
In
mathematics, an ellipse is a
plane curve surrounding two
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
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 ...
, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its
eccentricity , a number ranging from
(the
limiting case of a circle) to
(the limiting case of infinite elongation, no longer an ellipse but 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 descri ...
).
An ellipse has a simple
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
ic solution for its area, but only approximations for its
perimeter (also known as
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
), for which integration is required to obtain an exact solution.
Analytically, the equation of a standard ellipse centered at the origin with width
and height
is:
:
Assuming
, the foci are
for
. The standard parametric equation is:
:
Ellipses are the
closed type 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 ...
: a plane curve tracing the intersection of a cone with a
plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, c ...
s, both of which are
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (Y ...
and
unbounded. An angled
cross section of a
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 infi ...
is also an ellipse.
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the
directrix: for all points on the ellipse, the ratio between the distance to the
focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
:
Ellipses are common in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
. For example, the
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...
of each planet in the
Solar System
The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the
barycenter of the Sunplanet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
s. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under
parallel or
perspective projection. The ellipse is also the simplest
Lissajous figure
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
: x=A\sin(at+\delta),\quad y=B\sin(bt),
which describe the superposition of two perpendicular oscillations in x and y dire ...
formed when the horizontal and vertical motions are
sinusoids with the same frequency: a similar effect leads to
elliptical polarization of light in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
.
The name, (, "omission"), was given by
Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribut ...
in his ''Conics''.
Definition as locus of points

An ellipse can be defined geometrically as a set or
locus of points in the Euclidean plane:
: Given two fixed points
called the foci and a distance
which is greater than the distance between the foci, the ellipse is the set of points
such that the sum of the distances
is equal to
:
The midpoint
of the line segment joining the foci is called the ''center'' of the ellipse. The line through the foci is called the ''major axis'', and the line perpendicular to it through the center is the ''minor axis''. The major axis intersects the ellipse at two ''
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 case
yields a circle and is included as a special type of ellipse.
The equation
can be viewed in a different way (see figure):
: If
is the circle with center
and radius
, then the distance of a point
to the circle
equals the distance to the focus
:
::
is called the ''circular directrix'' (related to focus
) of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line below.
Using
Dandelin spheres In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plan ...
, one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.
In Cartesian coordinates
Standard equation
The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the ''x''-axis is the major axis, and:
: the foci are the points
,
: the vertices are
.
For an arbitrary point
the distance to the focus
is
and to the other focus
. Hence the point
is on the ellipse whenever:
:
Removing the
radicals
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
by suitable squarings and using
(see diagram) produces the standard equation of the ellipse:
:
or, solved for ''y:''
:
The width and height parameters
are called the
semi-major and semi-minor axes. The top and bottom points
are the ''co-vertices''. The distances from a point
on the ellipse to the left and right foci are
and
.
It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.
Parameters
Principal axes
Throughout this article, the
semi-major and semi-minor axes are denoted
and
, respectively, i.e.
In principle, the canonical ellipse equation
may have
(and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names
and
and the parameter names
and
Linear eccentricity
This is the distance from the center to a focus:
.
Eccentricity
The eccentricity can be expressed as:
:
assuming
An ellipse with equal axes (
) has zero eccentricity, and is a circle.
Semi-latus rectum
The length of the chord through one focus, perpendicular to the major axis, is called the ''latus rectum''. One half of it is the ''semi-latus rectum''
. A calculation shows:
:
The semi-latus rectum
is equal to the
radius of curvature at the vertices (see section
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
).
Tangent
An arbitrary line
intersects an ellipse at 0, 1, or 2 points, respectively called an ''exterior line'', ''tangent'' and ''secant''. Through any point of an ellipse there is a unique tangent. The tangent at a point
of the ellipse
has the coordinate equation:
:
A vector
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ...
of the tangent is:
:
with
Proof:
Let
be a point on an ellipse and
be the equation of any line
containing
. Inserting the line's equation into the ellipse equation and respecting
yields:
:
There are then cases:
#
Then line
and the ellipse have only point
in common, and
is a tangent. The tangent direction has
perpendicular vector , so the tangent line has equation
for some
. Because
is on the tangent and the ellipse, one obtains
.
#
Then line
has a second point in common with the ellipse, and is a secant.
Using (1) one finds that
is a tangent vector at point
, which proves the vector equation.
If
and
are two points of the ellipse such that
, then the points lie on two ''conjugate diameters'' (see
below
Below may refer to:
*Earth
* Ground (disambiguation)
* Soil
* Floor
* Bottom (disambiguation)
* Less than
*Temperatures below freezing
* Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fr ...
). (If
, the ellipse is a circle and "conjugate" means "orthogonal".)
Shifted ellipse
If the standard ellipse is shifted to have center
, its equation is
:
The axes are still parallel to the x- and y-axes.
General ellipse
In
analytic geometry, the ellipse is defined as a
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 ...
: the set of points
of the
Cartesian plane that, in non-degenerate cases, satisfy the
implicit equation
:
provided
To distinguish the
degenerate cases
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...
from the non-degenerate case, let ''∆'' be the
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 ...
:
Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.
[Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.]
The general equation's coefficients can be obtained from known semi-major axis
, semi-minor axis
, center coordinates
, and rotation angle
(the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
:
These expressions can be derived from the canonical equation
by an affine transformation of the coordinates
:
:
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:
:
Parametric representation
Standard parametric representation
Using
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 ...
s, a parametric representation of the standard ellipse
is:
:
The parameter ''t'' (called the ''
eccentric anomaly'' in astronomy) is not the angle of
with the ''x''-axis, but has a geometric meaning due to
Philippe de La Hire (see ''
Drawing ellipses'' below).
Rational representation
With the substitution
and trigonometric formulae one obtains
:
and the ''rational'' parametric equation of an ellipse
:
which covers any point of the ellipse
except the left vertex
.
For
this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing
The left vertex is the limit
Alternately, if the parameter
Tangent slope as parameter
A parametric representation, which uses the slope
m of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation
\vec x(t) = (a \cos t,\, b \sin t)^\mathsf:
:
\vec x'(t) = (-a\sin t,\, b\cos t)^\mathsf \quad \rightarrow \quad m = -\frac\cot t\quad \rightarrow \quad \cot t = -\frac.
With help of
trigonometric formulae one obtains:
:
\cos t = \frac = \frac\ ,\quad\quad
\sin t = \frac = \frac.
Replacing
\cos t and
\sin t of the standard representation yields:
:
\vec c_\pm(m) = \left(-\frac,\;\frac\right),\, m \in \R.
Here
m is the slope of the tangent at the corresponding ellipse point,
\vec c_+ is the upper and
\vec c_- the lower half of the ellipse. The vertices
(\pm a,\, 0), having vertical tangents, are not covered by the representation.
The equation of the tangent at point
\vec c_\pm(m) has the form
y = mx + n. The still unknown
n can be determined by inserting the coordinates of the corresponding ellipse point
\vec c_\pm(m):
:
y = mx \pm\sqrt\; .
This description of the tangents of an ellipse is an essential tool for the determination of the
orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.
General ellipse

Another definition of an ellipse uses
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
s:
: Any ''ellipse'' is an affine image of the unit circle with equation
x^2 + y^2 = 1.
;Parametric representation
An affine transformation of the Euclidean plane has the form
\vec x \mapsto \vec f\!_0 + A\vec x, where
A 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'' (franchi ...
(with non-zero
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
\vec f\!_0 is an arbitrary vector. If
\vec f\!_1, \vec f\!_2 are the column vectors of the matrix
A, the unit circle
(\cos(t), \sin(t)),
0 \leq t \leq 2\pi, is mapped onto the ellipse:
:
\vec x = \vec p(t) = \vec f\!_0 + \vec f\!_1 \cos t + \vec f\!_2 \sin t \ .
Here
\vec f\!_0 is the center and
\vec f\!_1,\; \vec f\!_2 are the directions of two
conjugate diameters, in general not perpendicular.
;Vertices
The four vertices of the ellipse are
\vec p(t_0),\;\vec p\left(t_0 \pm \tfrac\right),\; \vec p\left(t_0 + \pi\right), for a parameter
t = t_0 defined by:
:
\cot (2t_0) = \frac.
(If
\vec f\!_1 \cdot \vec f\!_2 = 0, then
t_0 = 0.) This is derived as follows. The tangent vector at point
\vec p(t) is:
:
\vec p\,'(t) = -\vec f\!_1\sin t + \vec f\!_2\cos t \ .
At a vertex parameter
t = t_0, the tangent is perpendicular to the major/minor axes, so:
:
0 = \vec p'(t) \cdot \left(\vec p(t) -\vec f\!_0\right) = \left(-\vec f\!_1\sin t + \vec f\!_2\cos t\right) \cdot \left(\vec f\!_1 \cos t + \vec f\!_2 \sin t\right).
Expanding and applying the identities
\; \cos^2 t -\sin^2 t=\cos 2t,\ \ 2\sin t \cos t = \sin 2t\; gives the equation for
t = t_0\; .
;Area
From Apollonios theorem (see below) one obtains:
The area of an ellipse
\;\vec x = \vec f_0 +\vec f_1 \cos t +\vec f_2 \sin t\; is
:
A=\pi, \det(\vec f_1, \vec f_2), \ .
;Semiaxes
With the abbreviations
\; M=\vec f_1^2+\vec f_2^2, \ N = \left, \det(\vec f_1,\vec f_2)\ the statements of Apollonios's theorem can be written as:
:
a^2+b^2=M, \quad ab=N \ .
Solving this nonlinear system for
a,b yields the semiaxes:
:
a=\frac(\sqrt+\sqrt)
:
b=\frac(\sqrt-\sqrt)\ .
;Implicit representation
Solving the parametric representation for
\; \cos t,\sin t\; by
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants ...
and using
\;\cos^2t+\sin^2t -1=0\; , one obtains the implicit representation
:
\det(\vec x\!-\!\vec f\!_0,\vec f\!_2)^2+\det(\vec f\!_1,\vec x\!-\!\vec f\!_0)^2-\det(\vec f\!_1,\vec f\!_2)^2=0.
Conversely: If the
equation
:
x^2+2cxy+d^2y^2-e^2=0\ , with
\; d^2-c^2 >0 \; ,
of an ellipse centered at the origin is given, then the two vectors
:
\vec f_1=,\quad \vec f_2=\frac\
point to two conjugate points and the tools developed above are applicable.
''Example'': For the ellipse with equation
\;x^2+2xy+3y^2-1=0\; the vectors are
:
\vec f_1=,\quad \vec f_2=\frac .

;Rotated Standard ellipse
For
\vec f_0= ,\;\vec f_1= a ,\;\vec f_2= b one obtains a parametric representation of the standard ellipse
rotated by angle
\theta:
:
x=x_\theta(t)=a\cos\theta\cos t-b\sin\theta\sin t\ ,
:
y=y_\theta(t)=a\sin\theta\cos t+b\cos\theta\sin t\ .
;Ellipse in space
The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows
\vec f\!_0, \vec f\!_1, \vec f\!_2 to be vectors in space.
Polar forms
Polar form relative to center

In
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...
, with the origin at the center of the ellipse and with the angular coordinate
\theta measured from the major axis, the ellipse's equation is
[
: r(\theta) = \frac=\frac
where e is the eccentricity, not Euler's number
]
Polar form relative to focus
If instead we use polar coordinates with the origin at one focus, with the angular coordinate \theta = 0 still measured from the major axis, the ellipse's equation is
: r(\theta)=\frac
where the sign in the denominator is negative if the reference direction \theta = 0 points towards the center (as illustrated on the right), and positive if that direction points away from the center.
In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate \phi, the polar form is
:r(\theta)=\frac.
The angle \theta in these formulas is called the true anomaly of the point. The numerator of these formulas is the semi-latus rectum \ell=a (1-e^2).
Eccentricity and the directrix property
Each of the two lines parallel to the minor axis, and at a distance of d = \frac = \frac from it, is called a ''directrix'' of the ellipse (see diagram).
: For an arbitrary point P of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
:: \frac = \frac = e = \frac\ .
The proof for the pair F_1, l_1 follows from the fact that \left, PF_1\^2 = (x - c)^2 + y^2,\ \left, Pl_1\^2 = \left(x - \tfrac\right)^2 and y^2 = b^2 - \tfracx^2 satisfy the equation
:\left, PF_1\^2 - \frac\left, Pl_1\^2 = 0\ .
The second case is proven analogously.
The converse is also true and can be used to define an ellipse (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 0 < e < 1, the ellipse is the locus of points for which the quotient of the distances to the point and to the line is e, that is:
:: E = \left\.
The extension to e = 0, which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the line at infinity in the projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
.
(The choice e = 1 yields a parabola, and if e > 1, a hyperbola.)
;Proof
Let F = (f,\, 0),\ e > 0, and assume (0,\, 0) is a point on the curve.
The directrix l has equation x = -\tfrac. With P = (x,\, y), the relation , PF, ^2 = e^2, Pl, ^2 produces the equations
:(x - f)^2 + y^2 = e^2\left(x + \frac\right)^2 = (ex + f)^2 and x^2\left(e^2 - 1\right) + 2xf(1 + e) - y^2 = 0.
The substitution p = f(1 + e) yields
: x^2\left(e^2 - 1\right) + 2px - y^2 = 0.
This is the equation of an ''ellipse'' (e < 1), or a ''parabola'' (e = 1), or a ''hyperbola'' (e > 1). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If e < 1, introduce new parameters a,\, b so that 1 - e^2 = \tfrac, \text\ p = \tfrac, and then the equation above becomes
:\frac + \frac = 1\ ,
which is the equation of an ellipse with center (a,\, 0), the ''x''-axis as major axis, and
the major/minor semi axis a,\, b.
;Construction of a directrix
Because of c\cdot\tfrac=a^2 point L_1 of directrix l_1 (see diagram) and focus F_1 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 cons ...
at circle x^2+y^2=a^2 (in diagram green). Hence L_1 can be constructed as shown in the diagram. Directrix l_1 is the perpendicular to the main axis at point L_1.
;General ellipse
If the focus is F = \left(f_1,\, f_2\right) and the directrix ux + vy + w = 0, one obtains the equation
:\left(x - f_1\right)^2 + \left(y - f_2\right)^2 = e^2 \frac\ .
(The right side of the equation uses the Hesse normal form of a line to calculate the distance , Pl, .)
Focus-to-focus reflection property
An ellipse possesses the following property:
: The normal at a point P bisects the angle between the lines \overline,\, \overline.
; Proof
Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too.
Let L be the point on the line \overline with the distance 2a to the focus F_2, a is the semi-major axis of the ellipse. Let line w be the bisector of the supplementary angle to the angle between the lines \overline,\, \overline. 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 ellipse. Hence w has only point P in common with the ellipse and is, therefore, the tangent at point P.
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, bu ...
one recognizes that 2a = \left, LF_2\ < \left, QF_2\ + \left, QL\ = \left, QF_2\ + \left, QF_1\ holds, which means: \left, QF_2\ + \left, QF_1\ > 2a . The equality \left, QL\ = \left, QF_1\ is true from the Angle bisector theorem because \frac=\frac and \left, PL\=\left, PF_1\ . But if Q is a point of the ellipse, the sum should be 2a.
; Application
The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).
Conjugate diameters
Definition of conjugate diameters
A circle has the following property:
: The midpoints of parallel chords lie on a diameter.
An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)
; Definition:
Two diameters d_1,\, d_2 of an ellipse are ''conjugate'' if the midpoints of chords parallel to d_1 lie on d_2\ .
From the diagram one finds:
: Two diameters \overline,\, \overline of an ellipse are conjugate whenever the tangents at P_1 and Q_1 are parallel to \overline.
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
In the parametric equation for a general ellipse given above,
: \vec x = \vec p(t) = \vec f\!_0 +\vec f\!_1 \cos t + \vec f\!_2 \sin t,
any pair of points \vec p(t),\ \vec p(t + \pi) belong to a diameter, and the pair \vec p\left(t + \tfrac\right),\ \vec p\left(t - \tfrac\right) belong to its conjugate diameter.
For the common parametric representation (a\cos t,b\sin t) of the ellipse with equation \tfrac+\tfrac=1 one gets: The points
:(x_1,y_1)=(\pm a\cos t,\pm b\sin t)\quad (signs: (+,+) or (-,-) )
:(x_2,y_2)=( a\sin t,\pm b\cos t)\quad (signs: (-,+) or (+,-) )
:are conjugate and
:\frac+\frac=0\ .
In case of a circle the last equation collapses to x_1x_2+y_1y_2=0\ .
Theorem of Apollonios on conjugate diameters
For an ellipse with semi-axes a,\, b the following is true:
: Let c_1 and c_2 be halves of two conjugate diameters (see diagram) then
:# c_1^2 + c_2^2 = a^2 + b^2.
:# The ''triangle'' O,P_1,P_2 with sides c_1,\, c_2 (see diagram) has the constant area A_\Delta = \fracab, which can be expressed by A_\Delta=\tfrac 1 2 c_2d_1=\tfrac 1 2 c_1c_2\sin\alpha, too. d_1 is the altitude of point P_1 and \alpha the angle between the half diameters. Hence the area of the ellipse (see section metric properties) can be written as A_=\pi ab=\pi c_2d_1=\pi c_1c_2\sin\alpha.
:# The parallelogram of tangents adjacent to the given conjugate diameters has the \text_ = 4ab\ .
; Proof:
Let the ellipse be in the canonical form with parametric equation
: \vec p(t) = (a\cos t,\, b\sin t).
The two points \vec c_1 = \vec p(t),\ \vec c_2 = \vec p\left(t + \frac\right) are on conjugate diameters (see previous section). From trigonometric formulae one obtains \vec c_2 = (-a\sin t,\, b\cos t)^\mathsf and
: \left, \vec c_1\^2 + \left, \vec c_2\^2 = \cdots = a^2 + b^2\ .
The area of the triangle generated by \vec c_1,\, \vec c_2 is
: A_\Delta = \frac\det\left(\vec c_1,\, \vec c_2\right) = \cdots = \fracab
and from the diagram it can be seen that the area of the parallelogram is 8 times that of A_\Delta. Hence
: \text_ = 4ab\ .
Orthogonal tangents
For the ellipse \tfrac+\tfrac=1 the intersection points of ''orthogonal'' tangents lie on the circle x^2+y^2=a^2+b^2.
This circle is called ''orthoptic'' or director circle of the ellipse (not to be confused with the circular directrix defined above).
Drawing ellipses
Ellipses appear in descriptive geometry as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools ('' ellipsographs'') to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...
and Proklos.
If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.
For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis).
If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.
de La Hire's point construction
The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation (a\cos t,\, b\sin t) of an ellipse:
# Draw the two ''circles'' centered at the center of the ellipse with radii a,b and the axes of the ellipse.
# Draw a ''line through the center'', which intersects the two circles at point A and B, respectively.
# Draw a ''line'' through A that is parallel to the minor axis and a ''line'' through B that is parallel to the major axis. These lines meet at an ellipse point (see diagram).
# Repeat steps (2) and (3) with different lines through the center.
Elliko-sk.svg, de La Hire's method
Parametric ellipse.gif, Animation of the method
Pins-and-string method
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string is tied at each end to the two pins; its length after tying is 2a. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the ''gardener's ellipse''.
A similar method for drawing confocal ellipses with a ''closed'' string is due to the Irish bishop Charles Graves.
Paper strip methods
The two following methods rely on the parametric representation (see section '' parametric representation'', above):
: (a\cos t,\, b\sin t)
This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes a,\, b have to be known.
;Method 1
The first method starts with
: a strip of paper of length a + b.
The point, where the semi axes meet is marked by P. If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point P has the parametric representation (a\cos t,\, b\sin t), where parameter t is the angle of the slope of the paper strip.
A technical realization of the motion of the paper strip can be achieved by a Tusi couple
The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and for ...
(see animation). The device is able to draw any ellipse with a ''fixed'' sum a + b, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.
Elliko-pap1.svg, Ellipse construction: paper strip method 1
Tusi couple vs Paper strip plus Ellipses horizontal.gif, Ellipses with Tusi couple. Two examples: red and cyan.
A variation of the paper strip method 1 uses the observation that the midpoint N of the paper strip is moving on the circle with center M (of the ellipse) and radius \tfrac. Hence, the paperstrip can be cut at point N into halves, connected again by a joint at N and the sliding end K fixed at the center M (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe.
Ellipse-papsm-1a.svg, Variation of the paper strip method 1
Ellipses with SliderCrank inner Ellipses.gif, Animation of the variation of the paper strip method 1
; Method 2:
The second method starts with
: a strip of paper of length a.
One marks the point, which divides the strip into two substrips of length b and a - b. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by (a\cos t,\, b\sin t), where parameter t is the angle of slope of the paper strip.
This method is the base for several ''ellipsographs'' (see section below).
Similar to the variation of the paper strip method 1 a ''variation of the paper strip method 2'' can be established (see diagram) by cutting the part between the axes into halves.
File:Archimedes Trammel.gif, Trammel of Archimedes (principle)
File:L-Ellipsenzirkel.png, Ellipsograph due to Benjamin Bramer
Benjamin Bramer (15 February 1588 – 17 March 1652) was a German mathematician, architect, inventor, and adviser.
Early life
Bramer was born on 15 February 1588 in Felsberg, Germany to a Protestant minister father. The minister later died wh ...
File:Ellipses with SliderCrank Ellipses at Slider Side.gif, Variation of the paper strip method 2
Most ellipsograph drafting
Drafting or draughting may refer to:
* Campdrafting, an Australian equestrian sport
* Drafting (aerodynamics), slipstreaming
* Drafting (writing), writing something that is likely to be amended
* Technical drawing, the act and discipline of compo ...
instruments are based on the second paperstrip method.
Approximation by osculating circles
From ''Metric properties'' below, one obtains:
* The radius of curvature at the vertices V_1,\, V_2 is: \tfrac
* The radius of curvature at the co-vertices V_3,\, V_4 is: \tfrac\ .
The diagram shows an easy way to find the centers of curvature C_1 = \left(a - \tfrac, 0\right),\, C_3 = \left(0, b - \tfrac\right) at vertex V_1 and co-vertex V_3, respectively:
# mark the auxiliary point H = (a,\, b) and draw the line segment V_1 V_3\ ,
# draw the line through H, which is perpendicular to the line V_1 V_3\ ,
# the intersection points of this line with the axes are the centers of the osculating circles.
(proof: simple calculation.)
The centers for the remaining vertices are found by symmetry.
With help of a French curve one draws a curve, which has smooth contact to the osculating circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve ...
s.
Steiner generation
The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:
: Given two pencils B(U),\, B(V) of lines at two points U,\, V (all lines containing U and V, respectively) and a projective but not perspective mapping \pi of B(U) onto B(V), then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the ellipse \tfrac + \tfrac = 1 one uses the pencils at the vertices V_1,\, V_2. Let P = (0,\, b) be an upper co-vertex of the ellipse and A = (-a,\, 2b),\, B = (a,\,2b).
P is the center of the rectangle V_1,\, V_2,\, B,\, A. The side \overline of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal AV_2 as direction onto the line segment \overline and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at V_1 and V_2 needed. The intersection points of any two related lines V_1 B_i and V_2 A_i are points of the uniquely defined ellipse. With help of the points C_1,\, \dotsc the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.
Steiner generation can also be defined for hyperbolas and parabolas. It 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.
As hypotrochoid
The ellipse is a special case of the hypotrochoid when R = 2r, as shown in the adjacent image. The special case of a moving circle with radius r inside a circle with radius R = 2r is called a Tusi couple
The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and for ...
.
Inscribed angles and three-point form
Circles
A circle with equation \left(x - x_\circ\right)^2 + \left(y - y_\circ\right)^2 = r^2 is uniquely determined by three points \left(x_1, y_1\right),\; \left(x_2,\,y_2\right),\; \left(x_3,\, y_3\right) not on a line. A simple way to determine the parameters x_\circ,y_\circ,r uses the '' inscribed angle theorem'' for circles:
: For four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\, (see diagram) the following statement is true:
: The four points are on a circle if and only if the angles at P_3 and P_4 are equal.
Usually one measures inscribed angles by a degree or radian ''θ,'' but here the following measurement is more convenient:
: In order to measure the angle between two lines with equations y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2, one uses the quotient:
:: \frac = \cot\theta\ .
Inscribed angle theorem for circles
For four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\, no three of them on a line, we have the following (see diagram):
: The four points are on a circle, if and only if the angles at P_3 and P_4 are equal. In terms of the angle measurement above, this means:
:: \frac
=
\frac
.
At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.
Three-point form of circle equation
: As a consequence, one obtains an equation for the circle determined by three non-colinear points P_i = \left(x_i,\, y_i\right):
::
\frac
=
\frac
.
For example, for P_1 = (2,\, 0),\; P_2 = (0,\, 1),\; P_3 = (0,\,0) the three-point equation is:
: \frac = 0, which can be rearranged to (x - 1)^2 + \left(y - \tfrac\right)^2 = \tfrac\ .
Using vectors, dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
s and 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 ...
s this formula can be arranged more clearly, letting \vec x = (x,\, y):
:
\frac
=
\frac
.
The center of the circle \left(x_\circ,\, y_\circ\right) satisfies:
: \begin
1 & \frac \\
\frac & 1
\end
\begin x_\circ \\ y_\circ \end
=
\begin
\frac \\
\frac
\end.
The radius is the distance between any of the three points and the center.
:
r = \sqrt
= \sqrt
= \sqrt.
Ellipses
This section, we consider the family of ellipses defined by equations \tfrac + \tfrac = 1 with a ''fixed'' eccentricity e. It is convenient to use the parameter:
: = \frac = \frac,
and to write the ellipse equation as:
: \left(x - x_\circ\right)^2 + \, \left(y - y_\circ\right)^2 = a^2,
where ''q'' is fixed and x_\circ,\, y_\circ,\, a vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if q < 1, the major axis is parallel to the ''x''-axis; if q > 1, it is parallel to the ''y''-axis.)
Like a circle, such an ellipse is determined by three points not on a line.
For this family of ellipses, one introduces the following q-analog angle measure, which is ''not'' a function of the usual angle measure ''θ'':
: In order to measure an angle between two lines with equations y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2 one uses the quotient:
:: \frac\ .
Inscribed angle theorem for ellipses
: Given four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4, no three of them on a line (see diagram).
: The four points are on an ellipse with equation (x - x_\circ)^2 + \, (y - y_\circ)^2 = a^2 if and only if the angles at P_3 and P_4 are equal in the sense of the measurement above—that is, if
:: \frac
=
\frac
\ .
At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.
Three-point form of ellipse equation
: A consequence, one obtains an equation for the ellipse determined by three non-colinear points P_i = \left(x_i,\, y_i\right):
:: \frac
=
\frac
\ .
For example, for P_1 = (2,\, 0),\; P_2 = (0,\,1),\; P_3 = (0,\, 0) and q = 4 one obtains the three-point form
: \frac = 0 and after conversion \frac + \frac = 1.
Analogously to the circle case, the equation can be written more clearly using vectors:
:
\frac
= \frac
,
where * is the modified dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
\vec u*\vec v = u_x v_x + \,u_y v_y.
Pole-polar relation
Any ellipse can be described in a suitable coordinate system by an equation \tfrac + \tfrac = 1. The equation of the tangent at a point P_1 = \left(x_1,\, y_1\right) of the ellipse is \tfrac + \tfrac = 1. If one allows point P_1 = \left(x_1,\, y_1\right) to be an arbitrary point different from the origin, then
: point P_1 = \left(x_1,\, y_1\right) \neq (0,\, 0) is mapped onto the line \tfrac + \tfrac = 1, not through the center of the ellipse.
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 ...
maps
* line y = mx + d,\ d \ne 0 onto the point \left(-\tfrac,\, \tfrac\right) and
* line x = c,\ c \ne 0 onto the point \left(\tfrac,\, 0\right).
Such a relation between points and lines generated by a conic is called '' pole-polar relation'' or ''polarity''. The pole is the point; the polar the line.
By calculation one can confirm the following properties of the pole-polar relation of the ellipse:
* For a point (pole) ''on'' the ellipse, the polar is the tangent at this point (see diagram: P_1,\, p_1).
* For a pole P ''outside'' the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing P (see diagram: P_2,\, p_2).
* For a point ''within'' the ellipse, the polar has no point with the ellipse in common (see diagram: F_1,\, l_1).
# The intersection point of two polars is the pole of the line through their poles.
# The foci (c,\, 0) and (-c,\, 0), respectively, and the directrices x = \tfrac and x = -\tfrac, respectively, belong to pairs of pole and polar. Because they are even polar pairs with respect to the circle x^2+y^2=a^2, the directrices can be constructed by compass and straightedge (see Inversive geometry).
Pole-polar relations exist for hyperbolas and parabolas as well.
Metric properties
All metric properties given below refer to an ellipse with equation
except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.() will be given.
Area
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 or planar lamina, while ''surface area'' refers to the area of an open su ...
A_\text enclosed by an ellipse is:
where a and b are the lengths of the semi-major and semi-minor axes, respectively. The area formula \pi a b is intuitive: start with a circle of radius b (so its area is \pi b^2) and stretch it by a factor a/b to make an ellipse. This scales the area by the same factor: \pi b^2(a/b) = \pi a b. However, using the same approach for the circumference would be fallacious – compare the integrals \int f(x)\, dx and \int \sqrt\, dx. It is also easy to rigorously prove the area formula using integration as follows. Equation () can be rewritten as y(x)= b \sqrt. For x\in a,a this curve is the top half of the ellipse. So twice the integral of y(x) over the interval a,a/math> will be the area of the ellipse:
: \begin
A_\text &= \int_^a 2b\sqrt\,dx\\
&= \frac ba \int_^a 2\sqrt\,dx.
\end
The second integral is the area of a circle of radius a, that is, \pi a^2. So
: A_\text = \frac\pi a^2 = \pi ab.
An ellipse defined implicitly by Ax^2+ Bxy + Cy^2 = 1 has area 2\pi / \sqrt.
The area can also be expressed in terms of eccentricity and the length of the semi-major axis as a^2\pi\sqrt (obtained by solving for flattening
Flattening is a measure of the compression of a circle or sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set o ...
, then computing the semi-minor axis).
So far we have dealt with ''erect'' ellipses, whose major and minor axes are parallel to the x and y axes. However, some applications require ''tilted'' ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its ''emittance''. In this case a simple formula still applies, namely
where y_, x_ are intercepts and x_, y_ are maximum values. It follows directly from Apollonios's theorem.
Circumference
The circumference C of an ellipse is:
: C \,=\, 4a\int_0^\sqrt \ d\theta \,=\, 4 a \,E(e)
where again a is the length of the semi-major axis, e=\sqrt is the eccentricity, and the function E is the complete elliptic integral of the second kind,
: E(e) \,=\, \int_0^\sqrt \ d\theta
which is in general not an elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, a ...
.
The circumference of the ellipse may be evaluated in terms of E(e) using Gauss's arithmetic-geometric mean; this is a quadratically converging iterative method (see here for details).
The exact infinite series is:
:\begin
C &= 2\pi a \leftright
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical th ...
\\
&= 2\pi a \left - \sum_^\infty \left(\frac\right)^2 \frac\right\\
&= -2\pi a \sum_^\infty \left(\frac\right)^2 \frac,
\end
where n!! is the double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
(extended to negative odd integers by the recurrence relation (2n-1)!! = (2n+1)!!/(2n+1), for n \le 0). This series converges, but by expanding in terms of h = (a-b)^2 / (a+b)^2, James Ivory
James Francis Ivory (born June 7, 1928) is an American film director, producer, and screenwriter. For many years, he worked extensively with Indian-born film producer Ismail Merchant, his domestic as well as professional partner, and with screen ...
and Bessel derived an expression that converges much more rapidly:
:\begin
C &= \pi (a+b) \sum_^\infty \left(\frac\right)^2 h^n \\
&= \pi (a+b) \left + \frac + \sum_^\infty \left(\frac\right)^2 h^n\right\\
&= \pi (a+b) \left + \sum_^\infty \left(\frac\right)^2 \frac\right
\end
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
gave two close approximations for the circumference in §16 of "Modular Equations and Approximations to \pi"; they are
: C \approx \pi \biggl (a + b) - \sqrt \biggr= \pi \biggl (a + b) - \sqrt\biggr/math>
and
: C\approx\pi\left(a+b\right)\left(1+\frac\right),
where h takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order h^3 and h^5, respectively.
Arc length
More generally, the arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
...
of a portion of the circumference, as a function of the angle subtended (or -coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by
: y=b\sqrt.
Then the arc length s from x_ to x_ is:
: s = -b\int_^ \sqrt \, dz.
This is equivalent to
: s = b\left \; 1 - \frac\right)\right_
where E(z \mid m) is the incomplete elliptic integral of the second kind with parameter m=k^.
Some lower and upper bounds on the circumference of the canonical ellipse x^2/a^2 + y^2/b^2 = 1 with a\geq b are
: \begin
2\pi b &\le C \le 2\pi a, \\
\pi (a+b) &\le C \le 4(a+b), \\
4\sqrt &\le C \le \sqrt \pi \sqrt .
\end
Here the upper bound 2\pi a is the circumference of a circumscribed
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every po ...
concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound 4\sqrt is the perimeter of an inscribed rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Th ...
with vertices at the endpoints of the major and the minor axes.
Curvature
The curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the can ...
is given by \kappa = \frac\left(\frac+\frac\right)^\ ,
radius of curvature at point (x,y):
: \rho = a^2 b^2 \left(\frac + \frac\right)^\frac = \frac \sqrt \ .
Radius of curvature at the two ''vertices'' (\pm a,0) and the centers of curvature:
: \rho_0 = \frac=p\ , \qquad \left(\pm\frac\,\bigg, \,0\right)\ .
Radius of curvature at the two ''co-vertices'' (0,\pm b) and the centers of curvature:
: \rho_1 = \frac\ , \qquad \left(0\,\bigg, \,\pm\frac\right)\ .
In triangle geometry
Ellipses appear in triangle geometry as
# Steiner ellipse
In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (ellipse that touches the triangle at its vertices) whose center is the triangle's ce ...
: ellipse through the vertices of the triangle with center at the centroid,
# inellipse
In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides ...
s: ellipses which touch the sides of a triangle. Special cases are the Steiner inellipse
In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse ins ...
and the Mandart inellipse.
As plane sections of quadrics
Ellipses 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 ...
s:
* Ellipsoid
* Elliptic cone
* Elliptic 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 infi ...
* Hyperboloid of one sheet
* Hyperboloid of two sheets
Ellipsoid Quadric.png, Ellipsoid
Quadric Cone.jpg, Elliptic cone
Elliptic Cylinder Quadric.png, Elliptic cylinder
Hyperboloid1.png, Hyperboloid of one sheet
Hyperboloid2.png, Hyperboloid of two sheets
Applications
Physics
Elliptical reflectors and acoustics
If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflecting off the walls, converge simultaneously to a single point: the ''second focus''. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.
Similarly, if a light source is placed at one focus of an elliptic mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
), this property holds for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp
A fluorescent lamp, or fluorescent tube, is a low-pressure mercury-vapor gas-discharge lamp that uses fluorescence to produce visible light. An electric current in the gas excites mercury vapor, which produces short-wave ultraviolet li ...
along a line of the paper; such mirrors are used in some document scanners.
Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a ''whisper chamber
The Whispering Gallery of St Paul's Cathedral, London
A whispering gallery is usually a circular, hemispherical, elliptical or ellipsoidal enclosure, often beneath a dome or a vault, in which whispers can be heard clearly in other parts of ...
''. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol
The United States Capitol, often called The Capitol or the Capitol Building, is the Seat of government, seat of the Legislature, legislative branch of the Federal government of the United States, United States federal government, which is form ...
(where John Quincy Adams
John Quincy Adams (; July 11, 1767 – February 23, 1848) was an American statesman, diplomat, lawyer, and diarist who served as the sixth president of the United States, from 1825 to 1829. He previously served as the eighth United States S ...
is said to have used this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt Lake City
Salt Lake City (often shortened to Salt Lake and abbreviated as SLC) is the capital and most populous city of Utah, United States. It is the seat of Salt Lake County, the most populous county in Utah. With a population of 200,133 in 2020, th ...
, Utah
Utah ( , ) is a state in the Mountain West subregion of the Western United States. Utah is a landlocked U.S. state bordered to its east by Colorado, to its northeast by Wyoming, to its north by Idaho, to its south by Arizona, and to its ...
; at an exhibit on sound at the Museum of Science and Industry in Chicago
(''City in a Garden''); I Will
, image_map =
, map_caption = Interactive Map of Chicago
, coordinates =
, coordinates_footnotes =
, subdivision_type = List of sovereign states, Count ...
; in front of the University of Illinois at Urbana–Champaign
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Unive ...
Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.
Planetary orbits
In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun pproximatelyat one focus, in his first law of planetary motion. Later, Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
explained this as a corollary of his law of universal gravitation.
More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.
Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible ...
and quantum effects
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 become significant when the particles are moving at high speed.)
For elliptical orbits, useful relations involving the eccentricity e are:
: \begin
e &= \frac = \frac \\
r_a &= (1 + e)a \\
r_p &= (1 - e)a
\end
where
* r_a is the radius at apoapsis (the farthest distance)
* r_p is the radius at periapsis (the closest distance)
* a is the length of the semi-major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
Also, in terms of r_a and r_p, the semi-major axis a is their arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
, the semi-minor axis b is their geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
, and the semi-latus rectum \ell is their harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipr ...
. In other words,
:\begin
a &= \frac \\ pt b &= \sqrt \\ pt \ell &= \frac = \frac
\end.
Harmonic oscillators
The general solution for a harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
in two or more dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.
Phase visualization
In electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the Lissajous figure
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
: x=A\sin(at+\delta),\quad y=B\sin(bt),
which describe the superposition of two perpendicular oscillations in x and y dire ...
display is an ellipse, rather than a straight line, the two signals are out of phase.
Elliptical gears
Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain
A chain is a wikt:series#Noun, serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression (physics), compression but line (g ...
or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed
Angular may refer to:
Anatomy
* Angular artery, the terminal part of the facial artery
* Angular bone, a large bone in the lower jaw of amphibians and reptiles
* Angular incisure, a small anatomical notch on the stomach
* Angular gyrus, a region ...
or torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...
from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for ...
.
Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.
An example gear application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.
Optics
* In a material that is optically anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...
( birefringent), the refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, o ...
depends on the direction of the light. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this ellipsoid is a sphere.)
* In lamp- pumped solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from the pump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal axis).
* In laser-plasma produced EUV light sources used in microchip lithography
Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone ( lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German ...
, EUV light is generated by plasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the input of the lithography machine.
Statistics and finance
In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours—loci of equal values of the density function—are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.
Computer graphics
Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.
In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties. These algorithms need only a few multiplications and additions to calculate each vector.
It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.
;Drawing with Bézier paths:
Composite Bézier curves may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bézier curves behave appropriately under such transformations.
Optimization theory
It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for solving this problem.
See also
* Cartesian oval, a generalization of the ellipse
* Circumconic and inconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld ...
* Distance of closest approach of ellipses
* Ellipse fitting
* Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, c ...
e
* Elliptic partial differential equation
* Elliptical distribution, in statistics
* Elliptical dome
* Geodesics on an ellipsoid
* Great ellipse
* Kepler's laws of planetary motion
In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular or ...
* ''n''-ellipse, a generalization of the ellipse for ''n'' foci
* Oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one o ...
* Spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
, the ellipsoid obtained by rotating an ellipse about its major or minor axis
* Stadium (geometry), a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides
* Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid
* Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy"
* True
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* ...
, eccentric, and mean anomaly
Notes
References
*
*
*
*
*
External links
*
*
*
*
*
Apollonius' Derivation of the Ellipse
at Convergence
''The Shape and History of The Ellipse in Washington, D.C.''
by Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer s ...
Ellipse circumference calculator
*
Trammel according Frans van Schooten
* by Matt Parker
{{Authority control
Conic sections
Plane curves
Elementary shapes
Algebraic curves