![Hyperbola (PSF)](https://upload.wikimedia.org/wikipedia/commons/0/0e/Hyperbola_%28PSF%29.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
, formed by the intersection of a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
and a double
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
. (The other conic sections are the
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
and the
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. A
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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
reciprocal function
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
in the
Cartesian plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
,
* as the path followed by the shadow of the tip of a
sundial
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
,
* as the shape of an
open orbit (as distinct from a closed elliptical orbit), such as the orbit of a
spacecraft
A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, p ...
during a
gravity assist
In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the p ...
ed swing-by of a planet or, more generally, any spacecraft (or celestial object) exceeding the
escape velocity
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
of the nearest planet or other gravitational body,
* as the
scattering trajectory of a
subatomic particle
In physical sciences, a subatomic particle is a particle that composes an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a pr ...
(acted on by repulsive instead of attractive forces but the principle is the same),
* in
radio navigation
Radio navigation or radionavigation is the application of radio frequencies to determine a position of an object on the Earth, either the vessel or an obstruction. Like radiolocation, it is a type of radiodetermination.
The basic principles a ...
, when the difference between distances to two points, but not the distances themselves, can be determined,
and so on.
Each
branch
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
of those two arms. So there are two asymptotes, whose intersection is at the center of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
the asymptotes are the two
coordinate axes
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
.
Hyperbolas share many of the ellipses' analytical properties such as
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
,
focus
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
, and
directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
s have their origin in the hyperbola, such as
hyperbolic paraboloids (saddle surfaces),
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
s ("wastebaskets"),
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
(
Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
's celebrated
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
),
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s (sinh, cosh, tanh, etc.), and
gyrovector space
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
s (a geometry proposed for use in both
relativity and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
which is not
Euclidean).
Etymology and history
The word "hyperbola" derives from the
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
, meaning "over-thrown" or "excessive", from which the English term
hyperbole
Hyperbole (; adj. hyperbolic ) is the use of exaggeration as a rhetorical device or figure of speech. In rhetoric, it is also sometimes known as auxesis (literally 'growth'). In poetry and oratory, it emphasizes, evokes strong feelings, and ...
also derives. Hyperbolae were discovered by
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.''
Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
in his investigations of the problem of
doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
, but were then called sections of obtuse cones. 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 section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
s, the ''Conics''.
The names of the other two general conic sections, the
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
and the
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
, derive from the corresponding Greek words for "deficient" and "applied"; all three names are borrowed from earlier Pythagorean terminology which referred to a comparison of the side of rectangles of fixed area with a given line segment. The rectangle could be "applied" to the segment (meaning, have an equal length), be shorter than the segment or exceed the segment.
Definitions
As locus of points
![Hyperbel-def-dc](https://upload.wikimedia.org/wikipedia/commons/1/14/Hyperbel-def-dc.svg)
A hyperbola can be defined geometrically as a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of points (
locus of points
In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions..
In ...
) in the Euclidean plane:
:A hyperbola is a set of points, such that for any point
of the set, the absolute difference of the distances
to two fixed points
(the ''foci'') is constant, usually denoted by
:
:
The midpoint
of the line segment joining the foci is called the ''center'' of the hyperbola. The line through the foci is called the ''major axis''. It contains the ''vertices''
, which have distance
to the center. The distance
of the foci to the center is called the ''focal distance'' or ''linear eccentricity''. The quotient
is the ''eccentricity''
.
The equation
can be viewed in a different way (see diagram):
If
is the circle with midpoint
and radius
, then the distance of a point
of the right branch to the circle
equals the distance to the focus
:
is called the ''circular directrix'' (related to focus
) of the hyperbola. In order to get the left branch of the hyperbola, one has to use the circular directrix related to
. This property should not be confused with the definition of a hyperbola with help of a directrix (line) below.
Hyperbola with equation ''y'' = ''A''/''x''
![Hyperbeln-gs-3](https://upload.wikimedia.org/wikipedia/commons/0/03/Hyperbeln-gs-3.svg)
If the ''xy''-coordinate system is
rotated about the origin by the angle
and new coordinates
are assigned, then
.
The rectangular hyperbola
(whose semi-axes are equal) has the new equation
.
Solving for
yields
Thus, in an ''xy''-coordinate system the graph of a function
with equation
:
is a ''rectangular hyperbola'' entirely in the first and third
quadrants with
*the coordinate axes as ''asymptotes'',
*the line
as ''major axis'' ,
*the ''center''
and the ''semi-axis''
*the ''vertices''
*the ''semi-latus rectum'' and ''radius of curvature '' at the vertices
*the ''linear eccentricity''
and the eccentricity
*the ''tangent''
at point
A rotation of the original hyperbola by
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
rotation, with equation
:
*the ''semi-axes''
*the line
as major axis,
*the ''vertices''
Shifting the hyperbola with equation
so that the new center is
, yields the new equation
:
and the new asymptotes are
and
.
The shape parameters
remain unchanged.
By the directrix property
![Hyperbel-ll-def](https://upload.wikimedia.org/wikipedia/commons/7/71/Hyperbel-ll-def.svg)
The two lines at distance
from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram).
For an arbitrary point
of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
:
The proof for the pair
follows from the fact that
and
satisfy the equation
:
The second case is proven analogously.
![Kegelschnitt-schar-ev](https://upload.wikimedia.org/wikipedia/commons/7/74/Kegelschnitt-schar-ev.svg)
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
(focus), any line
(directrix) not through
and any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
with
the set of points (locus of points), for which the quotient of the distances to the point and to the line is
:
:is a hyperbola.
(The choice
yields a
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
and if
an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
.)
;Proof:
Let
and assume
is a point on the curve.
The directrix
has equation
. With
, the relation
produces the equations
:
and
The substitution
yields
:
This is the equation of an ''ellipse'' (
) or a ''parabola'' (
) or a ''hyperbola'' (
). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If
, introduce new parameters
so that
, and then the equation above becomes
:
which is the equation of a hyperbola with center
, the ''x''-axis as major axis and
the major/minor semi axis
.
![Hyperbel-leitl-e](https://upload.wikimedia.org/wikipedia/commons/5/5b/Hyperbel-leitl-e.svg)
;Construction of a directrix
Because of
point
of directrix
(see diagram) and focus
are inverse with respect to the
circle inversion
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
at circle
(in diagram green). Hence point
can be constructed using the
theorem of Thales (not shown in the diagram). The directrix
is the perpendicular to line
through point
.
''Alternative construction of
'': Calculation shows, that point
is the intersection of the asymptote with its perpendicular through
(see diagram).
As plane section of a cone
![Dandelin-hyperbel](https://upload.wikimedia.org/wikipedia/commons/d/d4/Dandelin-hyperbel.svg)
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 , which are spheres that touch the cone along circles
,
and the intersecting (hyperbola) plane at points
and
. It turns out:
are the ''foci'' of the hyperbola.
# Let
be an arbitrary point of the intersection curve .
# The
generatrix
In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent.
Examples
A cone can be ...
of the cone containing
intersects circle
at point
and circle
at a point
.
# The line segments
and
are tangential to the sphere
and, hence, are of equal length.
# The line segments
and
are tangential to the sphere
and, hence, are of equal length.
# The result is:
is independent of the hyperbola point
, because no matter where point
is,
have to be on circles
,
, and line segment
has to cross the apex. Therefore, as point
moves along the red curve (hyperbola), line segment
simply rotates about apex without changing its length.
Pin and string construction
![Hyperbola-pin-string](https://upload.wikimedia.org/wikipedia/commons/6/6a/Hyperbola-pin-string.svg)
The definition of a hyperbola by its foci and its circular directrices (see above) can be used for drawing an arc of it with help of pins, a string and a ruler:
#
Choose the ''foci'' , the vertices and one of the ''circular directrices'' , for example (circle with radius )
# A ''ruler'' is fixed at point
free to rotate around
. Point
is marked at distance
.
# A ''string'' with length
is prepared.
# One end of the string is pinned at point
on the ruler, the other end is pinned to point
.
# Take a ''pen'' and hold the string tight to the edge of the ruler.
# ''Rotating'' the ruler around
prompts the pen to draw an arc of the right branch of the hyperbola, because of
(see the definition of a hyperbola by ''circular directrices'').
Steiner generation of a hyperbola
![Hyperbola construction - parallelogram method](https://upload.wikimedia.org/wikipedia/commons/1/13/Hyperbola_construction_-_parallelogram_method.gif)
The following method to construct single points of a hyperbola relies on the
Steiner generation of a non degenerate conic section:
:Given two
pencils
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
of lines at two points
(all lines containing
and
, respectively) and a projective but not perspective mapping
of
onto
, then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the hyperbola
one uses the pencils at the vertices
. Let
be a point of the hyperbola and
. The line segment
is divided into n equally-spaced segments and this division is projected parallel with the diagonal
as direction onto the line segment
(see diagram). The parallel projection is part of the projective mapping between the pencils at
and
needed. The intersection points of any two related lines
and
are points of the uniquely defined hyperbola.
''Remark:'' The subdivision could be extended beyond the points
and
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).
''Remark:''
# 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 ''y'' = ''a''/(''x'' − ''b'') + ''c'' and the 3-point-form
![Hyperbel-pws-s](https://upload.wikimedia.org/wikipedia/commons/8/8e/Hyperbel-pws-s.svg)
A hyperbola with equation
is uniquely determined by three points
with different ''x''- and ''y''-coordinates. A simple way to determine the shape parameters
uses the ''inscribed angle theorem'' for hyperbolas:
:In order to measure an angle between two lines with equations
in this context one uses the quotient
:
Analogous to the
inscribed angle
In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.
Equivalently, an in ...
theorem for circles one gets the
Inscribed angle theorem for hyperbolas:
: For four points
(see diagram) the following statement is true:
: The four points are on a hyperbola with equation
if and only if the angles at
and
are equal in the sense of the measurement above. That means if
::
(Proof: straightforward calculation. If the points are on a hyperbola, one can assume the hyperbola's equation is
.)
A consequence of the inscribed angle theorem for hyperbolas is the
3-point-form of a hyperbola's equation:
: The equation of the hyperbola determined by 3 points
is the solution of the equation
::
: for
.
As an affine image of the unit hyperbola ''x''² − ''y''² = 1
![Hyperbel-aff-s](https://upload.wikimedia.org/wikipedia/commons/9/9e/Hyperbel-aff-s.svg)
Another definition of a hyperbola uses
affine transformations:
:Any ''hyperbola'' is the affine image of the unit hyperbola with equation
.
;parametric representation
An affine transformation of the Euclidean plane has the form
, where
is a regular
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(its
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
is not 0) and
is an arbitrary vector. If
are the column vectors of the matrix
, the unit hyperbola
is mapped onto the hyperbola
:
is the center,
a point of the hyperbola and
a tangent vector at this point.
;vertices
In general the vectors
are not perpendicular. That means, in general
are ''not'' the vertices of the hyperbola. But
point into the directions of the asymptotes. The tangent vector at point
is
:
Because at a vertex the tangent is perpendicular to the major axis of the hyperbola one gets the parameter
of a vertex from the equation
:
and hence from
:
which yields
:
(The formulae
were used.)
The two ''vertices'' of the hyperbola are
;implicit representation
Solving the parametric representation for
by
Cramer's rule and using
, one gets the implicit representation
:
.
;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
to be vectors in space.
As an affine image of the hyperbola ''y'' = 1/''x''
![Hyperbel-aff2](https://upload.wikimedia.org/wikipedia/commons/d/df/Hyperbel-aff2.svg)
Because the unit hyperbola
is affinely equivalent to the hyperbola
, an arbitrary hyperbola can be considered as the affine image (see previous section) of the hyperbola
:
is the center of the hyperbola, the vectors
have the directions of the asymptotes and
is a point of the hyperbola. The tangent vector is
:
At a vertex the tangent is perpendicular to the major axis. Hence
:
and the parameter of a vertex is
:
is equivalent to
and
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
![Hyperbel-tang-s](https://upload.wikimedia.org/wikipedia/commons/3/3c/Hyperbel-tang-s.svg)
The tangent vector can be rewritten by factorization:
:
This means that
:the diagonal
of the parallelogram
is parallel to the tangent at the hyperbola point
(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
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined ...
.
;Area of the grey parallelogram:
The area of the grey parallelogram
in the above diagram is
:
and hence independent of point
. The last equation follows from a calculation for the case, where
is a vertex and the hyperbola in its canonical form
Point construction
![Hyperbel-pasc4-s](https://upload.wikimedia.org/wikipedia/commons/e/ee/Hyperbel-pasc4-s.svg)
For a hyperbola with parametric representation
(for simplicity the center is the origin) the following is true:
:For any two points
the points
:
:are collinear with the center of the hyperbola (see diagram).
The simple proof is a consequence of the equation
.
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
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined ...
.
Tangent-asymptotes-triangle
![Hyperbel-tad-s](https://upload.wikimedia.org/wikipedia/commons/2/28/Hyperbel-tad-s.svg)
For simplicity the center of the hyperbola may be the origin and the vectors
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
are the vertices,
span the minor axis and one gets
and
.
For the intersection points of the tangent at point
with the asymptotes one gets the points
:
The ''
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
'' of the triangle
can be calculated by a 2 × 2 determinant:
:
(see rules for
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s).
is the area of the rhombus generated by
. The area of a rhombus is equal to one half of the product of its diagonals. The diagonals are the semi-axes
of the hyperbola. Hence:
:The ''area'' of the triangle
is independent of the point of the hyperbola:
Reciprocation of a circle
The
reciprocation of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
''B'' in a circle ''C'' always yields a conic section such as a hyperbola. The process of "reciprocation in a circle ''C''" consists of replacing every line and point in a geometrical figure with their corresponding
pole and polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into it ...
, respectively. The ''pole'' of a line is the
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
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
:
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
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of the poles of the tangent lines to the circle ''B'', as well as the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
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
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
,
:
provided that the constants ''A''
''xx'', ''A''
''xy'', ''A''
''yy'', ''B''
''x'', ''B''
''y'', and ''C'' satisfy the determinant condition
:
This determinant is conventionally called the
discriminant of the conic section.
A special case of a hyperbola—the ''
degenerate hyperbola'' consisting of two intersecting lines—occurs when another determinant is zero:
:
This determinant Δ is sometimes called the discriminant of the conic section.
Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in
Conic section#Eccentricity in terms of coefficients.
The center (''x''
''c'', ''y''
''c'') of the hyperbola may be determined from the formulae
:
:
In terms of new coordinates, and , the defining equation of the hyperbola can be written
:
The principal axes of the hyperbola make an angle ''φ'' with the positive ''x''-axis that is given by
:
Rotating the coordinate axes so that the ''x''-axis is aligned with the transverse axis brings the equation into its canonical form
:
The major and minor semiaxes ''a'' and ''b'' are defined by the equations
:
:
where λ
1 and λ
2 are the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
:
For comparison, the corresponding equation for a degenerate hyperbola (consisting of two intersecting lines) is
:
The tangent line to a given point (''x''
0, ''y''
0) on the hyperbola is defined by the equation
:
where ''E'', ''F'' and ''G'' are defined by
:
:
:
The
normal line
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at t ...
to the hyperbola at the same point is given by the equation
:
The normal line is perpendicular to the tangent line, and both pass through the same point (''x''
0, ''y''
0).
From the equation
:
the left focus is
and the right focus is
where is the eccentricity. Denote the distances from a point (''x, y'') to the left and right foci as
and
For a point on the right branch,
:
and for a point on the left branch,
:
This can be proved as follows:
If (''x'',''y'') is a point on the hyperbola the distance to the left focal point is
:
To the right focal point the distance is
:
If (''x'',''y'') is a point on the right branch of the hyperbola then
and
:
:
Subtracting these equations one gets
:
If (''x,y'') is a point on the left branch of the hyperbola then
and
:
:
Subtracting these equations one gets
:
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
,
:the ''vertices'' are
.
For an arbitrary point
the distance to the focus
is
and to the second focus
. Hence the point
is on the hyperbola if the following condition is fulfilled
:
Remove the square roots by suitable squarings and use the relation
to obtain the equation of the hyperbola:
:
This equation is called the
canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an ...
of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to the original (see
below).
The axes of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
or ''principal axes'' are the ''transverse axis'' (containing the segment of length 2''a'' with endpoints at the vertices) and the ''conjugate axis'' (containing the segment of length 2''b'' perpendicular to the transverse axis and with midpoint at the hyperbola's center). As opposed to an ellipse, a hyperbola has only two vertices:
. The two points
on the conjugate axes are ''not'' on the hyperbola.
It follows from the equation that the hyperbola is ''symmetric'' with respect to both of the coordinate axes and hence symmetric with respect to the origin.
Eccentricity
For a hyperbola in the above canonical form, the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
is given by
:
Two hyperbolas are
geometrically similar to each other – meaning that they have the same shape, so that one can be transformed into the other by
rigid left and right movements,
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
,
taking a mirror image, and scaling (magnification) – if and only if they have the same eccentricity.
Asymptotes
![Hyperbola-3prop](https://upload.wikimedia.org/wikipedia/commons/5/5c/Hyperbola-3prop.svg)
Solving the equation (above) of the hyperbola for
yields
:
It follows from this that the hyperbola approaches the two lines
:
for large values of
. These two lines intersect at the center (origin) and are called ''asymptotes'' of the hyperbola
With the help of the second figure one can see that
:
The ''perpendicular distance from a focus to either asymptote'' is
(the semi-minor axis).
From the
Hesse normal form
The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: ''Geometry for Computer Graphics''. ...
of the asymptotes and the equation of the hyperbola one gets:
[Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", ''Mathematical Gazette'' 96, July 2012, 299–301.]
:
The ''product of the distances from a point on the hyperbola to both the asymptotes'' is the constant
which can also be written in terms of the eccentricity ''e'' as
From the equation
of the hyperbola (above) one can derive:
:
The ''product of the slopes of lines from a point P to the two vertices'' is the constant
In addition, from (2) above it can be shown that
[
: ''The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes'' is the constant
]
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'' . A calculation shows
:
The semi-latus rectum may also be viewed as the ''radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
'' at the vertices.
Tangent
The simplest way to determine the equation of the tangent at a point is to implicitly differentiate the equation of the hyperbola. Denoting ''dy/dx'' as ''y′'', this produces
:
With respect to , the equation of the tangent at point is
:
A particular tangent line distinguishes the hyperbola from the other conic sections. Let ''f'' be the distance from the vertex ''V'' (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2''f''. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.
Rectangular hyperbola
In the case the hyperbola is called ''rectangular'' (or ''equilateral''), because its asymptotes intersect at right angles. For this case, the linear eccentricity is , the eccentricity and the semi-latus rectum . The graph of the equation is a rectangular hyperbola.
Parametric representation with hyperbolic sine/cosine
Using the hyperbolic sine and cosine functions , a parametric representation of the hyperbola can be obtained, which is similar to the parametric representation of an ellipse:
:
which satisfies the Cartesian equation because
Further parametric representations are given in the section Parametric equations
Parametric may refer to:
Mathematics
*Parametric equation, a representation of a curve through equations, as functions of a variable
*Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribut ...
below.
Conjugate hyperbola
Exchange and to obtain the equation of the conjugate hyperbola (see diagram):
: also written as
:
In polar coordinates
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 is called true anomaly.
Relative to this coordinate system one has that
:
and
:
for pole = center:
With polar coordinates relative to the "canonical coordinate system" (see second diagram)
one has that
:
For the right branch of the hyperbola the range of is
:
Parametric equations
A hyperbola with equation can be described by several parametric equations:
#
# (''rational'' representation).
#
# Tangent slope as parameter:
#: A parametric representation, which uses the slope of the tangent at a point of the hyperbola can be obtained analogously to the ellipse case: Replace in the ellipse case by and use formulae for the hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s. One gets
#:
#: is the upper, and the lower half of the hyperbola. The points with vertical tangents (vertices ) are not covered by the representation.
#: The equation of the tangent at point is
#:
#: This description of the tangents of a hyperbola is an essential tool for the determination of the orthoptic of a hyperbola.
Hyperbolic functions
Just as the trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s are defined in terms of the unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, so also the hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s are defined in terms of the unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radi ...
, as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector
A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the large ...
which that angle subtends. The analogous hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
is likewise defined as twice the area of a hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and ...
.
Let be twice the area between the axis and a ray through the origin intersecting the unit hyperbola, and define 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 :
:
which simplifies to the area hyperbolic cosine
:
Solving for yields the exponential form of the hyperbolic cosine:
:
From one gets
:
and its inverse the area hyperbolic sine:
:
Other hyperbolic functions are defined according to the hyperbolic cosine and hyperbolic sine, so for example
:
Properties
The tangent bisects the angle between the lines to the foci
The tangent at a point bisects the angle between the lines .
;Proof:
Let be the point on the line with the distance to the focus (see diagram, is the semi major axis of the hyperbola). Line is the bisector of the angle between the lines . In order to prove that is the tangent line at point , one checks that any point on line which is different from cannot be on the hyperbola. Hence has only point in common with the hyperbola and is, therefore, the tangent at point .
From the diagram and the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
one recognizes that holds, which means: . But if is a point of the hyperbola, the difference should be .
Midpoints of parallel chords
The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).
The points of any chord may lie on different branches of the hyperbola.
The proof of the property on midpoints is best done for the hyperbola . Because any hyperbola is an affine image of the hyperbola (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:
For two points of the hyperbola
:the midpoint of the chord is
:the slope of the chord is
For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line
Consequence: for any pair of points 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 and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line , where all point-image pairs are on a line perpendicular to .
Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint of a chord divides the related line segment between the asymptotes into halves, too. This means that . This property can be used for the construction of further points of the hyperbola if a point and the asymptotes are given.
If the chord degenerates into a ''tangent'', then the touching point divides the line segment between the asymptotes in two halves.
Orthogonal tangents – orthoptic
For a hyperbola the intersection points of ''orthogonal'' tangents lie on the circle .
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 there are no pairs of orthogonal tangents.
Pole-polar relation for a hyperbola
Any hyperbola can be described in a suitable coordinate system by an equation . The equation of the tangent at a point of the hyperbola is If one allows point to be an arbitrary point different from the origin, then
:point is mapped onto the line , not through the center of the hyperbola.
This relation between points and lines is a bijection.
The inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
maps
:line onto the point and
:line onto the point
Such a relation between points and lines generated by a conic is called pole-polar relation or just ''polarity''. The pole is the point, the polar the line. See Pole and polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into it ...
.
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: ).
* For a pole ''outside'' the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing (see diagram: ).
* For a point ''within'' the hyperbola the polar has no point with the hyperbola in common. (see diagram: ).
''Remarks:''
# The intersection point of two polars (for example: ) is the pole of the line through their poles (here: ).
# The foci and respectively and the directrices and respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.
Other properties
*The following are concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.
*The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.[
]
Arc length
The arc length of a hyperbola does not have an elementary expression. The upper half of a hyperbola can be parameterized as
:
Then the integral giving the arc length from to can be computed as:
:
After using the substitution , this can also be represented using the incomplete elliptic integral of the second kind with parameter :
:
Using only real numbers, this becomes
:
where is the incomplete elliptic integral of the first kind with parameter and is the Gudermannian function
In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...
.
Derived curves
Several other curves can be derived from the hyperbola by inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
, the so-called inverse curve
In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray ...
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
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...
or a strophoid, respectively.
Elliptic coordinates
A family of confocal hyperbolas is the basis of the system of elliptic coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
in two dimensions. These hyperbolas are described by the equation
:
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''2 transforms the Cartesian coordinate system into two families of orthogonal hyperbolas.
Conic section analysis of the hyperbolic appearance of circles
Besides providing a uniform description of circles, ellipses, parabolas, and hyperbolas, conic sections can also be understood as a natural model of the geometry of perspective in the case where the scene being viewed consists of circles, or more generally an ellipse. The viewer is typically a camera or the human eye and the image of the scene a central projection
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a proje ...
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.
Applications
Sundials
Hyperbolas may be seen in many sundial
A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
s. On any given day, the sun revolves in a circle on the celestial sphere
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...
, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the ''declination line''). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a ''pelekinon'' by the Greeks, since it resembles a double-bladed axe.
Multilateration
A hyperbola is the basis for solving multilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning).
When more than three distances are involved, it may be called multilateration, for e ...
problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points. Such problems are important in navigation, particularly on water; a ship can locate its position from the difference in arrival times of signals from a LORAN
LORAN, short for long range navigation, was a hyperbolic radio navigation system developed in the United States during World War II. It was similar to the UK's Gee system but operated at lower frequencies in order to provide an improved range u ...
or GPS
The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...
transmitters. Conversely, a homing beacon or any transmitter can be located by comparing the arrival times of its signals at two separate receiving stations; such techniques may be used to track objects and people. In particular, the set of possible positions of a point that has a distance difference of 2''a'' from two given points is a hyperbola of vertex separation 2''a'' whose foci are the two given points.
Path followed by a particle
The path followed by any particle in the classical Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
is a conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
. 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 Rutherford may refer to:
Places Australia
* Rutherford, New South Wales, a suburb of Maitland
* Rutherford (Parish), New South Wales, a civil parish of Yungnulgra County
Canada
* Mount Rutherford, Jasper National Park
* Rutherford, Edmonto ...
demonstrated the existence of an atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutron i ...
by examining the scattering of alpha particle
Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be pr ...
s from gold
Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile met ...
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 appears as one solution to the Korteweg–de Vries equation which describes the motion of a soliton wave in a canal.
Angle trisection
As shown first by Apollonius of Perga, 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 . Construct a hyperbola of eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
''e''=2 with 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 obtaining the point P' as the image of P. Segment AP' has the same length as segment BP due to the reflection, while segment PP' has the same length as segment BP due to the eccentricity of the hyperbola. As OA, OP', OP and OB are all radii of the same circle (and so, have the same length), the triangles OAP', OPP' and OPB are all congruent. Therefore, the angle has been trisected, since 3×POB = AOB.[This construction is due to ]Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
(circa 300 A.D.) and the proof comes from .
Efficient portfolio frontier
In portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificatio ...
, the locus of mean-variance efficient portfolios (called the efficient frontier) is the upper half of the east-opening branch of a hyperbola drawn with the portfolio return's standard deviation plotted horizontally and its expected value plotted vertically; according to this theory, all rational investors would choose a portfolio characterized by some point on this locus.
Biochemistry
In biochemistry
Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
and pharmacology
Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemica ...
, the Hill equation and Hill-Langmuir equation
The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressu ...
respectively describe biological responses and the formation of protein–ligand complex
A protein–ligand complex is a complex of a protein bound with a ligand that is formed following molecular recognition between proteins that interact with each other or with various other molecules. Formation of a protein-ligand complex is based ...
es as functions of ligand concentration. They are both rectangular hyperbolae.
Hyperbolas as plane sections of quadrics
Hyperbolas appear as plane sections of the following quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s:
* Elliptic cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
* Hyperbolic cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
* Hyperbolic paraboloid
* Hyperboloid of one sheet
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...
* Hyperboloid of two sheets
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
File:Quadric Cone.jpg, Elliptic cone
File:Hyperbolic Cylinder Quadric.png, Hyperbolic cylinder
File:Hyperbol Paraboloid.pov.png, Hyperbolic paraboloid
File:Hyperboloid1.png, Hyperboloid of one sheet
File:Hyperboloid2.png, Hyperboloid of two sheets
See also
Other conic sections
*Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
*Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
*Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
*Degenerate conic
In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible variety, irreducible curve. This means that the defining equation is factorable over the comp ...
Other related topics
*Elliptic coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...
, an orthogonal coordinate system based on families of ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s and hyperbolas.
*Hyperbolic growth
When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meani ...
*Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
*Hyperbolic sector
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and ...
*Hyperboloid structure
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the gro ...
*Hyperbolic trajectory
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the f ...
*Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
*Multilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning).
When more than three distances are involved, it may be called multilateration, for e ...
*Rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
*Translation of axes
In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
*Unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radi ...
Notes
References
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External links
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Apollonius' Derivation of the Hyperbola
a
Convergence
Frans van Schooten: ''Mathematische Oeffeningen'', 1659
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{{Authority control
Conic sections
Analytic geometry
Algebraic curves