, 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. As such, 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 algebra
ic solution for its area, but only approximations for its perimeter, for which integration is required to obtain an exact solution.
, the equation of a standard ellipse centered at the origin with width
, the foci are
. The standard parametric equation is:
Ellipses are the closed
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, parabola
s and hyperbola
s, both of which are open
. 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
. 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 ellipsoids
. 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
formed when the horizontal and vertical motions are sinusoid
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
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
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''.
yields a circle and is included as a special type of ellipse.
can be viewed in a different way (see figure):
is the circle with midpoint
, 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
, one can prove that any plane 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
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
and to the other focus
. Hence the point
is on the ellipse whenever:
Removing the radicals
by suitable squarings and using
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
It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.
Throughout this article, the semi-major and semi-minor axes
, respectively, i.e.
In principle, the canonical ellipse equation
(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 the parameter names
This is the distance from the center to a focus:
The eccentricity can be expressed as:
An ellipse with equal axes (
) has zero eccentricity, and is a circle.
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
An arbitrary line
intersects an ellipse at
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
of the tangent is:
be a point on an ellipse and
be the equation of any line
. Inserting the line's equation into the ellipse equation and respecting
: There are then cases:
and the ellipse have only point
in common, and
is a tangent. The tangent direction has perpendicular vector
, so the tangent line has equation
is on the tangent and the ellipse, one obtains
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.
are two points of the ellipse such that
, then the points lie on two ''conjugate diameters'' (see below
, the ellipse is a circle and "conjugate" means "orthogonal".)
If the standard ellipse is shifted to have center
, its equation is
The axes are still parallel to the x- and y-axes.
* The center is the origin
is the angle measured from ''x''-axis.
* The parameter
(called the ''eccentric anomaly
'' in astronomy) is not the angle of
with the ''x''-axis.
are the semi-axes in the x and y directions, respectively.
is fixed (constant value)
* ''t'' is a parameter = independent variable used to parametrize the ellipse
In analytic geometry
, the ellipse is defined as a quadric
: the set of points
of the Cartesian plane
that, in non-degenerate cases, satisfy the implicit
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:
Standard parametric representation
Using trigonometric function
s, a parametric representation of the standard ellipse
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
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
this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing
The left vertex is the limit
Rational representations of conic sections are commonly used in computer-aided design
(see Bezier curve
Tangent slope as parameter
A parametric representation, which uses the slope
of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation
With help of trigonometric formulae
of the standard representation yields:
is the slope of the tangent at the corresponding ellipse point,
is the upper and
the lower half of the ellipse. The vertices
, having vertical tangents, are not covered by the representation.
The equation of the tangent at point
has the form
. The still unknown
can be determined by inserting the coordinates of the corresponding ellipse point
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.
Another definition of an ellipse uses affine transformation
: Any ''ellipse'' is an affine image of the unit circle with equation
An affine transformation of the Euclidean plane has the form
is a regular matrix
(with non-zero determinant
is an arbitrary vector. If
are the column vectors of the matrix
, the unit circle
, is mapped onto the ellipse:
is the center and
are the directions of two conjugate diameter
s, in general not perpendicular.
The four vertices of the ellipse are
, for a parameter
.) This is derived as follows. The tangent vector at point
At a vertex parameter
, the tangent is perpendicular to the major/minor axes, so:
Expanding and applying the identities
gives the equation for
Solving the parametric representation for
by Cramer's rule
, one gets the implicit representation
;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
to be vectors in space.
Polar form relative to center
In polar coordinates
, with the origin at the center of the ellipse and with the angular coordinate
measured from the major axis, the ellipse's equation is
Polar form relative to focus
If instead we use polar coordinates with the origin at one focus, with the angular coordinate
still measured from the major axis, the ellipse's equation is
where the sign in the denominator is negative if the reference direction
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
, the polar form is
in these formulas is called the true anomaly
of the point. The numerator of these formulas is the semi-latus rectum
Eccentricity and the directrix property
Each of the two lines parallel to the minor axis, and at a distance of
from it, is called a ''directrix'' of the ellipse (see diagram).
: For an arbitrary point
of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
The proof for the pair
follows from the fact that
satisfy the equation
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
(focus), any line
(directrix) not through
, and any real number
the ellipse is the locus of points for which the quotient of the distances to the point and to the line is
, which is the eccentricity of a circle, is not allowed in this context. One may consider the directrix of a circle to be the line at infinity.
yields a parabola
, and if
, a hyperbola
, and assume
is a point on the curve.
, the relation
produces the equations
This is the equation of an ''ellipse'' (
), or a ''parabola'' (
), or a ''hyperbola'' (
). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
, introduce new parameters
, and then the equation above becomes
which is the equation of an ellipse with center
, the ''x''-axis as major axis, and
the major/minor semi axis
If the focus is
and the directrix
, one obtains the equation
(The right side of the equation uses the Hesse normal form
of a line to calculate the distance
Focus-to-focus reflection property
An ellipse possesses the following property:
: The normal at a point
bisects the angle between the lines
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.
be the point on the line
with the distance
to the focus
is the semi-major axis of the ellipse. Let line
be the bisector of the supplementary angle to the angle between the lines
. In order to prove that
is the tangent line at point
, one checks that any point
which is different from
cannot be on the ellipse. Hence
has only point
in common with the ellipse and is, therefore, the tangent at point
From the diagram and the triangle inequality
one recognizes that
holds, which means:
. But if
is a point of the ellipse, the sum should be
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
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.)
of an ellipse are ''conjugate'' if the midpoints of chords parallel to
From the diagram one finds:
: Two diameters
of an ellipse are conjugate whenever the tangents at
are parallel to
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
In the parametric equation for a general ellipse given above,
any pair of points
belong to a diameter, and the pair
belong to its conjugate diameter.
Theorem of Apollonios on conjugate diameters
For an ellipse with semi-axes
the following is true:
be halves of two conjugate diameters (see diagram) then
:# The ''triangle''
(see diagram) has the constant area
, which can be expressed by
is the altitude of point
the angle between the half diameters. Hence the area of the ellipse (see section metric properties
) can be written as
:# The parallelogram of tangents adjacent to the given conjugate diameters has the
Let the ellipse be in the canonical form with parametric equation
The two points
are on conjugate diameters (see previous section). From trigonometric formulae one obtains
The area of the triangle generated by
and from the diagram it can be seen that the area of the parallelogram is 8 times that of
For the ellipse
the intersection points of ''orthogonal'' tangents lie on the circle
This circle is called ''orthoptic'' or director circle
of the ellipse (not to be confused with the circular directrix defined above).
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 (''ellipsograph
s'') to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as Archimedes
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
of an ellipse:
# Draw the two ''circles'' centered at the center of the ellipse with radii
and the axes of the ellipse.
# Draw a ''line through the center'', which intersects the two circles at point
# Draw a ''line'' through
that is parallel to the minor axis and a ''line'' through
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
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 pin
s, 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
. 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
This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes
have to be known.
The first method starts with
: a strip of paper of length
The point, where the semi axes meet is marked by
. If the strip slides with both ends on the axes of the desired ellipse, then point
traces the ellipse. For the proof one shows that point
has the parametric representation
, where parameter
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
(see animation). The device is able to draw any ellipse with a ''fixed'' sum
, 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
of the paper strip is moving on the circle with center
(of the ellipse) and radius
. Hence, the paperstrip can be cut at point
into halves, connected again by a joint at
and the sliding end
fixed at the center
(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
One marks the point, which divides the strip into two substrips of length
. 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
, where parameter
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
File:Ellipses with SliderCrank Ellipses at Slider Side.gif|Variation of the paper strip method 2
Most ellipsograph drafting
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
* The radius of curvature at the co-vertices
The diagram shows an easy way to find the centers of curvature
# mark the auxiliary point
and draw the line segment
# draw the line through
, which is perpendicular to the line
# 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
The following method to construct single points of an ellipse relies on the Steiner generation of a conic section
: Given two pencils
of lines at two points
(all lines containing
, respectively) and a projective but not perspective mapping
, then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the ellipse
one uses the pencils at the vertices
be an upper co-vertex of the ellipse and
is the center of the rectangle
. The side
of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal
as direction onto the line segment
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
needed. The intersection points of any two related lines
are points of the uniquely defined ellipse. With help of the points
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.
The ellipse is a special case of the hypotrochoid
, as shown in the adjacent image. The special case of a moving circle with radius
inside a circle with radius
is called a Tusi couple
Inscribed angles and three-point form
A circle with equation
is uniquely determined by three points
not on a line. A simple way to determine the parameters
uses the ''inscribed angle theorem
'' for circles:
: For four points
(see diagram) the following statement is true:
: The four points are on a circle if and only if the angles at
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
one uses the quotient:
Inscribed angle theorem for circles
For four points
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
are equal. In terms of the angle measurement above, this means:
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
For example, for
the three-point equation is:
, which can be rearranged to
Using vectors, dot product
s and determinant
s this formula can be arranged more clearly, letting
The center of the circle
The radius is the distance between any of the three points and the center.
This section, we consider the family of ellipses defined by equations
with a ''fixed'' eccentricity
. It is convenient to use the parameter:
and to write the ellipse equation as:
where ''q'' is fixed and
vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if
, the major axis is parallel to the ''x''-axis; if
, 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
one uses the quotient:
Inscribed angle theorem for ellipses
: Given four points
, no three of them on a line (see diagram).
: The four points are on an ellipse with equation
if and only if the angles at
are equal in the sense of the measurement above—that is, if
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
For example, for
one obtains the three-point form
and after conversion
Analogously to the circle case, the equation can be written more clearly using vectors:
is the modified dot product
Any ellipse can be described in a suitable coordinate system by an equation
. The equation of the tangent at a point
of the ellipse is
If one allows point
to be an arbitrary point different from the origin, then
is mapped onto the line
, not through the center of the ellipse.
This relation between points and lines is a bijection
The inverse function
onto the point
onto the point
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:
* For a pole
''outside'' the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing
* For a point ''within'' the ellipse, the polar has no point with the ellipse in common (see diagram:
# The intersection point of two polars is the pole of the line through their poles.
# The foci
, respectively, and the directrices
, respectively, belong to pairs of pole and polar. Because they are even polar pairs with respect to the circle
, the directrices can be constructed by compass and straightedge (see Inversive geometry
Pole-polar relations exist for hyperbolas and parabolas as well.
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.
enclosed by an ellipse is:
are the lengths of the semi-major and semi-minor axes, respectively. The area formula
is intuitive: start with a circle of radius
(so its area is
) and stretch it by a factor
to make an ellipse. This scales the area by the same factor:
It is also easy to rigorously prove the area formula using integration
as follows. Equation () can be rewritten as
this curve is the top half of the ellipse. So twice the integral of
over the interval