![Ellipse-var](https://upload.wikimedia.org/wikipedia/commons/7/78/Ellipse-var.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 ...
, an ellipse is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
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
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-center, in geometry
* Eccentricity (graph theory) of a v ...
, 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 descript ...
).
An ellipse has a simple
algebra
Algebra () is one of the 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 mathematics.
Elementary a ...
ic solution for its area, but only approximations for its
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
(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 to ...
), 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 specia ...
: a plane curve tracing the intersection of a cone with 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' ...
(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, cal ...
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'' (YF ...
and
unbounded. An angled
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
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 infin ...
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
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 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 r ...
,
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 as a p ...
of each planet in the
Solar System
The Solar SystemCapitalization 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 S ...
is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the
barycenter
In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important conc ...
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
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IBM ...
or
perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
. 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 dir ...
formed when the horizontal and vertical motions are
sinusoid
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
s with the same frequency: a similar effect leads to
elliptical polarization
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elli ...
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, ultraviole ...
.
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 contribution ...
in his ''Conics''.
Definition as locus of points
![Ellipse-def-dc](https://upload.wikimedia.org/wikipedia/commons/c/c7/Ellipse-def-dc.svg)
An ellipse can be defined geometrically as a set or
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:
: 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 plane ...
, 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 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
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 longe ...
. 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
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 longe ...
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
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 (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 canonic ...
).
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 obj ...
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)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
). (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
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
, 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 de ...
: the set of points
of 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 t ...
that, in non-degenerate cases, satisfy the
implicit
Implicit may refer to:
Mathematics
* Implicit function
* Implicit function theorem
* Implicit curve
* Implicit surface
* Implicit differential equation
Other uses
* Implicit assumption, in logic
* Implicit-association test, in social psychology
...
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 descri ...
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 and ...
:
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 all ...
s, a parametric representation of the standard ellipse
is:
:
The parameter ''t'' (called the ''
eccentric anomaly In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position alo ...
'' 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