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hyperbola
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{{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 planeplane(File:Hyperbel-def-ass-e.svg|300px|thumb|Hyperbola (red): features)In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.Hyperbolas arise in many ways:
  • as the curve representing the function f(x) = 1/x in the Cartesian plane,{{harvtxt|Oakley|1944|p=17}}
  • as the path followed by the shadow of the tip of a sundial,
  • as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet,
  • as the path of a single-apparition comet (one travelling too fast ever to return to the solar system),
  • as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same),
  • in radio navigation, when the difference between distances to two points, but not the distances themselves, can be determined,
and so on.Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve f(x) = 1/x the asymptotes are the two coordinate axes.{{harvtxt|Oakley|1944|p=17}}Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

Etymology and history

The word "hyperbola" derives from the Greek , meaning "over-thrown" or "excessive", from which the English term hyperbole also derives. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones.{{citation |title=Apollonius of Perga: Treatise on Conic Sections with Introductions Including an Essay on Earlier History on the Subject |first=Sir Thomas Little |last=Heath |publisher=Cambridge University Press |year=1896 |contribution=Chapter I. The discovery of conic sections. Menaechmus |pages=xvii–xxx |url=https://books.google.com/books?hl=en&lr=&id=B0k0AQAAMAAJ&pg=PR17}}. The term hyperbola is believed to have been coined by Apollonius of Perga (c. 262–c. 190 BC) in his definitive work on the conic sections, the Conics.{{citation |title=A History of Mathematics |first=Carl B. |last=Boyer |first2=Uta C. |last2=Merzbach |author2-link=Uta Merzbach|publisher=Wiley |year=2011 |isbn=9780470630563 |url=https://books.google.com/books?id=bR9HAAAAQBAJ&pg=RA2-PT73 |page=73 |quote=It was Apollonius (possibly following up a suggestion of Archimedes) who introduced the names "ellipse" and "hyperbola" in connection with these curves.}} The names of the other two general conic sections, the ellipse and the parabola, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.{{citation|pages=30–31|first=Howard|last=Eves|title=A Survey of Geometry (Vol. One)|year=1963|publisher=Allyn and Bacon}}

Definition of a hyperbola as locus of points

(File:Hyperbel-def-e.svg|thumb|Hyperbola: definition by the distances of points to two fixed points (foci))(File:Hyperbel-def-dc.svg|thumb|Hyperbola: definition with circular directrix)A hyperbola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
A hyperbola is a set of points, such that for any point P of the set, the absolute difference of the distances |PF_1|, |PF_2| to two fixed points F_1,F_2 (the foci), is constant, usually denoted by 2a, a>0 :{{harvtxt|Protter|Morrey|1970|pp=308–310}}
H = {P mid ||PF_2| - |PF_1 || = 2a } .
The midpoint M of the line segment joining the foci is called the center of the hyperbola.{{harvtxt|Protter|Morrey|1970|p=310}} The line through the foci is called the major axis. It contains the vertices V_1,V_2, which have distance a to the center. The distance c of the foci to the center is called the focal distance or linear eccentricity. The quotient tfrac c a is the eccentricity e.The equation ||PF_2| - |PF_1 || = 2a can be viewed in a different way (see diagram):If c_2 is the circle with midpoint F_2 and radius 2a, then the distance of a point P of the right branch to the circle c_2 equals the distance to the focus F_1:
|PF_1|=|Pc_2|.
c_2 is called the circular directrix (related to focus F_2) of the hyperbola.{{citation|first1=Tom M.|last1=Apostol|first2=Mamikon A.|last2=Mnatsakanian|title=New Horizons in Geometry|year=2012|publisher=The Mathematical Association of America|series=The Dolciani Mathematical Expositions #47|isbn=978-0-88385-354-2|page=251}}The German term for this circle is Leitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle). In order to get the left branch of the hyperbola, one has to use the circular directrix related to F_1. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.

Hyperbola in Cartesian coordinates

Equation

If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the x-axis is the major axis, then the hyperbola is called east-west-opening and
the foci are the points F_1=(c,0), F_2=(-c,0),{{harvtxt|Protter|Morrey|1970|p=310}} the vertices are V_1=(a, 0), V_2=(-a,0).{{harvtxt|Protter|Morrey|1970|p=310}}
For an arbitrary point (x,y) the distance to the focus (c,0) is sqrt{ (x-c)^2 + y^2 } and to the second focus sqrt{ (x+c)^2 + y^2 }. Hence the point (x,y) is on the hyperbola if the following condition is fulfilled
sqrt{(x-c)^2 + y^2} - sqrt{(x+c)^2 + y^2} = pm 2a .
Remove the square roots by suitable squarings and use the relation b^2 = c^2-a^2 to obtain the equation of the hyperbola:
frac{x^2}{a^2}-frac{y^2}{b^2}= 1 .
This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).The axes of symmetry or principal axes are the transverse axis (containing the segment of length 2a with endpoints at the vertices) and the conjugate axis (containing the segment of length 2b perpendicular to the transverse axis and with midpoint at the hyperbola's center).{{harvtxt|Protter|Morrey|1970|p=310}} As opposed to an ellipse, a hyperbola has only two vertices: (a,0),; (-a,0). The two points (0,b),; (0,-b) on the conjugate axis are not on the hyperbola.It follows from the equation that the hyperbola is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.

Eccentricity

For a hyperbola in the above canonical form, the eccentricity is given by
e=sqrt{1+frac{b^2}{a^2}}.
Two hyperbolas are geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by rigid left and right movements, rotation, taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.

Asymptotes

(File:Hyperbel-param-e.svg|250px|thumb|Hyperbola: semi-axes a,b, linear eccentricity c, semi latus rectum p)(File:Hyperbola-3prop.svg|300px|thumb|Hyperbola: 3 properties)Solving the equation (above) of the hyperbola for y yields
y=pmfrac{b}{a}sqrt{x^2-a^2}.
It follows from this that the hyperbola approaches the two lines
y=pm frac{b}{a}x
for large values of |x|. These two lines intersect at the center (origin) and are called asymptotes of the hyperbola tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1 .{{harvtxt|Protter|Morrey|1970|pp=APP-29–APP-30}}With help of the figure one can see that
{color{blue}{(1)}} The distance from a focus to either asymptote is b (the semi-minor axis).
From the Hesse normal form tfrac{bxpm ay}{sqrt{a^2+b^2}}=0 of the asymptotes and the equation of the hyperbola one gets:Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", Mathematical Gazette 96, July 2012, 299–301.
{color{magenta}{(2)}} The product of the distances from a point on the hyperbola to both the asymptotes is the constant tfrac{a^2b^2}{a^2+b^2} , which can also be written in terms of the eccentricity e as left( tfrac{b}{e}right) ^2.
From the equation y=pmfrac{b}{a}sqrt{x^2-a^2} of the hyperbola (above) one can derive:
{color{green}{(3)}} The product of the slopes of lines from a point P to the two vertices is the constant b^2/a^2 .
In addition, from (2) above it can be shown that
{color{red}{(4)}} The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes is the constant tfrac{a^2+b^2}{4}.

Semi-latus rectum

The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the latus rectum. One half of it is the semi-latus rectum p. A calculation shows
p = frac{b^2}a.
The semi-latus rectum p may also be viewed as the radius of curvature of the osculating circles at the vertices.

Tangent

The simplest way to determine the equation of the tangent at a point (x_0,y_0) is to implicitly differentiate the equation tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1 of the hyperbola. Denoting dy/dx as y′, this produces
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.
With respect to tfrac{x_0^2}{a^2}-tfrac{y_0^2}{b^2}= 1, the equation of the tangent at point (x_0,y_0) is
frac{x_0}{a^2}x-frac{y_0}{b^2}y = 1.
A particular tangent line distinguishes the hyperbola from the other conic sections.J. W. Downs, Practical Conic Sections, Dover Publ., 2003 (orig. 1993): p. 26. Let f be the distance from the vertex V (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2f. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.

Rectangular hyperbola

In the case a = b the hyperbola is called rectangular (or equilateral), because its asymptotes intersect rectangularly (that is, are perpendicular). For this case, the linear eccentricity is c=sqrt{2}a, the eccentricity e=sqrt{2} and the semi-latus rectum p=a.

Parametric representation with hyperbolic sine/cosine

Using the hyperbolic sine and cosine functions cosh,sinh, a parametric representation of the hyperbola tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1 can be obtained, which is similar to the parametric representation of an ellipse:
(pm a cosh t, b sinh t),, t in R ,
which satisfies the Cartesian equation because cosh^2 t -sinh^2 t =1 .Further parametric representations are given in the section Parametric equations below.File:Drini-conjugatehyperbolas.svg|thumb|Here {{nowrap|a {{=}} b {{=}} 1}} giving the unit hyperbolaunit hyperbola

Conjugate hyperbola

Exchange x and y to obtain the equation of the conjugate hyperbola (see diagram):
frac{y^2}{a^2}-frac{x^2}{b^2}= 1 , also written as frac{x^2}{b^2}-frac{y^2}{a^2}= -1 .

Hyperbolic functions

Image:Hyperbolic functions-2.svg|thumb|296px|right|A ray through the unit hyperbolaunit hyperbolaJust as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram.Let a be twice the area between the x axis and a ray through the origin intersecting the unit hyperbola and define b=cosh a as the horizontal coordinate of the intersection point.Then, by the Pythagorean theorem,
frac{a}{2}=frac{bsqrt{b^{2}-1}}{2}-displaystyleint_1^b sqrt{x^{2}-1} , mathrm dx=frac{bsqrt{b^{2}-1}}{2}-frac{bsqrt{b^{2}-1}-ln left(b+sqrt{b^{2}-1}right)}{2},
which simplifies to
a=ln left(b+sqrt{b^{2}-1}right).
Solving for b yields the exponential form of the hyperbolic cosine:
x=b=cosh a=frac{e^{a}+e^{-a}}{2}.
By defining y as sinh a, from x^{2}-y^{2}=1 one gets
sinh a=sqrt{cosh ^{2}a-1}.
This implies the exponential form of the hyperbolic sine (h is the vertical coordinate of the intersection point):
y=h=sinh a=frac{e^{a}-e^{-a}}{2}.
Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example
operatorname{tanh}a=frac{sinh a}{cosh a}=frac{e^{2a}-1}{e^{2a}+1}.
The area hyperbolic cosine is defined as follows:
a=operatorname{arcosh}b=ln left(b+sqrt{b^{2}-1}right).
Solving for a in the hyperbolic sine equation yields the area hyperbolic sine:
a=operatorname{arsinh}h=ln left(h+sqrt{h^{2}+1}right).

Hyperbola with equation 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 y=A/x with the coordinate axes as asymptotesred: A = 1; magenta: A = 4; blue: A = 9)If the xy-coordinate system is rotated about the origin by the angle -45^circ and new coordinates xi,eta are assigned, then x=tfrac{xi+eta}{sqrt{2}},; y=tfrac{-xi+eta}{sqrt{2}} . The rectangular hyperbola tfrac{x^2-y^2}{a^2}=1 (whose semi-axes are equal) has the new equation tfrac{2xieta}{a^2}=1.Solving for eta yields eta=tfrac{a^2/2}{xi} . Thus, in an xy-coordinate system the graph of a function f: xmapsto tfrac{A}{x},; A>0; , with equation
y=frac{A}{x};, A>0; , is a rectangular hyperbola entirely in the first and third quadrants with
  • the coordinate axes as asymptotes,
  • the line y=x as major axis ,
  • the center (0,0) and the semi-axis a=b=sqrt{2A} ; ,
  • the vertices left(sqrt{A},sqrt{A}right), left(-sqrt{A},-sqrt{A}right) ; ,
  • the semi-latus rectum and radius of curvature at the vertices p=a=sqrt{2A} ; ,
  • the linear eccentricity c=2sqrt{A} and the eccentricity e=sqrt{2} ; ,
  • the tangent y=-tfrac{A}{x_0^2}x+2tfrac{A}{x_0} at point (x_0,A/x_0); .
A rotation of the original hyperbola by +45^circ results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for the case of -45^circ rotation, with equation
y=frac{-A}{x} ; , A>0; ,
  • the semi-axes a=b=sqrt{2A} ; ,
  • the line y=-x as major axis,
  • the vertices left(-sqrt{A},sqrt{A}right), left(sqrt{A},-sqrt{A}right) ; .
Shifting the hyperbola with equation y=frac{A}{x}, Ane 0 , so that the new center is (c_0,d_0), yields the new equation
y=frac{A}{x-c_0}+d_0; ,
and the new asymptotes are x=c_0 and y=d_0. The shape parameters a,b,p,c,e remain unchanged.

Definition of a hyperbola by the directrix property

(File:Hyperbel-ll-e.svg|300px|thumb|Hyperbola: directrix property)(File:Hyperbel-ll-def.svg|300px|thumb|Hyperbola: definition with directrix property)The two lines at distance d = frac{a^2}c and parallel to the minor axis are called directrices of the hyperbola (see diagram).For an arbitrary point P of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
frac{|PF_1|}{|Pl_1|} = frac{|PF_2|}{|Pl_2|} = e= frac{c}{a} .
The proof for the pair F_1, l_1 follows from the fact that |PF_1|^2=(x-c)^2+y^2, |Pl_1|^2=left(x-tfrac{a^2}{c}right)^2 and y^2=tfrac{b^2}{a^2}x^2-b^2 satisfy the equation
|PF_1|^2-frac{c^2}{a^2}|Pl_1|^2=0 .
The second case is proven analogously.(File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex and common semi latus rectum)The inverse statement is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola):For any point F (focus), any line l (directrix) not through F and any real number e with e > 1 the set of points (locus of points), for which the quotient of the distances to the point and to the line is e
H = left{P , Biggr| , frac{|PF|}{|Pl|} = eright} is a hyperbola.
(The choice e = 1 yields a parabola and if e < 1 an ellipse.)
Proof:
Let F=(f,0) , e >0 and assume (0,0) is a point on the curve. The directrix l has equation x=-tfrac{f}{e}. With P=(x,y), the relation |PF|^2=e^2|Pl|^2 produces the equations
(x-f)^2+y^2=e^2left(x+tfrac{f}{e}right)^2=(e x+f)^2 and x^2(e^2-1)+2xf(1+e)-y^2=0.
The substitution p=f(1+e) yields
x^2(e^2-1)+2px-y^2=0.
This is the equation of an ellipse (e1). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).If e>1, introduce new parameters a,b so thate^2-1 =tfrac{b^2}{a^2}, text { and } p=tfrac{b^2}{a}, and then the equation above becomes
tfrac{(x+a)^2}{a^2}-tfrac{y^2}{b^2}=1 ,
which is the equation of a hyperbola with center (-a,0), the x-axis as major axis andthe major/minor semi axis a,b.

Hyperbola as plane section of a cone

(File:Hyperbel-dandel-s.svg|450px|thumb|Hyperbola (red): two views of a cone and two Dandelin spheres d1, d2)The intersection of an upright double cone by a plane not through the vertex with slope greater than the slope of the lines on the cone is a hyperbola (see diagram: red curve). In order to prove the defining property of a hyperbola (see above) one uses two Dandelin spheres d_1, d_2, which are spheres that touch the cone along circles c_1 , c_2 and the intersecting (hyperbola) plane at points F_1 and F_2. It turns out: F_1, F_2 are the foci of the hyperbola.
  1. Let P be an arbitrary point of the intersection curve .
  2. The generator (line) of the cone containing P intersects circle c_1 at point A and circle c_2 at a point B.
  3. The line segments overline{PF_1} and overline{PA} are tangential to the sphere d_1 and, hence, are of equal length.
  4. The line segments overline{PF_2} and overline{PB} are tangential to the sphere d_2 and, hence, are of equal length.
  5. The result is: |PF_1|-|PF_2|=|PA|-|PB|=|AB| is independent of the hyperbola point P.

Pin and string construction

(File:Hyperbola-pin-string.svg|300px|thumb|Hyperbola: Pin and string construction)The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler Frans van Schooten: Mathematische Oeffeningen, Leyden, 1659, p. 327:(0) Choose the foci F_1,F_2, the vertices V_1,V_2 and one of the circular directrices , for example c_2 (circle with radius 2a) (1) A ruler is fixed at point F_2 free to rotate around F_2. Point B is marked at distance 2a.(2) A string with length |AB| is prepared.(3) One end of the string is pinned at point A on the ruler, the other end is pinned to point F_1.(4) Take a pen and hold the string tight to the edge of the ruler. (5) Rotating the ruler around F_2 prompts the pen to draw an arc of the right branch of the hyperbola, because of |PF_1|=|PB| (see the definition of a hyperbola by circular directrices).

The tangent bisects the angle between the lines to the foci

(File:Hyperbel-wh-s.svg|300px|thumb|Hyperbola: the tangent bisects the lines through the foci)The tangent at a point P bisects the angle between the lines overline{PF_1}, overline{PF_2}.
Proof:
Let L be the point on the line overline{PF_2} with the distance 2a to the focus F_2 (see diagram, a is the semi major axis of the hyperbola). Line w is the bisector of the angle between the lines overline{PF_1}, overline{PF_2}. In order to prove that w is the tangent line at point P, one checks that any point Q on line w which is different from P cannot be on the hyperbola. Hence w has only point P in common with the hyperbola and is, therefore, the tangent at point P. From the diagram and the triangle inequality one recognizes that |QF_2|b the intersection points of orthogonal tangents lie on the circle x^2+y^2=a^2-b^2. This circle is called the orthoptic of the given hyperbola.The tangents may belong to points on different branches of the hyperbola.In case of ale b 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 tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1. The equation of the tangent at a point P_0=(x_0,y_0) of the hyperbola is tfrac{x_0x}{a^2}-tfrac{y_0y}{b^2}=1. If one allows point P_0=(x_0,y_0) to be an arbitrary point different from the origin, then
point P_0=(x_0,y_0)ne(0,0) is mapped onto the line frac{x_0x}{a^2}-frac{y_0y}{b^2}=1 , not through the center of the hyperbola.
This relation between points and lines is a bijection.The inverse function maps
line y=mx+d, dne 0 onto the point left(-frac{ma^2}{d},-frac{b^2}{d}right) and
line x=c, cne 0 onto the point left(frac{a^2}{c},0right) .
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: P_1, p_1).
  • For a pole P outside the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing P (see diagram: P_2, p_2, P_3, p_3).
  • For a point within the hyperbola the polar has no point with the hyperbola in common. (see diagram: P_4, p_4).
Remarks:
  1. The intersection point of two polars (for example: p_2,p_3) is the pole of the line through their poles (here: P_2,P_3).
  2. The foci (c,0), and (-c,0) respectively and the directrices x=tfrac{a^2}{c} and x=-tfrac{a^2}{c} respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.

Hyperbola as an affine image of the unit hyperbola x² – y² 1

(File:Hyperbel-aff-s.svg|300px|thumb|Hyperbola as an affine image of the unit hyperbola)Another definition of a hyperbola uses affine transformations:
Any hyperbola is the affine image of the unit hyperbola with equation x^2-y^2=1.
An affine transformation of the Euclidean plane has the form vec x to vec f_0+Avec x, where A is a regular matrix (its determinant is not 0) and vec f_0 is an arbitrary vector. If vec f_1, vec f_2 are the column vectors of the matrix A, the unit hyperbola (pmcosh(t),sinh(t)), t in R, is mapped onto the hyperbola
vec x = vec p(t)=vec f_0 pmvec f_1 cosh t +vec f_2 sinh t .
vec f_0 is the center, vec f_0+ vec f_1 a point of the hyperbola and vec f_2 a tangent vector at this point. In general the vectors vec f_1, vec f_2 are not perpendicular. That means, in general vec f_0pm vec f_1 are not the vertices of the hyperbola. But vec f_1pm vec f_2 point into the directions of the asymptotes. The tangent vector at point vec p(t) is
vec p'(t) = vec f_1sinh t + vec f_2cosh t .
Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter t_0 of a vertex from the equation
vec p'(t)cdot left(vec p(t) -vec f_0right) = left(vec f_1sinh t + vec f_2cosh tright)cdotleft(vec f_1 cosh t +vec f_2 sinh tright) =0
and hence from
coth (2t_0)= -tfrac{vec f_1^{, 2}+vec f_2^{, 2}}{2vec f_1 cdot vec f_2} ,
which yields
t_0=tfrac{1}{4}lntfrac{left(vec f_1-vec f_2right)^2}{left(vec f_1+vec f_2right)^2}.
(The formulae cosh^2 x +sinh^2 x=cosh 2x, 2sinh x cosh x = sinh 2x, operatorname{arcoth} x = tfrac{1}{2}lntfrac{x+1}{x-1} were used.)The two vertices of the hyperbola are vec f_0pmleft(vec f_1cosh t_0 +vec f_2 sinh t_0right).The advantage of this definition is that one gets a simple parametric representation of an arbitrary hyperbola, even in the space, if the vectors vec f_0, vec f_1, vec f_2 are vectors of the Euclidean space.

Hyperbola as an affine image of the hyperbola y 1/x

(File:Hyperbel-aff2.svg|thumb|300px|Hyperbola as affine image of y = 1/x)Because the unit hyperbola x^2-y^2=1 is affinely equivalent to the hyperbola y=1/x, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola y=1/x :
vec x= vec p(t)=vec f_0 + vec f_1 t+ vec f_2 tfrac{1}, quad tne 0 .
M: vec f_0 is the center of the hyperbola, the vectors vec f_1 , vec f_2 have the directions of the asymptotes and vec f_1 + vec f_2 is a point of the hyperbola. The tangent vector is
vec p'(t)=vec f_1 - vec f_2 tfrac{1}{t^2}.
At a vertex the tangent is perpendicular to the major axis. Hence
vec p'(t)cdot left(vec p(t) -vec f_0right) = left(vec f_1 - vec f_2 tfrac{1}{t^2}right)cdotleft(vec f_1 t+ vec f_2 tfrac{1}right) = vec f_1^2t-vec f_2^2 tfrac{1}{t^3} = 0
and the parameter of a vertex is
t_0= pm sqrt[4]{tfrac{vec f_2^2}{vec f_1^2}}.
left(vec f_1t - vec f_2 tfrac{1}right) .This means that
the diagonal AB of the parallelogram M: vec f_0, A=vec f_0+vec f_1t, B: vec f_0+ vec f_2 tfrac{1}, P: vec f_0+vec f_1t+vec f_2 tfrac{1} is parallel to the tangent at the hyperbola point P (see diagram).
This property provides a way to construct the tangent at a point on the hyperbola.This property of a hyperbola is an affine version of the 3-point-degeneration of Pascal's theorem.Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, S. 33, (PDF; 757 kB)
Area of the grey parallelogram:
The area of the grey parallelogram MAPB in the above diagram is
text{Area}=Big|detleft( tvec f_1, tfrac{1}vec f_2right)Big|=Big|detleft(vec f_1,vec f_2right)Big|= cdots = frac{a^2+b^2}{4}
and hence independent of point P. The last equation follows from a calculation for the case, where P is a vertex and the hyperbola in its canonical form tfrac{x^2}{a^2}-tfrac{y^2}{b^2}=1 .

Point construction

(File:Hyperbel-pasc4-s.svg|thumb|Point construction: asymptotes and P1 are given → P2)For a hyperbola with parametric representation vec x= vec p(t)=vec f_1 t+ vec f_2 tfrac{1} (for simplicity the center is the origin) the following is true:
For any two points P_1: vec f_1 t_1+ vec f_2 tfrac{1}{t_1}, P_2: vec f_1 t_2+ vec f_2 tfrac{1}{t_2} the points 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} are collinear with the center of the hyperbola (see diagram).
The simple proof is a consequence of the equation tfrac{1}{t_1}vec a=tfrac{1}{t_2}vec b.This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given.This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem.Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes, S. 32, (PDF; 757 kB)

Tangent-asymptotes-triangle

(File:Hyperbel-tad-s.svg|thumb|Hyperbola: tangent-asymptotes-triangle)For simplicity the center of the hyperbola may be the origin and the vectors vec f_1,vec f_2 have equal length. If the last assumption is not fulfilled one can first apply a parameter transformation (see above) in order to make the assumption true. Hence pm(vec f_1+vec f_2) are the vertices, pm(vec f_1-vec f_2) span the minor axis and one gets |vec f_1+vec f_2|=a and |vec f_1-vec f_2|=b.For the intersection points of the tangent at point vec p(t_0)=vec f_1 t_0+ vec f_2 tfrac{1}{t_0} with the asymptotes one gets the points
C=2t_0vec f_1, D=tfrac{2}{t_0}vec f_2.
The area of the triangle M,C,D can be calculated by a 2x2-determinant:
A=tfrac{1}{2}Big|detleft( 2t_0vec f_1, tfrac{2}{t_0}vec f_2right)Big|=2Big|detleft(vec f_1,vec f_2right)Big|
(see rules for determinants).< varphi < arccos left(-frac 1 eright). for pole = center:With polar coordinates relative to the "canonical coordinate system" (see second diagram)one has that
r =frac{b}{sqrt{e^2 cos^2 varphi -1}} .,
For the right branch of the hyperbola the range of varphi is
-arccos left(frac 1 eright) < varphi < arccos left(frac 1 eright).

Parametric equations

A hyperbola with equation tfrac{x^2}{a^2}-tfrac{y^2}{b^2}= 1 can be described by several parametric equations:
1: left{begin{matrix}quad ; x , = , pm a cosh t y , = , b sinh t end{matrix}right. quad , t in R . 2: left{begin{matrix} quad x , = , pm a, tfrac{t^2+1}{2t} y , = , b, tfrac{t^2-1}{2t} end{matrix}right. quad , t >0 . (rational representation) 3: left{begin{matrix}quad x , = , frac{a}{cos t}=asec t y , = , pm b tan t end{matrix}right. , quad 0 le t < 2pi; ; t ne frac{pi}{2}; ; t ne frac{3}{2}pi .
4: Tangent slope as parameter:
A parametric representation, which uses the slope m of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case b^2 by -b^2and use formulae for the hyperbolic functions. One gets
vec c_pm(m) = left(-frac{ma^2}{pmsqrt{m^2a^2-b^2}};,;frac{-b^2}{pmsqrt{m^2a^2-b^2}}right) , |m|>b/a .
vec c_- is the upper and vec c_+ the lower half of the hyperbola. The points with vertical tangents (vertices (pm a,0)) are not covered by the representation.The equation of the tangent at point vec c_pm(m) is
y = m x pmsqrt{m^2a^2-b^2}.
This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.

Other mathematical definitions

Reciprocation of a circle

The reciprocation of a circleB in a circle C always yields a conic section such as a hyperbola. The process of "reciprocation in a circle C" consists of replacing every line and point in a geometrical figure with their corresponding pole and polar, respectively. The pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point.The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of reciprocation circle C. If B and C represent the points at the centers of the corresponding circles, then
e = frac{overline{BC}}{r}.Since the eccentricity of a hyperbola is always greater than one, the center B must lie outside of the reciprocating circle C.This definition implies that the hyperbola is both the locus of the poles of the tangent lines to the circle B, as well as the envelope of the polar lines of the points on B. Conversely, the circle B is the envelope of polars of points on the hyperbola, and the locus of poles of tangent lines to the hyperbola. Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.

Quadratic equation

A hyperbola can also be defined as a second-degree equation in the Cartesian coordinates (x, y) in the plane,
A_{xx} x^2 + 2 A_{xy} xy + A_{yy} y^2 + 2 B_x x + 2 B_y y + C = 0,provided that the constants A'xx, A'xy, A'yy, B'x, By, and C satisfy the determinant condition
D := begin{vmatrix} A_{xx} & A_{xy}A_{xy} & A_{yy} end{vmatrix} < 0.,This determinant is conventionally called the discriminant of the conic section.{{citation|title=Math refresher for scientists and engineers|first1=John R.|last1=Fanchi|publisher=John Wiley and Sons|year=2006|isbn=0-471-75715-2|pages=44–45|url=https://books.google.com/books?id=75mAJPcAWT8C}}, Section 3.2, page 45A special case of a hyperbola—the degenerate hyperbola consisting of two intersecting lines—occurs when another determinant is zero:
Delta := begin{vmatrix} A_{xx} & A_{xy} & B_x A_{xy} & A_{yy} & B_y B_x & B_y & C end{vmatrix} = 0.This determinant Δ is sometimes called the discriminant of the conic section.Korn, Granino A. and Korn, Theresa M. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover Publ., second edition, 2000: p. 40.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 parameters of the quadratic form.The center (x'c, y'c) of the hyperbola may be determined from the formulae
x_c = -frac{1}{D} begin{vmatrix} B_x & A_{xy} B_y & A_{yy} end{vmatrix};
y_c = -frac 1 D begin{vmatrix} A_{xx} & B_x A_{xy} & B_y end{vmatrix}.In terms of new coordinates, {{nowrap|ξ {{=}} xx'c}} and {{nowrap|η {{=}} yy'c}}, the defining equation of the hyperbola can be written
A_{xx} xi^2 + 2A_{xy} xieta + A_{yy} eta^2 + frac Delta D = 0.The principal axes of the hyperbola make an angle φ with the positive x-axis that is given by
tan 2varphi = frac{2A_{xy}}{A_{xx} - A_{yy}}.Rotating the coordinate axes so that the x-axis is aligned with the transverse axis brings the equation into its canonical form
frac{x^2}{a^2} - frac{y^2}{b^2} = 1.The major and minor semiaxes a and b are defined by the equations
a^2 = -frac Delta {lambda_1 D} = -frac Delta {lambda_1^2 lambda_2},
b^{2} = -frac{Delta}{lambda_{2}D} = -frac{Delta}{lambda_{1}lambda_{2}^{2}},where λ1 and λ2 are the roots of the quadratic equation
lambda^2 - left( A_{xx} + A_{yy} right)lambda + D = 0.For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is
frac{x^2}{a^2} - frac{y^2}{b^2} = 0.The tangent line to a given point (x0, y0) on the hyperbola is defined by the equation
E x + F y + G = 0where E, F and G are defined by
E = A_{xx} x_0 + A_{xy} y_0 + B_x,
F = A_{xy} x_0 + A_{yy} y_0 + B_y,
G = B_x x_0 + B_y y_0 + C.The normal line to the hyperbola at the same point is given by the equation
F(x - x_0) - E(y - y_0) = 0.The normal line is perpendicular to the tangent line, and both pass through the same point (x0, y0).From the equation
frac{x^2}{a^2} - frac{y^2}{b^2} = 1, qquad 0 < b leq a,
the left focus is (-ae,0) and the right focus is (ae,0), where {{math|e}} is the eccentricity. Denote the distances from a point (x, y) to the left and right foci as r_1 ,! and r_2 . ,! For a point on the right branch,
r_1 - r_2 =2 a, , !
and for a point on the left branch,
r_2 - r_1 =2 a. , !
This can be proved as follows:If (x,y) is a point on the hyperbola the distance to the left focal point is
r_1^2 =(x+a e)^2 + y^2 = x^2 + 2 x a e + a^2 e^2 + left(x^2-a^2right)left(e^2-1right)=
(e x + a)^2.To the right focal point the distance is
r_2^2 = (x-a e)^2 + y^2 = x^2 - 2 x a e + a^2 e^2 + left(x^2-a^2right)left(e^2-1right)=
(e x - a)^2.If (x,y) is a point on the right branch of the hyperbola then e x > a,! and
r_1 =e x + a,,! r_2 =e x - a.,!
Subtracting these equations one gets
r_1 - r_2 =2 a.,!
If (x,y) is a point on the left branch of the hyperbola then e x < -a,! and
r_1 = -e x - a,,! r_2 = -e x + a.,!
Subtracting these equations one gets
r_2 - r_1 =2 a.,!

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. 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.(If center O were on the sphere, all images of the circles would be circles or lines; see stereographic projectionstereographic projectionBesides 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
a) a circle, if circle c is in a special position, for example parallel to the image plane and others (see stereographic projection), b) an ellipse, if c has no point with the lens plane in common, c) a parabola, if c has one point with the lens plane in common and d) 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.

Arc length

The arc length of a hyperbola does not have a closed-form expression. The upper half of a hyperbola can be parameterized as
y=bsqrt{frac{x^{2}}{a^{2}}-1}.
Then the integral giving the arc length s from x_{1} to x_{2} can be computed numerically:
s=bint_{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.
After using the substitution z=iv, this can also be represented using the elliptic integral of the second kind with parameter m=k^{2}:
s=-ibBiggr[Eleft(iz , Biggr| , 1+frac{a^{2}}{b^{2}}right)Biggr]^{operatorname{arcosh}frac{x_{2}}{a}}_{operatorname{arcosh}frac{x_{1}}{a}}.

Derived curves

{{Sinusoidal_spirals.svg}}Several other curves can be derived from the hyperbola by inversion, the so-called inverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli; 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
left(frac x {c costheta}right)^2 - left(frac y {c sintheta}right)^2 = 1where 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 = z2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.

Other properties of hyperbolas

  • The following are concurrent: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.WEB,weblink Hyperbola, Mathafou.free.fr, 26 August 2018, weblink
  • 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.

Applications

(File:Akademia Ekonomiczna w Krakowie Pawilon C.JPG|thumb|right|Hyperbolas as declination lines on a sundial)

Sundials

Hyperbolas may be seen in many sundials. 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 2a from two given points is a hyperbola of vertex separation 2a 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 Rutherford experiment demonstrated the existence of an atomic nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb force, which satisfies the inverse square law requirement for a Kepler problem.

Korteweg–de Vries equation

The hyperbolic trig function operatorname{sech}, x 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 trisect any angle, a well studied problem of geometry. Given an angle, first draw a circle centered at its vertex O, which intersects the sides of the angle at points A and B. Next draw the line segment with endpoints A and B and its perpendicular bisector ell. Construct a hyperbola of eccentricitye=2 with ell as directrix and B as a focus. Let P be the intersection (upper) of the hyperbola with the circle. Angle POB trisects angle AOB.To prove this, reflect the line segment OP about the line ell obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.This construction is due to Pappus of Alexandria (circa 300 A.D.) and the proof comes from {{harvtxt|Kazarinoff|1970|loc=pg. 62}}.

Efficient portfolio frontier

In portfolio theory, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.

Biochemistry

In biochemistry and pharmacology, the Hill equation and Hill-Langmuir equation respectively describe biological responses and the formation of protein–ligand complexes as functions of ligand concentration. They are both rectangular hyperbolae.

Hyperbolas as plane sections of quadrics

Hyperbolas appear as plane sections of the following quadrics: Quadric Cone.jpg|Elliptic coneHyperbolic Cylinder Quadric.png|Hyperbolic cylinderHyperbol Paraboloid.pov.png|Hyperbolic paraboloidHyperboloid1.png|Hyperboloid of one sheetHyperboloid2.png|Hyperboloid of two sheets

See also

Other conic sections

{{div col|colwidth=25em}} {{div col end}}

Other related topics

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Notes

{{Reflist}}

References

  • {{citation |last=Kazarinoff |first=Nicholas D. |title=Ruler and the Round |year=2003 |publisher=Dover |location=Mineola, N.Y. |isbn=0-486-42515-0}}

External links

{{commons category|Hyperbolas}}{{EB1911 poster|Hyperbola}}
  • {{springer|title=Hyperbola|id=p/h048230}}
  • weblink" title="web.archive.org/web/20070625162103weblink">Apollonius' Derivation of the Hyperbola at weblink" title="web.archive.org/web/20070713083148weblink">Convergence
  • Frans van Schooten: Mathematische Oeffeningen, 1659
  • {{planetmath reference|id=5996|title=Unit hyperbola}}
  • {{planetmath reference|id=3584|title=Conic section}}
  • {{planetmath reference|id=6241|title=Conjugate hyperbola}}
  • {{MathWorld |id=Hyperbola}}
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= is equivalent to t_0=pm 1 and vec f_0pm(vec f_1+vec f_2) are the vertices of the hyperbola.The following properties of a hyperbola are easily proven using the representation of a hyperbola introduced in this section.

Tangent construction

(File:Hyperbel-tang-s.svg|thumb|Tangent construction: asymptotes and P given → tangent)The tangent vector can be rewritten by factorization:
vec p'(t)=tfrac{1}
is the area of the rhombus generated by vec f_1,vec f_2. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes a,b of the hyperbola. Hence:
The area of the triangle MCD is independent of the point of the hyperbola: A=ab.

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)For pole = focus: The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its origin in a focus and its x-axis pointing towards the origin of the "canonical coordinate system" as illustrated in the first diagram.In this case the angle varphi is called true anomaly.Relative to this coordinate system one has that
r = frac{p}{1 mp e cos varphi}, quad p=tfrac{b^2}{a}
and
-arccos left(-frac 1 eright)