, y_\circ)</math>, and rotation angle <math>\theta</math> (the angle from the positive horizontal axis to the hyperbola's major axis) using the formulae:

<math display=block>\begin{align}
  A_{xx} &=   -a^2 \sin^2\theta + b^2 \cos^2\theta, &
  B_{x} &= -A_{xx} x_\circ   -  A_{xy} y_\circ, \\[1ex]
  A_{yy} &=   -a^2 \cos^2\theta + b^2 \sin^2\theta, &
  B_{y} &= - A_{xy} x_\circ   - A_{yy} y_\circ, \\[1ex]
  A_{xy} &=  \left(a^2 + b^2\right) \sin\theta \cos\theta, &
  C &=   A_{xx} x_\circ^2 +  2A_{xy} x_\circ y_\circ + A_{yy} y_\circ^2 - a^2 b^2.
\end{align}</math>

These expressions can be derived from the canonical equation

<math display=block>\frac{X^2}{a^2} - \frac{Y^2}{b^2} = 1</math>

by a [[rigid transformation|translation and rotation]] of the coordinates {{nobr|<math>(x, y)</math>:}}

<math display=block>\begin{alignat}{2}
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{alignat}</math>

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 <math>(x_c, y_c)</math> of the hyperbola may be determined from the formulae

<math display=block>\begin{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}</math>

In terms of new coordinates, <math>\xi = x - x_c</math> and <math>\eta = y - y_c,</math> the defining equation of the hyperbola can be written

<math display=block>
A_{xx} \xi^2 + 2A_{xy} \xi\eta + A_{yy} \eta^2 + \frac \Delta D = 0.
</math>

The principal axes of the hyperbola make an angle <math>\varphi</math> with the positive <math>x</math>-axis that is given by

<math display=block>\tan (2\varphi) = \frac{2A_{xy}}{A_{xx} - A_{yy}}.</math>

Rotating the coordinate axes so that the <math>x</math>-axis is aligned with the transverse axis brings the equation into its '''canonical form'''

<math display=block>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.</math>

The major and minor semiaxes <math>a</math> and <math>b</math> are defined by the equations

<math display=block>\begin{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}</math>

where <math>\lambda_1</math> and <math>\lambda_2</math> are the [[root of a function|roots]] of the [[quadratic equation]]

<math display=block>\lambda^2 - \left( A_{xx} + A_{yy} \right)\lambda + D = 0.</math>

For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is

<math display=block>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0.</math>

The tangent line to a given point <math>(x_0, y_0)</math> on the hyperbola is defined by the equation

<math display=block>E x + F y + G = 0</math>

where <math>E,</math> <math>F,</math> and <math>G</math> are defined by

<math display=block>\begin{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}</math>

The [[normal (geometry)|normal line]] to the hyperbola at the same point is given by the equation

<math display=block>F(x - x_0) - E(y - y_0) = 0.</math>

The normal line is perpendicular to the tangent line, and both pass through the same point <math>(x_0, y_0).</math>

From the equation

<math display=block>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \qquad 0 < b \leq a,</math>

the left focus is <math>(-ae,0)</math> and the right focus is <math>(ae,0), </math> where <math>e</math> is the eccentricity. Denote the distances from a point <math>(x, y)</math> to the left and right foci as <math>r_1</math> and <math>r_2.</math> For a point on the right branch,

<math display=block> r_1 - r_2 = 2 a, </math>

and for a point on the left branch,

<math display=block> r_2 - r_1 = 2 a. </math>

This can be proved as follows:

If <math>(x, y)</math> is a point on the hyperbola the distance to the left focal point is

<math display=block>
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.
</math>

To the right focal point the distance is

<math display=block>
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.
</math>

If <math>(x, y)</math> is a point on the right branch of the hyperbola then <math>ex > a</math> and

<math display=block>\begin{align}
r_1 &= e x + a, \\
r_2 &= e x - a.
\end{align}</math>

Subtracting these equations one gets

<math display=block>r_1 - r_2 = 2a.</math>

If <math>(x, y)</math> is a point on the left branch of the hyperbola then <math>ex < -a</math> and

<math display=block>\begin{align}
r_1 &= - e x - a, \\
r_2 &= - e x + a.
\end{align}</math>

Subtracting these equations one gets

<math display=block>r_2 - r_1 = 2a.</math>

==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 <math>F_1=(c,0),\ F_2=(-c,0)</math>,{{sfn|Protter|Morrey|1970|p=310}}
:the ''vertices'' are <math>V_1=(a, 0),\ V_2=(-a,0)</math>.{{sfn|Protter|Morrey|1970|p=310}}
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is <math display="inline">\sqrt{(x-c)^2 + y^2}</math> and to the second focus <math display="inline">\sqrt{(x+c)^2 + y^2}</math>. Hence the point <math>(x,y)</math> is on the hyperbola if the following condition is fulfilled
<math display="block">\sqrt{(x-c)^2 + y^2} - \sqrt{(x+c)^2 + y^2} = \pm 2a \ .</math>
Remove the square roots by suitable squarings and use the relation <math>b^2 = c^2-a^2</math> to obtain the equation of the hyperbola:

<math display="block">\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ .</math>

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 [[congruence (geometry)|congruent]] to the original (see [[#Quadratic equation|below]]).

The axes of [[symmetry (geometry)|symmetry]] 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).{{sfn|Protter|Morrey|1970|p=310}} As opposed to an ellipse, a hyperbola has only two vertices: <math>(a,0),\; (-a,0)</math>. The two points <math>(0,b),\; (0,-b)</math> 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 (mathematics)|eccentricity]] is given by

<math display="block">e=\sqrt{1+\frac{b^2}{a^2}}.</math>

Two hyperbolas are [[similarity (geometry)|geometrically similar]] to each other – meaning that they have the same shape, so that one can be transformed into the other by [[translation (geometry)|rigid left and right movements]], [[rotation (mathematics)|rotation]], [[reflection (mathematics)|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 <math>y</math> yields
<math display="block">y=\pm\frac{b}{a} \sqrt{x^2-a^2}.</math>
It follows from this that the hyperbola approaches the two lines
<math display="block">y=\pm \frac{b}{a}x </math>
for large values of <math>|x|</math>. These two lines intersect at the center (origin) and are called ''asymptotes'' of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1 \ .</math>{{sfn|Protter|Morrey|1970|pp=APP-29–APP-30}}

With the help of the second figure one can see that
:<math>{\color{blue}{(1)}}</math> The ''perpendicular distance from a focus to either asymptote'' is <math>b</math> (the semi-minor axis).

From the [[Hesse normal form]] <math>\tfrac{bx\pm ay}{\sqrt{a^2+b^2}}=0 </math> of the asymptotes and the equation of the hyperbola one gets:<ref name=Mitchell>Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", ''Mathematical Gazette'' 96, July 2012, 299–301.</ref>
:<math>{\color{magenta}{(2)}}</math> The ''product of the distances from a point on the hyperbola to both the asymptotes'' is the constant <math>\tfrac{a^2b^2}{a^2+b^2}\ , </math> which can also be written in terms of the eccentricity ''e'' as <math>\left( \tfrac{b}{e}\right) ^2.</math>

From the equation <math>y=\pm\frac{b}{a}\sqrt{x^2-a^2}</math> of the hyperbola (above) one can derive:
:<math>{\color{green}{(3)}}</math> The ''product of the slopes of lines from a point P to the two vertices'' is the constant <math>b^2/a^2\ .</math>

In addition, from (2) above it can be shown that<ref name=Mitchell/>
:<math>{\color{red}{(4)}}</math> ''The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes'' is the constant <math>\tfrac{a^2+b^2}{4}.</math>

===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'' <math>p</math>. A calculation shows
<math display="block">p = \frac{b^2}a.</math>
The semi-latus rectum <math>p</math> 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 <math>(x_0,y_0)</math> is to [[implicit differentiation|implicitly differentiate]] the equation <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math> of the hyperbola. Denoting ''dy/dx'' as ''y′'', this produces
<math display="block">\frac{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.</math>
With respect to <math>\tfrac{x_0^2}{a^2}-\tfrac{y_0^2}{b^2}= 1</math>, the equation of the tangent at point <math>(x_0,y_0)</math> is
<math display="block">\frac{x_0}{a^2}x-\frac{y_0}{b^2}y = 1.</math>

A particular tangent line distinguishes the hyperbola from the other conic sections.<ref>J. W. Downs, ''Practical Conic Sections'', Dover Publ., 2003 (orig. 1993): p. 26.</ref> 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 <math>a = b</math> the hyperbola is called ''rectangular'' (or ''equilateral''), because its asymptotes intersect at right angles. For this case, the linear eccentricity is <math>c=\sqrt{2}a</math>, the eccentricity <math>e=\sqrt{2}</math> and the semi-latus rectum <math>p=a</math>. The graph of the equation <math>y=1/x</math> is a rectangular hyperbola.

===Parametric representation with hyperbolic sine/cosine===
Using the [[hyperbolic function|hyperbolic sine and cosine functions]] <math>\cosh,\sinh</math>, a parametric representation of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math> can be obtained, which is similar to the parametric representation of an ellipse:
<math display="block">(\pm a \cosh t, b \sinh t),\, t \in \R \ ,</math>
which satisfies the Cartesian equation because <math>\cosh^2 t -\sinh^2 t =1 .</math>

Further parametric representations are given in the section [[#Parametric equations|Parametric equations]] below.

[[File:Conjugate-unit-hyperbolas.svg|thumb|Here {{nowrap|''a'' {{=}} ''b'' {{=}} 1}} giving the [[unit hyperbola]] in blue and its conjugate hyperbola in green, sharing the same red asymptotes.]]
=== Conjugate hyperbola ===
{{Main|Conjugate hyperbola}}
For the hyperbola <math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math>, change the sign on the right to obtain the equation of the '''conjugate hyperbola''':
:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1</math> (which can also be written as <math>\frac{y^2}{b^2}-\frac{x^2}{a^2} = 1</math>).

A hyperbola and its conjugate may have [[conjugate diameters#Of hyperbola|diameters which are conjugate]]. In the theory of [[special relativity]], such diameters may represent axes of time and space, where one hyperbola represents [[event (relativity)|event]]s at a given spatial distance from the [[centre (geometry)#Projective conics|center]], and the other represents events at a corresponding temporal distance from the center.

:<math>xy = c^2</math> and <math>xy = -c^2</math> also specify conjugate hyperbolas.

==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 <math>r = \frac{p}{1 - e \cos \theta}</math>]]

===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 <math>\varphi</math> is called '''true anomaly'''.

Relative to this coordinate system one has that

<math display="block">r = \frac{p}{1 \mp e \cos \varphi}, \quad p = \frac{b^2}{a}</math>

and

<math display="block">-\arccos \left(-\frac 1 e\right) < \varphi < \arccos \left(-\frac 1 e\right). </math>

===Origin at the center===
With polar coordinates relative to the "canonical coordinate system" (see second diagram)
one has that

<math display="block">r =\frac{b}{\sqrt{e^2 \cos^2 \varphi -1}} .\,</math>

For the right branch of the hyperbola the range of <math> \varphi </math> is
<math display="block">-\arccos \left(\frac 1 e\right) < \varphi < \arccos \left(\frac 1 e\right).</math>

===Eccentricity===
{{Anchor|Polar coordinate eccentricity}}
When using polar coordinates, the eccentricity of the hyperbola can be expressed as <math>\sec\varphi_\text{max}</math> where <math>\varphi_\text{max}</math> is the limit of the angular coordinate. As <math>\varphi</math> approaches this limit, ''r'' approaches infinity and the denominator in either of the equations noted above approaches zero, hence:<ref name=Casey1885>Casey, John, (1885) [https://archive.org/details/cu31924001520455/page/n219/mode/2up "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>{{rp|219}}

<math display="block">e^2 \cos^2 \varphi_\text{max} - 1 = 0</math>

<math display="block">1 \pm e \cos \varphi_\text{max} = 0</math>

<math display="block">\implies e = \sec\varphi_\text{max}</math>

==Parametric equations==
A hyperbola with equation <math>\tfrac{x^2}{a^2} - \tfrac{y^2}{b^2} = 1</math> can be described by several parametric equations:
# Through hyperbolic trigonometric functions <math display="block">
 \begin{cases}
  x = \pm a \cosh t, \\
  y = b \sinh t,
 \end{cases} \qquad t \in \R.
</math>
# As a ''rational'' representation <math display="block">
 \begin{cases}
  x = \pm a \dfrac{t^2 + 1}{2t}, \\[1ex]
  y = b \dfrac{t^2 - 1}{2t},
 \end{cases} \qquad t > 0</math>
# Through circular trigonometric functions <math display="block">
 \begin{cases}
  x = \frac{a}{\cos t} = a \sec t, \\
  y = \pm b \tan t,
 \end{cases} \qquad 0 \le t < 2\pi,\ t \ne \frac{\pi}{2},\ t \ne \frac{3}{2} \pi.</math>
# With the tangent slope as parameter: {{pb}} A parametric representation, which uses the slope <math>m</math> of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case <math>b^2</math> by <math>-b^2</math> and use formulae for the [[hyperbolic function]]s. One gets <math display="block">\vec c_\pm(m) = \left(-\frac{ma^2}{\pm\sqrt{m^2a^2 - b^2}}, \frac{-b^2}{\pm\sqrt{m^2a^2 - b^2}}\right),\quad |m| > b/a.</math> Here, <math>\vec c_-</math> is the upper, and <math>\vec c_+</math> the lower half of the hyperbola. The points with vertical tangents (vertices <math>(\pm a, 0)</math>) are not covered by the representation. {{pb}} The equation of the tangent at point <math>\vec c_\pm(m)</math> is <math display="block">y = m x \pm\sqrt{m^2a^2 - b^2}.</math> This description of the tangents of a hyperbola is an essential tool for the determination of the [[orthoptic (geometry)|orthoptic]] of a hyperbola.

==Hyperbolic functions==
{{Main|Hyperbolic functions}}
[[File:Hyperbolic functions-2.svg|thumb|296px|right|A ray through the [[unit hyperbola]] <math>x^2\ -\ y^2\ =\ 1</math> at the point <math> (\cosh\,a,\,\sinh\,a)</math>, where <math>a</math> is twice the area between the ray, the hyperbola, and the <math>x</math>-axis. For points on the hyperbola below the <math>x</math>-axis, the area is considered negative.]]

Just as the [[trigonometric function]]s are defined in terms of the [[unit circle]], so also the [[hyperbolic function]]s 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 <math>a</math> be twice the area between the <math>x</math> axis and a ray through the origin intersecting the unit hyperbola, and define <math display=inline>(x,y) = (\cosh a,\sinh a) = (x, \sqrt{x^2-1})</math> 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 <math>(1,0)</math>:
<math display="block">\begin{align}
\frac{a}{2} &= \frac{xy}{2} - \int_1^x \sqrt{t^{2}-1} \, dt \\[1ex]
            &= \frac{1}{2} \left(x\sqrt{x^2-1}\right) - \frac{1}{2} \left(x\sqrt{x^2-1} - \ln \left(x+\sqrt{x^2-1}\right)\right),
\end{align}</math>
which simplifies to the [[inverse hyperbolic functions|area hyperbolic cosine]]
<math display="block">a=\operatorname{arcosh}x=\ln \left(x+\sqrt{x^2-1}\right).</math>
Solving for <math>x</math> yields the exponential form of the hyperbolic cosine:
<math display="block">x=\cosh a=\frac{e^a+e^{-a}}{2}.</math>
From <math>x^2-y^2=1</math> one gets
<math display="block">y=\sinh a=\sqrt{\cosh^2 a - 1}=\frac{e^a-e^{-a}}{2},</math>
and its inverse the [[inverse hyperbolic functions|area hyperbolic sine]]:
<math display="block">a=\operatorname{arsinh}y=\ln \left(y+\sqrt{y^2+1}\right).</math>
Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example
<math display="block">\operatorname{tanh}a=\frac{\sinh a}{\cosh a}=\frac{e^{2a}-1}{e^{2a}+1}.</math>

==Properties==

===Reflection property===
[[File:Hyperbel-wh-s.svg|300px|thumb|Hyperbola: the tangent bisects the lines through the foci]]
The tangent at a point <math>P</math> bisects the angle between the lines <math>\overline{PF_1}, \overline{PF_2}.</math> This is called the ''optical property'' or ''reflection property'' of a hyperbola.<ref> {{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 |jstor=2688124 |doi=10.1080/0025570X.1963.11975375 }} {{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 |jstor=2313439 |doi=10.1080/00029890.1968.11970997 }} {{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 |jstor=2686519 |doi=10.1080/00494925.1984.11972763 <!-- Deny Citation Bot--> |doi-broken-date=2024-12-16 |url=https://www.tandfonline.com/doi/abs/10.1080/00494925.1984.11972763 |url-access=subscription }} </ref>
;Proof:
Let <math>L</math> be the point on the line <math>\overline{PF_2}</math> with the distance <math>2a</math> to the focus <math>F_2</math> (see diagram, <math>a</math> is the semi major axis of the hyperbola). Line <math>w</math> is the bisector of the angle between the lines <math>\overline{PF_1}, \overline{PF_2}</math>. In order to prove that <math>w</math> is the tangent line at point <math>P</math>, one checks that any point <math>Q</math> on line <math>w</math> which is different from <math>P</math> cannot be on the hyperbola. Hence <math>w</math> has only point <math>P</math> in common with the hyperbola and is, therefore, the tangent at point <math>P</math>. <br/>
From the diagram and the [[triangle inequality]] one recognizes that <math>|QF_2|<|LF_2|+|QL|=2a+|QF_1|</math> holds, which means: <math>|QF_2|-|QF_1|<2a</math>. But if <math>Q</math> is a point of the hyperbola, the difference should be <math>2a</math>.

===Midpoints of parallel chords===
[[File:Hyperbel-psehnen-s.svg|thumb|Hyperbola: the midpoints of parallel chords lie on a line.]]
[[File:Hyperbel-sa-s.svg|thumb|Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes.]]
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 <math>y=1/x</math>. Because any hyperbola is an affine image of the hyperbola <math>y=1/x</math> (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:<br/>
For two points <math>P=\left(x_1,\tfrac {1}{x_1}\right), \ Q=\left(x_2,\tfrac {1}{x_2}\right)</math> of the hyperbola <math>y=1/x</math>

:the midpoint of the chord is <math>M=\left(\tfrac{x_1+x_2}{2},\cdots\right)=\cdots =\tfrac{x_1+x_2}{2}\; \left(1,\tfrac{1}{x_1x_2}\right) \ ;</math>

:the slope of the chord is <math>\frac{\tfrac {1}{x_2}-\tfrac {1}{x_1}}{x_2-x_1}=\cdots =-\tfrac{1}{x_1x_2} \ .</math>

For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line <math>y=\tfrac{1}{x_1x_2} \; x \ .</math>

Consequence: for any pair of points <math>P,Q</math> 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 <math>P,Q</math> and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line <math>m</math>, where all point-image pairs are on a line perpendicular to <math>m</math>.

Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint <math>M</math> of a chord <math>P Q</math> divides the related line segment <math>\overline P \, \overline Q</math> between the asymptotes into halves, too. This means that <math>|P\overline P|=|Q\overline Q|</math>. This property can be used for the construction of further points <math>Q</math> of the hyperbola if a point <math>P</math> 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===
[[File:Orthoptic-hyperbola-s.svg|thumb|Hyperbola with its orthoptic (magenta)]]
{{Main|Orthoptic (geometry)}}
For a hyperbola <math display="inline">\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \, a>b</math> the intersection points of ''orthogonal'' tangents lie on the circle <math>x^2+y^2=a^2-b^2</math>. <br/>
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 <math>a\le b</math> there are no pairs of orthogonal tangents.

===Pole-polar relation for a hyperbola===
[[File:Hyperbel-pol-s.svg|250px|thumb|Hyperbola: pole-polar relation]]
Any hyperbola can be described in a suitable coordinate system by an equation <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math>. The equation of the tangent at a point <math>P_0=(x_0,y_0)</math> of the hyperbola is <math>\tfrac{x_0x}{a^2}-\tfrac{y_0y}{b^2}=1.</math> If one allows point <math>P_0=(x_0,y_0)</math> to be an arbitrary point different from the origin, then

:point <math>P_0=(x_0,y_0)\ne(0,0)</math> is mapped onto the line <math>\frac{x_0x}{a^2}-\frac{y_0y}{b^2}=1 </math>, not through the center of the hyperbola.

This relation between points and lines is a [[bijection]].

The [[inverse function]] maps

:line <math>y=mx+d,\ d\ne 0</math> onto the point <math>\left(-\frac{ma^2}{d},-\frac{b^2}{d}\right)</math> and

:line <math>x=c,\ c\ne 0</math> onto the point <math>\left(\frac{a^2}{c},0\right)\ .</math>

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]].

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: <math>P_1,\ p_1</math>).
* For a pole <math>P</math> ''outside'' the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing <math>P</math> (see diagram: <math>P_2,\ p_2,\ P_3,\ p_3</math>).
* For a point ''within'' the hyperbola the polar has no point with the hyperbola in common. (see diagram: <math>P_4,\ p_4</math>).

''Remarks:''
# The intersection point of two polars (for example: <math>p_2,p_3</math>) is the pole of the line through their poles (here: <math>P_2,P_3</math>).
# The foci <math>(c,0),</math> and <math> (-c,0)</math> respectively and the directrices <math>x=\tfrac{a^2}{c}</math> and <math>x=-\tfrac{a^2}{c}</math> respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.

===Other properties===
* The following are [[concurrent lines|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.<ref name=web4>{{cite web |title=Hyperbola |website=Mathafou.free.fr |url=http://mathafou.free.fr/themes_en/hyperb.html |access-date=26 August 2018 |url-status=dead |archive-url=https://web.archive.org/web/20160304061843/http://mathafou.free.fr/themes_en/hyperb.html |archive-date=4 March 2016}}</ref><ref name="web1">{{cite web |title=Properties of a Hyperbola |url=http://www.ul.ie/~rynnet/swconics/HP%27s.htm |access-date=2011-06-22 |url-status=dead |archive-url=https://web.archive.org/web/20170202180210/http://www3.ul.ie/~rynnet/swconics/HP's.htm |archive-date=2017-02-02}}</ref>
* 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.<ref name=web1/>
* Since both the transverse axis and the conjugate axis are axes of symmetry, the [[symmetry group]] of a hyperbola is the [[Klein four-group]].
* The rectangular hyperbolas ''xy'' = [[constant (mathematics)|constant]] admit [[group action]]s by [[squeeze mapping]]s which have the hyperbolas as [[invariant set]]s.

==Arc length==
The arc length of a hyperbola does not have an [[elementary function|elementary expression]]. The upper half of a hyperbola can be parameterized as

<math display="block">y = b\sqrt{\frac{x^{2}}{a^{2}}-1}.</math>

Then the integral giving the arc length <math>s</math> from <math>x_{1}</math> to <math>x_{2}</math> can be computed as:

<math display="block">s = b\int_{\operatorname{arcosh}\frac{x_{1}}{a}}^{\operatorname{arcosh}\frac{x_{2}}{a}} \sqrt{1+\left(1+\frac{a^{2}}{b^{2}}\right) \sinh ^{2}v} \, \mathrm dv.</math>

After using the substitution <math>z = iv</math>, this can also be represented using the [[elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integral of the second kind]] <math>E</math> with parameter <math>m = k^2</math>:

<math display="block">s = ib \Biggr[E\left(iv \, \Biggr| \, 1 + \frac{a^2}{b^2}\right)\Biggr]^{\operatorname{arcosh}\frac{x_1}{a}}_{\operatorname{arcosh}\frac{x_2}{a}}.</math>

Using only real numbers, this becomes<ref>{{dlmf |last=Carlson |first=B. C. |id=19.7.E7 |title=Elliptic Integrals}}</ref>

<math display="block">s=b\left[F\left(\operatorname{gd}v\,\Biggr|-\frac{a^2}{b^2}\right) - E\left(\operatorname{gd}v\,\Biggr|-\frac{a^2}{b^2}\right) + \sqrt{1+\frac{a^2}{b^2}\tanh^2 v}\,\sinh v\right]_{\operatorname{arcosh}\tfrac{x_1}{a}}^{\operatorname{arcosh}\tfrac{x_2}{a}}</math>

where <math>F</math> is the [[elliptic integral#Incomplete elliptic integral of the first kind|incomplete elliptic integral of the first kind]] with parameter <math>m = k^2</math> and <math>\operatorname{gd}v=\arctan\sinh v</math> is the [[Gudermannian function]].

==Derived curves==
{{Sinusoidal_spirals.svg}}
Several other curves can be derived from the hyperbola by [[inversive geometry#Circle inversion|inversion]], the so-called [[inverse curve]]s of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the [[lemniscate of Bernoulli]]; 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]] 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

<math display="block">
\left(\frac x {c \cos\theta}\right)^2 - \left(\frac y {c \sin\theta}\right)^2 = 1
</math>

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

Other orthogonal two-dimensional coordinate systems involving hyperbolas may be obtained by other conformal mappings. For example, the mapping ''w'' = ''z''<sup>2</sup> transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.

==Conic section analysis of the hyperbolic appearance of circles==
[[File:Zp-Kugel-Augp-innen.svg|350px|thumb|[[Central projection]] of circles on a sphere: The center ''O'' of projection is inside the sphere, the image plane is red. <br/>
As images of the circles one gets a circle (magenta), ellipses, hyperbolas and lines. The special case of a parabola does not appear in this example.<br/>
(If center ''O'' were ''on'' the sphere, all images of the circles would be circles or lines; see [[stereographic projection]]).]]
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]] 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
{{ordered list | list-style-type = lower-alpha
| a '''circle''', if circle ''c'' is in a special position, for example parallel to the image plane and others (see stereographic projection),
| an '''ellipse''', if ''c'' has ''no'' point with the lens plane in common,
| a '''parabola''', if ''c'' has ''one'' point with the lens plane in common and
| a '''hyperbola''', if ''c'' has ''two'' points with the lens plane in common.
}}
(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==
[[File:Akademia Ekonomiczna w Krakowie Pawilon C.JPG|thumb|right|Hyperbolas as declination lines on a sundial]]
[[File:supersonic shockwave cone.svg|thumb|The contact zone of a level supersonic aircraft's [[shockwave]] on flat ground (yellow) is a part of a hyperbola as the ground intersects the cone parallel to its axis.]]

===Sundials===
Hyperbolas may be seen in many [[sundial]]s. On any given day, the sun revolves in a circle on 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]] 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]] 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]] is a [[conic section]]. 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 [[Geiger–Marsden experiment|Rutherford experiment]] demonstrated the existence of an [[atomic nucleus]] by examining the scattering of [[alpha particle]]s from [[gold]] atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive [[Coulomb's law|Coulomb force]], which satisfies the [[inverse square law]] requirement for a Kepler problem.<ref name=Heilbron1968>{{cite journal |last=Heilbron |first=John L. |title=The Scattering of α and β Particles and Rutherford's Atom |date=1968 |journal=Archive for History of Exact Sciences |volume=4 |issue=4 |pages=247–307 |jstor=41133273 |doi=10.1007/BF00411591 }}</ref>

===Korteweg–de Vries equation===
The hyperbolic trig function <math>\operatorname{sech}\, x</math> appears as one solution to the [[Korteweg–de Vries equation]] which describes the motion of a soliton wave in a canal.

===Angle trisection===
[[File:Hyperbola angle trisection.svg|thumb|Trisecting an angle (AOB) using a hyperbola of eccentricity 2 (yellow curve)]]

As shown first by [[Apollonius of Perga]], a hyperbola can be used to [[angle trisection|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 <math>\ell</math>. Construct a hyperbola of [[eccentricity (mathematics)|eccentricity]] ''e''=2 with <math>\ell</math> as [[directrix (conic section)|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 <math>\ell</math> 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.<ref>Since 2 times the distance of '''P''' to <math>\ell</math> is '''PP'''' which is equal to '''BP''' by directrix-focus property</ref> 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'''.<ref>This construction is due to [[Pappus of Alexandria]] (circa 300 A.D.) and the proof comes from {{harvnb|Kazarinoff|1970|loc=[https://archive.org/details/rulerround0000unse/page/62/ {{p.|62}}]}}.</ref>

===Efficient portfolio frontier===
In [[modern portfolio theory#The efficient frontier with no risk-free asset|portfolio theory]], the locus of [[mean variance efficiency|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]] and [[pharmacology]], the [[Hill equation (biochemistry)|Hill equation]] and [[Hill equation (biochemistry)|Hill-Langmuir equation]] respectively describe biological [[stimulus–response model|responses]] and the formation of [[protein–ligand complex]]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]]s:
* Elliptic [[cone]]
* Hyperbolic [[cylinder]]
* [[Hyperbolic paraboloid]]
* [[Hyperboloid of one sheet]]
* [[Hyperboloid of two sheets]]
<gallery>
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
</gallery>

==See also==

===Other conic sections===
{{Div col|colwidth=25em}}
*[[Circle]]
*[[Ellipse]]
*[[Parabola]]
*[[Degenerate conic]]
{{Div col end}}

===Other related topics===
{{Div col|colwidth=25em}}
*[[Elliptic coordinates]], an orthogonal coordinate system based on families of [[ellipse]]s and hyperbolas.
*[[Hyperbolic growth]]
*[[Hyperbolic partial differential equation]]
*[[Hyperbolic sector]]
*[[Hyperboloid structure]]
*[[Hyperbolic trajectory]]
*[[Hyperboloid]]
*[[Multilateration]]
*[[Rotation of axes]]
*[[Translation of axes]]
*[[Unit hyperbola]]
{{Div col end}}

==Notes==
{{Reflist}}

==References==
* {{citation |last=Kazarinoff |first=Nicholas D. |author-link=Nicholas D. Kazarinoff |title=Ruler and the Round |year=1970 |publisher=Prindle, Weber & Schmidt |location=Boston |isbn=0-87150-113-9 |url=https://archive.org/details/rulerround0000unse/page/n4/}}
* {{citation |last1=Oakley |first1=C. O. |title=An Outline of the Calculus |location=New York |publisher=[[Barnes & Noble]] |year=1944}}
* {{citation |last1=Protter |first1=Murray H. |author-link=Murray Protter |last2=Morrey |first2=Charles B. Jr. |author-link2=Charles B. Morrey Jr. |year=1970 |lccn=76087042 |title=College Calculus with Analytic Geometry |edition=2nd |publisher=[[Addison-Wesley]] |location=Reading}}

==External links==
{{Commons category|Hyperbolas}}
{{EB1911 poster|Hyperbola}}
* {{Springer|title=Hyperbola|id=p/h048230}}
* [https://web.archive.org/web/20070625162103/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=204 Apollonius' Derivation of the Hyperbola] at [https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/ Convergence]
* [https://objects.library.uu.nl/reader/index.php?obj=1874-20606&lan=en#page//16/67/58/166758852278065092993411613632659493149.jpg/mode/2up ''Mathematische Oeffeningen''], Frans van Schooten, 1659
* {{MathWorld|id=Hyperbola}}

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[[Category:Algebraic curves]]
[[Category:Analytic geometry]]
[[Category:Conic sections]]