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 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, 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).
An ellipse has a simple
algebraic solution for its area, but only approximations for its
perimeter (also known as
circumference), 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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
type of
conic section: 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
hyperbolas, both of which are
open and
unbounded. An angled
cross section of a
cylinder 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,
astronomy and
engineering. For example, the
orbit of each planet in the
Solar System 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
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 dire ...
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 of light in
optics.
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
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
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 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: the set of points
of the
Cartesian plane 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 from the non-degenerate case, let ''∆'' be the
determinant
:
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 functions, 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