Editing Hyperbola (section)

==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>