View source for Ellipse

{{About|the geometric figure}}
{{confusion|Ellipsis|Eclipse|Ecliptic}}
{{Short description|Plane curve: conic section}}
[[File:Ellipse-conic.svg|thumb|An ellipse (red) obtained as the intersection of a cone with an inclined plane.]]
[[File:Ellipse-def0.svg|300px|thumb|Ellipse: notations]]
[[File:Ellipse-var.svg|thumb|Ellipses: examples with increasing eccentricity]]
In [[mathematics]], an '''ellipse''' is a [[plane curve]] surrounding two [[focus (geometry)|focal points]], such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a [[circle]], which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its [[eccentricity (mathematics)|eccentricity]] <math>e</math>, a number ranging from <math>e = 0</math> (the [[Limiting case (mathematics)|limiting case]] of a circle) to <math>e = 1</math> (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 for its [[perimeter]] (also known as [[circumference]]), [[Integral|integration]] is required to obtain an exact solution.
[[Analytic geometry|Analytically]], the equation of a standard ellipse centered at the origin with width <math>2a</math> and height <math>2b</math> is:
<math display="block">\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 .</math>
Assuming <math>a \ge b</math>, the foci are <math>(\pm c, 0)</math> for <math display="inline">c = \sqrt{a^2-b^2}</math>. The standard parametric equation is:
<math display="block">(x,y) = (a\cos(t),b\sin(t)) \quad \text{for} \quad 0\leq t\leq 2\pi.</math>
Ellipses are the [[closed curve|closed]] type of [[conic section]]: a plane curve tracing the intersection of a [[cone]] with a [[plane (mathematics)|plane]] (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and [[hyperbola]]s, both of which are [[open curve|open]] and [[unbounded set|unbounded]]. An angled [[Cross section (geometry)|cross section]] of a right circular [[cylinder (geometry)#Cylindric section|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 [[#Eccentricity and the directrix property|directrix]]: for all points on the ellipse, the ratio between the distance to the [[focus (geometry)|focus]] and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
<math display="block">e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}.</math>
Ellipses are common in [[physics]], [[astronomy]] and [[engineering]]. For example, the [[Kepler orbit|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 [[Barycentric coordinates (astronomy)|barycenter]] of the Sun{{ndash}}planet 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]]s. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under [[parallel projection|parallel]] or [[perspective projection]]. The ellipse is also the simplest [[Lissajous figure]] formed when the horizontal and vertical motions are [[Sine wave|sinusoid]]s with the same frequency: a similar effect leads to [[elliptical polarization]] of light in [[optics]].
The name, {{lang|grc|ἔλλειψις}} ({{transl|grc|élleipsis}}, "omission"), was given by [[Apollonius of Perga]] in his ''Conics''.
== Definition as locus of points ==
[[File:Ellipse-def-e.svg|thumb|Ellipse: definition by sum of distances to foci]]
[[File:Ellipse-def-dc.svg|thumb|Ellipse: definition by focus and circular directrix]]
An ellipse can be defined geometrically as a set or [[locus of points]] in the Euclidean plane:
{{block indent | em = 1.5 | text = Given two fixed points <math>F_1, F_2</math> called the foci and a distance  <math>2a</math> which is greater than the distance between the foci, the ellipse is the set of points <math>P</math> such that the sum of the distances <math>|PF_1|,\ |PF_2|</math> is equal to <math>2a</math>: <math display="block">E = \left\{P\in \R^2 \,\mid\, \left|PF_2\right| + \left|PF_1\right| = 2a \right\} .</math>}}
The midpoint <math>C</math> 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''. {{anchor|Vertex}}The major axis intersects the ellipse at two ''[[vertex (curve)|vertices]]'' <math>V_1,V_2</math>, which have distance <math>a</math> to the center. The distance <math>c</math> of the foci to the center is called the ''focal distance'' or linear eccentricity. The quotient <math>e = \tfrac{c}{a}</math> is the ''eccentricity''.
The case <math>F_1 = F_2</math> yields a circle and is included as a special type of ellipse.
The equation <math>\left|PF_2\right| + \left|PF_1\right| = 2a</math> can be viewed in a different way (see figure):
{{block indent | em = 1.5 | text = If <math>c_2</math> is the circle with center <math>F_2</math> and radius <math>2a</math>, then the distance of a point <math>P</math> to the circle <math>c_2</math> equals the distance to the focus <math>F_1</math>: <math display="block">\left|PF_1\right| = \left|Pc_2\right|.</math>}}
<math>c_2</math> is called the ''circular directrix'' (related to focus {{nowrap|<math>F_2</math>)}} of the ellipse.<ref>{{citation | first1=Tom M.|last1=Apostol | first2=Mamikon A.|last2=Mnatsakanian | title=New Horizons in Geometry | year = 2012 | publisher=The Mathematical Association of America|series=The Dolciani Mathematical Expositions #47 | isbn = 978-0-88385-354-2 | page=251}}</ref><ref>The German term for this circle is ''Leitkreis'' which can be translated as "Director circle", but that term has a different meaning in the English literature (see [[Director circle]]).</ref> 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 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 ==
[[File:Ellipse-param.svg|thumb|Shape parameters:{{unbulleted list
| ''a'': semi-major axis,
| ''b'': semi-minor axis,
| ''c'': linear eccentricity,
| ''p'': semi-latus rectum (usually <math>\ell</math>).
}}]]
=== 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:
{{unbulleted list | style = padding-left:1.5em
| the foci are the points <math>F_1 = (c,\, 0),\ F_2=(-c,\, 0)</math>,
| the vertices are <math>V_1 = (a,\, 0),\ V_2 = (-a,\, 0)</math>.
}}
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is <math display="inline">\sqrt{(x - c)^2 + y^2 }</math> and to the other focus <math display="inline">\sqrt{(x + c)^2 + y^2}</math>. Hence the point <math>(x,\, y)</math> is on the ellipse whenever:
<math display="block">\sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} = 2a\ .</math>
Removing the [[radical expression|radicals]] by suitable squarings and using <math>b^2 = a^2-c^2</math> (see diagram) produces the standard equation of the ellipse:<ref name="mathworld">{{cite web | url=http://mathworld.wolfram.com/Ellipse.html |title=Ellipse - from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2020-09-10 |access-date=2020-09-10}}</ref>
<math display="block">\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
or, solved for ''y'':
<math display="block">y = \pm\frac{b}{a}\sqrt{a^2 - x^2} = \pm \sqrt{\left(a^2 - x^2\right)\left(1 - e^2\right)}.</math>
The width and height parameters <math>a,\; b</math> are called the [[semi-major and semi-minor axes]]. The top and bottom points <math>V_3 = (0,\, b),\; V_4 = (0,\, -b)</math> are the ''co-vertices''. The distances from a point <math>(x,\, y)</math> on the ellipse to the left and right foci are <math>a + ex</math> and <math>a - ex</math>.
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 <math>a</math> and <math>b</math>, respectively, i.e. <math>a \ge b > 0 \ .</math>
In principle, the canonical ellipse equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1 </math> may have <math>a < b</math> (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 <math>x</math> and <math> y</math> and the parameter names <math>a</math> and <math> b.</math>
==== Linear eccentricity ====
This is the distance from the center to a focus: <math>c = \sqrt{a^2 - b^2}</math>.
==== Eccentricity ====
[[File:Pythagorean_theorem_ellipse_eccentricity.svg|thumb|upright|Eccentricity ''e'' in terms of semi-major ''a'' and semi-minor ''b'' axes: {{nowrap|1=''e''&sup2; + (''b/a'')&sup2; = 1}}]]
The eccentricity can be expressed as:
<math display="block">e = \frac{c}{a} = \sqrt{1 - \left(\frac{b}{a}\right)^2},</math>
assuming <math>a > b.</math> An ellipse with equal axes (<math>a = b</math>) 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'' <math>\ell</math>. A calculation shows:<ref>{{harvtxt|Protter|Morrey|1970|pp=304,APP-28}}</ref>
<math display="block">\ell = \frac{b^2}a = a \left(1 - e^2\right).</math>
The semi-latus rectum <math>\ell</math> is equal to the [[radius of curvature]] at the vertices (see section [[#Curvature|curvature]]).
=== Tangent ===
An arbitrary line <math>g</math> 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 <math>(x_1,\, y_1)</math> of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> has the coordinate equation:
<math display="block">\frac{x_1}{a^2}x + \frac{y_1}{b^2}y = 1.</math>
A vector [[parametric equation]] of the tangent is:
<math display="block">\vec x = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \left(\begin{array}{r}
-y_1 a^2 \\
x_1 b^2
\end{array}\right) , \quad s \in \R. </math>
'''Proof:'''
Let <math>(x_1,\, y_1)</math> be a point on an ellipse and <math display="inline">\vec{x} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \begin{pmatrix} u \\ v \end{pmatrix}</math> be the equation of any line <math>g</math> containing <math>(x_1,\, y_1)</math>. Inserting the line's equation into the ellipse equation and respecting <math display="inline">\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1</math> yields:
<math display="block">
\frac{\left(x_1 + su\right)^2}{a^2} + \frac{\left(y_1 + sv\right)^2}{b^2} = 1\ \quad\Longrightarrow\quad
2s\left(\frac{x_1u}{a^2} + \frac{y_1v}{b^2}\right) + s^2\left(\frac{u^2}{a^2} + \frac{v^2}{b^2}\right) = 0\ .</math>
There are then cases:
# <math>\frac{x_1}{a^2}u + \frac{y_1}{b^2}v = 0.</math> Then line <math>g</math> and the ellipse have only point <math>(x_1,\, y_1)</math> in common, and <math>g</math> is a tangent. The tangent direction has [[normal (geometry)|perpendicular vector]] <math>\begin{pmatrix} \frac{x_1}{a^2} & \frac{y_1}{b^2} \end{pmatrix}</math>, so the tangent line has equation <math display="inline">\frac{x_1}{a^2}x + \tfrac{y_1}{b^2}y = k</math> for some <math>k</math>. Because <math>(x_1,\, y_1)</math> is on the tangent and the ellipse, one obtains <math>k = 1</math>.
# <math>\frac{x_ 1}{a^2}u + \frac{y_1}{b^2}v \ne 0.</math> Then line <math>g</math> has a second point in common with the ellipse, and is a secant.
Using (1) one finds that <math>\begin{pmatrix} -y_1 a^2 & x_1 b^2 \end{pmatrix}</math> is a tangent vector at point <math>(x_1,\, y_1)</math>, which proves the vector equation.
If <math>(x_1, y_1)</math> and <math>(u, v)</math> are two points of the ellipse such that <math display="inline">\frac{x_1u}{a^2} + \tfrac{y_1v}{b^2} = 0</math>, then the points lie on two ''conjugate diameters'' (see [[#Conjugate diameters|below]]). (If <math>a = b</math>, the ellipse is a circle and "conjugate" means "orthogonal".)
=== Shifted ellipse ===
If the standard ellipse is shifted to have center <math>\left(x_\circ,\, y_\circ\right)</math>, its equation is
<math display="block">\frac{\left(x - x_\circ\right)^2}{a^2} + \frac{\left(y - y_\circ\right)^2}{b^2} = 1 \ .</math>
The axes are still parallel to the ''x''- and ''y''-axes.
=== General ellipse ===
{{Main|Matrix representation of conic sections}}
[[File:General ellipse.png|thumb|right|upright=1.25|A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle]]
In [[analytic geometry]], the ellipse is defined as a [[Quadratic form|quadric]]: the set of points <math>(x,\, y)</math> of the [[Cartesian plane]] that, in non-degenerate cases, satisfy the [[Implicit and explicit functions|implicit]] equation<ref>{{cite book|url=https://books.google.com/books?id=yMdHnyerji8C | title=Precalculus with Limits|last1=Larson|first1=Ron| last2=Hostetler|first2=Robert P. | last3=Falvo|first3=David C.| publisher=Cengage Learning|year=2006 | isbn=978-0-618-66089-6|page=767 | chapter=Chapter 10 | chapter-url=https://books.google.com/books?id=yMdHnyerji8C&pg=PA767}}
</ref><ref>{{cite book| url=https://books.google.com/books?id=9HRLAn326zEC | title=Precalculus| last1=Young|first1=Cynthia Y.|author-link=Cynthia Y. Young| publisher=John Wiley and Sons| year=2010| isbn=978-0-471-75684-2|page=831| chapter=Chapter 9| chapter-url=https://books.google.com/books?id=9HRLAn326zEC&pg=PA831}}
</ref>
<math display="block">Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0</math>
provided <math>B^2 - 4AC < 0.</math>
To distinguish the [[degenerate conic|degenerate cases]] from the non-degenerate case, let ''∆'' be the [[determinant]]
<math display="block">\Delta = \begin{vmatrix}
A & \frac{1}{2}B & \frac{1}{2}D \\
\frac{1}{2}B &            C & \frac{1}{2}E \\
\frac{1}{2}D & \frac{1}{2}E &            F
\end{vmatrix} = ACF + \tfrac14 BDE - \tfrac14(AE^2 + CD^2 + FB^2).
</math>
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.<ref name="Lawrence">Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.</ref>{{rp|p=63}}
The general equation's coefficients can be obtained from known semi-major axis <math>a</math>, semi-minor axis <math>b</math>, center coordinates <math>\left(x_\circ,\, y_\circ\right)</math>, and rotation angle <math>\theta</math> (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
<math display="block">\begin{align}
A &=   a^2 \sin^2\theta + b^2 \cos^2\theta &
B &=  2\left(b^2 - a^2\right) \sin\theta \cos\theta \\[1ex]
C &=   a^2 \cos^2\theta + b^2 \sin^2\theta &
D &= -2A x_\circ   -  B y_\circ \\[1ex]
E &= - B x_\circ   - 2C y_\circ &
F &=   A x_\circ^2 +  B x_\circ y_\circ + C y_\circ^2 - a^2 b^2.
\end{align}</math>
These expressions can be derived from the canonical equation
<math display="block">\frac{X^2}{a^2} + \frac{Y^2}{b^2} = 1</math>
by a Euclidean transformation of the coordinates <math>(X,\, Y)</math>:
<math display="block">\begin{align}
X &=  \left(x - x_\circ\right) \cos\theta + \left(y - y_\circ\right) \sin\theta, \\
Y &= -\left(x - x_\circ\right) \sin\theta + \left(y - y_\circ\right) \cos\theta.
\end{align}</math>
Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:<ref name="mathworld"/>
<math display="block">\begin{align}
a, b    &= \frac{-\sqrt{2 \big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) F\big)\big((A + C) \pm \sqrt{(A - C)^2 + B^2}\big)}}{B^2 - 4 A C}, \\
x_\circ  &= \frac{2CD - BE}{B^2 - 4AC}, \\[5mu]
y_\circ  &= \frac{2AE - BD}{B^2 - 4AC}, \\[5mu]
\theta &= \tfrac12 \operatorname{atan2}(-B,\, C-A),
\end{align}</math>
where {{math|[[atan2]]}} is the 2-argument arctangent function.
== Parametric representation ==
[[File:Elliko-sk.svg|thumb|The construction of points based on the parametric equation and the interpretation of parameter ''t'', which is due to de la Hire]]
[[File:Ellipse-ratpar.svg|thumb|Ellipse points calculated by the rational representation with equally spaced parameters (<math>\Delta u = 0.2</math>).]]
===Standard parametric representation===
Using [[trigonometric function]]s, a parametric representation of the standard ellipse <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2} = 1</math> is:
<math display="block">(x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\, .</math>
The parameter ''t'' (called the ''[[eccentric anomaly]]'' in astronomy) is not the angle of <math>(x(t),y(t))</math> with the ''x''-axis, but has a geometric meaning due to [[Philippe de La Hire]] (see ''{{slink||Drawing ellipses}}'' below).<ref>K. Strubecker: ''Vorlesungen über Darstellende Geometrie'', GÖTTINGEN, VANDENHOECK & RUPRECHT, 1967, p. 26</ref>
===Rational representation===
With the substitution <math display="inline">u = \tan\left(\frac{t}{2}\right)</math> and trigonometric formulae one obtains
<math display="block">\cos t = \frac{1 - u^2}{1 + u^2}\ ,\quad \sin t = \frac{2u}{1 + u^2}</math>
and the ''rational'' parametric equation of an ellipse
<math display="block">\begin{cases}
x(u) = a \, \dfrac{1 - u^2}{1 + u^2} \\[10mu]
y(u) = b \, \dfrac{2u}{1 + u^2} \\[10mu]
-\infty < u < \infty
\end{cases}</math>
which covers any point of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> except the left vertex <math>(-a,\, 0)</math>.
For <math>u \in [0,\, 1],</math> this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing <math>u.</math> The left vertex is the limit <math display="inline">\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.</math>
Alternately, if the parameter <math>[u:v]</math> is considered to be a point on the [[real projective line]] <math display="inline">\mathbf{P}(\mathbf{R})</math>, then the corresponding rational parametrization is
<math display="block">
[u:v] \mapsto \left(a\frac{v^2 - u^2}{v^2 + u^2}, b\frac{2uv}{v^2 + u^2} \right).
</math>
Then <math display="inline">[1:0] \mapsto (-a,\, 0).</math>
Rational representations of conic sections are commonly used in [[computer-aided design]] (see [[Bezier curve#Rational Bézier curves|Bezier curve]]).
===Tangent slope as parameter===
A parametric representation, which uses the slope <math>m</math> of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation <math>\vec x(t) = (a \cos t,\, b \sin t)^\mathsf{T}</math>:
<math display="block">\vec x'(t) = (-a\sin t,\, b\cos t)^\mathsf{T} \quad \rightarrow \quad m = -\frac{b}{a}\cot t\quad \rightarrow \quad \cot t = -\frac{ma}{b}.</math>
With help of [[List of trigonometric identities#Pythagorean identities|trigonometric formulae]] one obtains:
<math display="block">\cos t = \frac{\cot t}{\pm\sqrt{1 + \cot^2t}} = \frac{-ma}{\pm\sqrt{m^2 a^2 + b^2}}\ ,\quad\quad
\sin t = \frac{1}{\pm\sqrt{1 + \cot^2t}} = \frac{b}{\pm\sqrt{m^2 a^2 + b^2}}.</math>
Replacing <math>\cos t</math> and <math>\sin t</math> of the standard representation yields:
<math display="block">\vec c_\pm(m) = \left(-\frac{ma^2}{\pm\sqrt{m^2 a^2 + b^2}},\;\frac{b^2}{\pm\sqrt{m^2a^2 + b^2}}\right),\, m \in \R.</math>
Here <math>m</math> is the slope of the tangent at the corresponding ellipse point, <math>\vec c_+</math> is the upper and <math>\vec c_-</math> the lower half of the ellipse. The vertices<math>(\pm a,\, 0)</math>, having vertical tangents, are not covered by the representation.
The equation of the tangent at point <math>\vec c_\pm(m)</math> has the form <math>y = mx + n</math>. The still unknown <math>n</math> can be determined by inserting the coordinates of the corresponding ellipse point <math>\vec c_\pm(m)</math>:
<math display="block">y = mx \pm \sqrt{m^2 a^2 + b^2}\, .</math>
This description of the tangents of an ellipse is an essential tool for the determination of the [[orthoptic (geometry)|orthoptic]] of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.
===General ellipse===
[[File:ellipse-aff.svg|300px|thumb|Ellipse as an affine image of the unit circle]]
Another definition of an ellipse uses [[affine transformation]]s:
: Any ''ellipse'' is an affine image of the unit circle with equation <math>x^2 + y^2 = 1</math>.
;Parametric representation
An affine transformation of the Euclidean plane has the form <math>\vec x \mapsto \vec f\!_0 + A\vec x</math>, where <math>A</math> is a regular [[matrix (mathematics)|matrix]] (with non-zero [[determinant]]) and <math>\vec f\!_0</math> is an arbitrary vector. If <math>\vec f\!_1, \vec f\!_2</math> are the column vectors of the matrix <math>A</math>, the unit circle <math>(\cos(t), \sin(t))</math>, <math>0 \leq t \leq 2\pi</math>, is mapped onto the ellipse:
<math display="block">\vec x = \vec p(t) = \vec f\!_0 + \vec f\!_1 \cos t + \vec f\!_2 \sin t \, .</math>
Here <math>\vec f\!_0</math> is the center and <math>\vec f\!_1,\; \vec f\!_2</math> are the directions of two [[conjugate diameter]]s, in general not perpendicular.
;Vertices
The four vertices of the ellipse are <math>\vec p(t_0),\;\vec p\left(t_0 \pm \tfrac{\pi}{2}\right),\; \vec p\left(t_0 + \pi\right)</math>, for a parameter <math>t = t_0</math> defined by:
<math display="block">\cot (2t_0) = \frac{\vec f\!_1^{\,2} - \vec f\!_2^{\,2}}{2\vec f\!_1 \cdot \vec f\!_2}.</math>
(If <math>\vec f\!_1 \cdot \vec f\!_2 = 0</math>, then <math>t_0 = 0</math>.) This is derived as follows. The tangent vector at point <math>\vec p(t)</math> is:
<math display="block">\vec p\,'(t) = -\vec f\!_1\sin t + \vec f\!_2\cos t \ .</math>
At a vertex parameter <math>t = t_0</math>, the tangent is perpendicular to the major/minor axes, so:
<math display="block">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).</math>
Expanding and applying the identities <math>\; \cos^2 t -\sin^2 t=\cos 2t,\ \ 2\sin t \cos t = \sin 2t\;</math> gives the equation for <math>t = t_0\; .</math>
;Area
From Apollonios theorem (see below) one obtains:<br>
The area of an ellipse <math>\;\vec x = \vec f_0 +\vec f_1 \cos t +\vec f_2 \sin t\; </math> is
<math display="block">A=\pi \left|\det(\vec f_1, \vec f_2)\right| .</math>
;Semiaxes
With the abbreviations
<math>\; M=\vec f_1^2+\vec f_2^2, \ N = \left|\det(\vec f_1,\vec f_2)\right| </math>  the statements of Apollonios's theorem can be written as:
<math display="block">a^2+b^2=M, \quad ab=N \ .</math>
Solving this nonlinear system for <math>a,b</math> yields the semiaxes:
<math display="block">\begin{align}
a &= \frac{1}{2}(\sqrt{M+2N}+\sqrt{M-2N}) \\[1ex]
b &= \frac{1}{2}(\sqrt{M+2N}-\sqrt{M-2N})\, .
\end{align}</math>
;Implicit representation
Solving the parametric representation for <math>\; \cos t,\sin t\;</math> by [[Cramer's rule]] and using <math>\;\cos^2t+\sin^2t -1=0\; </math>, one obtains the implicit representation
<math display="block">\det{\left(\vec x\!-\!\vec f\!_0,\vec f\!_2\right)^2} + \det{\left(\vec f\!_1,\vec x\!-\!\vec f\!_0\right)^2} - \det{\left(\vec f\!_1,\vec f\!_2\right)^2} = 0.</math>
Conversely: If the [[Matrix representation of conic sections|equation]]
:<math>x^2+2cxy+d^2y^2-e^2=0\ ,</math> with <math>\; d^2-c^2 >0 \; ,</math>
of an ellipse centered at the origin is given, then the two vectors
<math display="block">\vec f_1={e \choose 0},\quad \vec f_2=\frac{e}{\sqrt{d^2-c^2}}{-c\choose 1} </math>
point to two conjugate points and the tools developed above are applicable.
''Example'': For the ellipse with equation <math>\;x^2+2xy+3y^2-1=0\; </math> the vectors are
<math display="block">\vec f_1={1 \choose 0},\quad \vec f_2=\frac{1}{\sqrt{2}}{-1\choose 1} .</math>
[[File:Nested Ellipses.svg|thumb|upright=1.2|Whirls: nested, scaled and rotated ellipses.  The spiral is not drawn: we see it as the [[Locus (mathematics)|locus]] of points where the ellipses are especially close to each other.]]
;Rotated standard ellipse
For <math>\vec f_0= {0\choose 0},\;\vec f_1= a {\cos \theta\choose \sin \theta},\;\vec f_2= b{-\sin \theta\choose \;\cos \theta}</math> one obtains a parametric representation of the standard ellipse [[Rotation matrix|rotated]] by angle <math>\theta</math>:
<math display="block">\begin{align}
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 \, .
\end{align}</math>
;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 <math>\vec f\!_0, \vec f\!_1, \vec f\!_2</math> to be vectors in space.
== Polar forms ==
=== Polar form relative to center ===
[[File:Ellipse Polar center.svg|thumb|right|Polar coordinates centered at the center.]]
In [[polar coordinates]], with the origin at the center of the ellipse and with the angular coordinate <math>\theta</math> measured from the major axis, the ellipse's equation is<ref name="Lawrence" />{{rp||page=75}}
<math display="block">r(\theta) = \frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}=\frac{b}{\sqrt{1 - (e\cos\theta)^2}}</math>
where <math>e</math> is the eccentricity, not Euler's number.
=== Polar form relative to focus ===
[[File:Ellipse Polar.svg|thumb|right|Polar coordinates centered at focus.]]
If instead we use polar coordinates with the origin at one focus, with the angular coordinate <math>\theta = 0</math> still measured from the major axis, the ellipse's equation is
<math display="block">r(\theta)=\frac{a (1-e^2)}{1\pm e\cos\theta }</math>
where the sign in the denominator is negative if the reference direction <math>\theta = 0</math> points towards the center (as illustrated on the right), and positive if that direction points away from the center.
The angle <math>\theta</math> is called the [[true anomaly]] of the point. The numerator <math>\ell=a (1-e^2)</math> is the [[semi-latus rectum]].
==Eccentricity and the directrix property==
[[File:Ellipse-ll-e.svg|300px|thumb|Ellipse: directrix property]]
Each of the two lines parallel to the minor axis, and at a distance of <math display="inline">d = \frac{a^2}{c} = \frac{a}{e}</math> from it, is called a ''directrix'' of the ellipse (see diagram).
: For an arbitrary point <math>P</math> of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: <math display="block">\frac{\left|PF_1\right|}{\left|Pl_1\right|} = \frac{\left|PF_2\right|}{\left|Pl_2\right|} = e = \frac{c}{a}\ .</math>
The proof for the pair <math>F_1, l_1</math> follows from the fact that <math display="inline">\left|PF_1\right|^2 = (x - c)^2 + y^2,\ \left|Pl_1\right|^2 = \left(x - \tfrac{a^2}{c}\right)^2</math> and <math>y^2 = b^2 - \tfrac{b^2}{a^2}x^2</math> satisfy the equation
<math display="block">\left|PF_1\right|^2 - \frac{c^2}{a^2}\left|Pl_1\right|^2 = 0\, .</math>
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 <math>F</math> (focus), any line <math>l</math> (directrix) not through <math>F</math>, and any real number <math>e</math> with <math>0 < e < 1,</math> the ellipse is the locus of points for which the quotient of the distances to the point and to the line is <math>e,</math> that is: <math display="block">E = \left\{P\ \left|\ \frac{|PF|}{|Pl|} = e\right.\right\}.</math>
The extension to <math>e = 0</math>, 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]].
(The choice <math>e = 1</math> yields a parabola, and if <math>e > 1</math>, a hyperbola.)
[[File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex and common semi-latus rectum]]
;Proof
Let <math>F = (f,\, 0),\ e > 0</math>, and assume <math>(0,\, 0)</math> is a point on the curve. The directrix <math>l</math> has equation <math>x = -\tfrac{f}{e}</math>. With <math>P = (x,\, y)</math>, the relation <math>|PF|^2 = e^2|Pl|^2</math> produces the equations
:<math>(x - f)^2 + y^2 = e^2\left(x + \frac{f}{e}\right)^2 = (ex + f)^2</math> and <math>x^2\left(e^2 - 1\right) + 2xf(1 + e) - y^2 = 0.</math>
The substitution <math>p = f(1 + e)</math> yields
<math display="block">x^2\left(e^2 - 1\right) + 2px - y^2 = 0.</math>
This is the equation of an ''ellipse'' (<math>e < 1</math>), or a ''parabola'' (<math>e = 1</math>), or a ''hyperbola'' (<math>e > 1</math>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).
If <math>e < 1</math>, introduce new parameters <math>a,\, b</math> so that <math>1 - e^2 = \tfrac{b^2}{a^2}, \text{ and }\ p = \tfrac{b^2}{a}</math>, and then the equation above becomes
<math display="block">\frac{(x - a)^2}{a^2} + \frac{y^2}{b^2} = 1\, ,</math>
which is the equation of an ellipse with center <math>(a,\, 0)</math>, the ''x''-axis as major axis, and
the major/minor semi axis <math>a,\, b</math>.
[[File:Leitlinien-konstr-e.svg|thumb|Construction of a directrix]]
;Construction of a directrix
Because of <math>c\cdot\tfrac{a^2}{c}=a^2</math> point <math>L_1</math> of directrix <math>l_1</math> (see diagram) and focus <math>F_1</math> are inverse with respect to the [[circle inversion]] at circle <math>x^2+y^2=a^2</math> (in diagram green). Hence <math>L_1</math> can be constructed as shown in the diagram. Directrix <math>l_1</math> is the perpendicular to the main axis at point <math>L_1</math>.
;General ellipse
If the focus is <math>F = \left(f_1,\, f_2\right)</math> and the directrix <math>ux + vy + w = 0</math>, one obtains the equation
<math display="block">\left(x - f_1\right)^2 + \left(y - f_2\right)^2 = e^2 \frac{\left(ux + vy + w\right)^2}{u^2 + v^2}\ .</math>
(The right side of the equation uses the [[Hesse normal form]] of a line to calculate the distance <math>|Pl|</math>.)
== Focus-to-focus reflection property ==
[[File:Ellipse-reflex.svg|250px|thumb|Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.]]
[[File:Elli-norm-tang-n.svg|250px|thumb|Rays from one focus reflect off the ellipse to pass through the other focus.]]
An ellipse possesses the following property:
: The normal at a point <math>P</math> bisects the angle between the lines <math>\overline{PF_1},\, \overline{PF_2}</math>.
;Proof
Because the tangent line is perpendicular to the normal, an equivalent statement is that the tangent is the external angle bisector of the lines to the foci (see diagram).
Let <math>L</math> be the point on the line <math>\overline{PF_2}</math> with distance <math>2a</math> to the focus <math>F_2</math>, where <math>a</math> is the semi-major axis of the ellipse. Let line <math>w</math> be the external angle bisector of the lines <math>\overline{PF_1}</math> and <math>\overline{PF_2}.</math> Take any other point <math>Q</math> on <math>w.</math> By the [[triangle inequality]] and the [[angle bisector theorem]], <math>2a = \left|LF_2\right| < {}</math><math>\left|QF_2\right| + \left|QL\right| = {}</math><math>\left|QF_2\right| + \left|QF_1\right|,</math> therefore <math>Q</math> must be outside the ellipse. As this is true for every choice of <math>Q,</math> <math>w</math> only intersects the ellipse at the single point <math>P</math> so must be the tangent line.
; 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]]).
Additionally, because of the focus-to-focus reflection property of ellipses, if the rays are allowed to continue propagating, reflected rays will eventually align closely with the major axis.
== Conjugate diameters ==
=== Definition of conjugate diameters ===
[[File:Parallelproj-kreis-ellipse.svg|400px|thumb|Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.]]
{{Main|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 <math>d_1,\, d_2</math> of an ellipse are ''conjugate'' if the midpoints of chords parallel to <math>d_1</math> lie on <math>d_2\ .</math>
From the diagram one finds:
: Two diameters <math>\overline{P_1 Q_1},\, \overline{P_2 Q_2}</math> of an ellipse are conjugate whenever the tangents at <math>P_1</math> and <math>Q_1</math> are parallel to <math>\overline{P_2 Q_2}</math>.
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
In the parametric equation for a general ellipse given above,
<math display="block">\vec x = \vec p(t) = \vec f\!_0 +\vec f\!_1 \cos t + \vec f\!_2 \sin t,</math>
any pair of points <math>\vec p(t),\ \vec p(t + \pi)</math> belong to a diameter, and the pair <math>\vec p\left(t + \tfrac{\pi}{2}\right),\ \vec p\left(t - \tfrac{\pi}{2}\right)</math> belong to its conjugate diameter.
For the common parametric representation <math>(a\cos t,b\sin t)</math> of the ellipse with equation <math>\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1</math> one gets: The points
:<math>(x_1,y_1)=(\pm a\cos t,\pm b\sin t)\quad </math> (signs: (+,+) or (−,−) )
:<math>(x_2,y_2)=({\color{red}{\mp}} a\sin t,\pm b\cos t)\quad </math> (signs: (−,+) or (+,−) )
:are conjugate and
:<math>\frac{x_1x_2}{a^2}+\frac{y_1y_2}{b^2}=0\ .</math>
In case of a circle the last equation collapses to <math>x_1x_2+y_1y_2=0\ . </math>
=== Theorem of Apollonios on conjugate diameters ===
[[File:Elli-apoll-cd.svg|upright=1.2|thumb|Theorem of Apollonios]]
[[File:Elli-apoll-area-altern.svg|thumb|upright=1.2|For the alternative area formula]]
For an ellipse with semi-axes <math>a,\, b</math> the following is true:<ref>Bronstein&Semendjajew: ''Taschenbuch der Mathematik'', Verlag Harri Deutsch, 1979, {{ISBN|3871444928}}, p. 274.</ref><ref>''Encyclopedia of Mathematics'', Springer,  URL: http://encyclopediaofmath.org/index.php?title=Apollonius_theorem&oldid=17516 .</ref>
: Let <math>c_1 </math> and  <math> c_2</math> be halves of two conjugate diameters (see diagram) then
:# <math>c_1^2 + c_2^2 = a^2 + b^2</math>.
:# The ''triangle'' <math>O,P_1,P_2</math> with sides <math>c_1,\, c_2</math> (see diagram) has the constant area <math display="inline">A_\Delta = \frac{1}{2}ab</math>, which can be expressed by <math>A_\Delta=\tfrac 1 2 c_2d_1=\tfrac 1 2 c_1c_2\sin\alpha</math>, too. <math>d_1</math> is the altitude of point <math>P_1</math> and <math>\alpha</math> the angle between the half diameters. Hence the area of the ellipse (see section [[#Metric properties|metric properties]]) can be written as <math>A_{el}=\pi ab=\pi c_2d_1=\pi c_1c_2\sin\alpha</math>.
:# The parallelogram of tangents adjacent to the given conjugate diameters has the <math>\text{Area}_{12} = 4ab\ .</math>
; Proof:
Let the ellipse be in the canonical form with parametric equation
<math display="block">\vec p(t) = (a\cos t,\, b\sin t).</math>
The two points <math display="inline">\vec c_1 = \vec p(t),\ \vec c_2 = \vec p\left(t + \frac{\pi}{2}\right)</math> are on conjugate diameters (see previous section). From trigonometric formulae one obtains <math>\vec c_2 = (-a\sin t,\, b\cos t)^\mathsf{T}</math> and
<math display="block">\left|\vec c_1\right|^2 + \left|\vec c_2\right|^2 = \cdots = a^2 + b^2\, .</math>
The area of the triangle generated by <math>\vec c_1,\, \vec c_2</math> is
<math display="block">A_\Delta = \tfrac{1}{2} \det\left(\vec c_1,\, \vec c_2\right) = \cdots = \tfrac{1}{2}ab</math>
and from the diagram it can be seen that the area of the parallelogram is 8 times that of <math>A_\Delta</math>. Hence
<math display="block">\text{Area}_{12} = 4ab\, .</math>
== Orthogonal tangents ==
[[File:Orthoptic-ellipse-s.svg|thumb|Ellipse with its orthoptic]]
{{main|Orthoptic (geometry)}}
For the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> the intersection points of ''orthogonal'' tangents lie on the circle <math>x^2 + y^2 = a^2 + b^2</math>.
This circle is called ''orthoptic'' or [[director circle]] of the ellipse (not to be confused with the circular directrix defined above).
== Drawing ellipses ==
[[File:Zp-turm-tor.svg|thumb|Central projection of circles (gate)]]
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 was known to the 5th century mathematician [[Proclus]], and the tool now known as an [[elliptical trammel]] was invented by [[Leonardo da Vinci]].<ref>{{cite journal |last=Blake |first=E. M. |year=1900 |title=The Ellipsograph of Proclus |journal=American Journal of Mathematics |volume=22 |number=2 |pages=146–153 |doi=10.2307/2369752 |jstor=2369752 |jstor-access=free }}</ref>
If there is no ellipsograph available, one can draw an ellipse using an [[#Approximation by osculating circles|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 [[Philippe de La Hire|de La Hire]].<ref>K. Strubecker: ''Vorlesungen über Darstellende Geometrie.'' Vandenhoeck & Ruprecht, Göttingen 1967, S. 26.</ref>  It is based on the [[#Standard parametric representation|standard parametric representation]] <math>(a\cos t,\, b\sin t)</math> of an ellipse:
# Draw the two ''circles'' centered at the center of the ellipse with radii <math>a,b</math> and the axes of the ellipse.
# Draw a ''line through the center'', which intersects the two circles at point <math>A</math> and <math>B</math>, respectively.
# Draw a ''line'' through <math>A</math> that is parallel to the minor axis and a ''line'' through <math>B</math> 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.
<gallery widths="220" heights="220" class="float-left">
Elliko-sk.svg|de La Hire's method
Parametric ellipse.gif|Animation of the method
</gallery>
[[File:Elliko-g.svg|250px|thumb|Ellipse: gardener's 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 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 <math>2a</math>. 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''. The Byzantine architect [[Anthemius of Tralles]] ({{c.|600}}) described how this method could be used to construct an elliptical reflector,<ref>From {{lang|el|Περί παραδόξων μηχανημάτων}} [''Concerning Wondrous Machines'']: "If, then, we stretch a string surrounding the points A, B tightly around the first point from which the rays are to be reflected, the line will be drawn which is part of the so-called ellipse, with respect to which the surface of the mirror must be situated." {{pb}} {{cite book |last=Huxley |first=G. L. |year=1959 |title=Anthemius of Tralles: A Study in Later Greek Geometry |lccn=59-14700 |location=Cambridge, MA |pages=8–9 |url=https://archive.org/details/anthemiusoftrall0000huxl/page/8/ |url-access=limited }}</ref> and it was elaborated in a now-lost 9th-century treatise by [[Al-Ḥasan ibn Mūsā ibn Shākir|Al-Ḥasan ibn Mūsā]].<ref>Al-Ḥasan's work was titled {{transliteration|ar|Kitāb al-shakl al-mudawwar al-mustaṭīl}} [''The Book of the Elongated Circular Figure'']. {{pb}} {{cite book |last1=Rashed |first1=Roshdi |translator-last=Shank |translator-first=Michael H. |title=Classical Mathematics from Al-Khwarizmi to Descartes |date=2014 |publisher=Routledge |location=New York |isbn=978-13176-2-239-0 |page=559 }}</ref>
A similar method for drawing [[Confocal conic sections#Graves's theorem: the construction of confocal ellipses by a string|confocal ellipses]] with a ''closed'' string is due to the Irish bishop [[Charles Graves (bishop)|Charles Graves]].
=== Paper strip methods ===
The two following methods rely on the parametric representation (see ''{{slink||Standard parametric representation}}'', above):
<math display="block">(a\cos t,\, b\sin t)</math>
This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes <math> a,\, b</math> have to be known.
;Method 1
The first method starts with
: a strip of paper of length <math>a + b</math>.
The point, where the semi axes meet is marked by <math>P</math>. If the strip slides with both ends on the axes of the desired ellipse, then point <math>P</math> traces the ellipse. For the proof one shows that point <math>P</math> has the parametric representation <math>(a\cos t,\, b\sin t)</math>, where parameter <math>t</math> 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 <math>a + b</math>, 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.
<gallery widths="250" heights="250">
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.
</gallery>
A variation of the paper strip method 1 uses the observation that the midpoint <math>N</math> of the paper strip is moving on the circle with center <math>M</math> (of the ellipse) and radius <math>\tfrac{a + b}{2}</math>. Hence, the paperstrip can be cut at point <math>N</math> into halves, connected again by a joint at <math>N</math> and the sliding end <math>K</math> fixed at the center <math>M</math> (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.<ref>J. van Mannen: ''Seventeenth century instruments for drawing conic sections.'' In: ''The Mathematical Gazette.'' Vol. 76, 1992, p. 222–230.</ref> This variation requires only one sliding shoe.
<gallery widths="300" heights="200">
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
</gallery>
[[File:Elliko-pap2.svg|250px|thumb|Ellipse construction: paper strip method 2]]
; Method 2:
The second method starts with
: a strip of paper of length <math>a</math>.
One marks the point, which divides the strip into two substrips of length <math>b</math> and <math>a - b</math>. 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 <math>(a\cos t,\, b\sin t)</math>, where parameter <math>t</math> 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.
<gallery widths="200" heights="150">
File:Archimedes Trammel.gif|[[Elliptical trammel]] (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
</gallery>Most ellipsograph [[Drafting machine|drafting]] instruments are based on the second paperstrip method.[[File:Elliko-skm.svg|250px|thumb|Approximation of an ellipse with osculating circles]]
=== Approximation by osculating circles ===
From ''Metric properties'' below, one obtains:
* The radius of curvature at the vertices <math>V_1,\, V_2</math> is: <math>\tfrac{b^2}{a}</math>
* The radius of curvature at the co-vertices <math>V_3,\, V_4</math> is: <math>\tfrac{a^2}{b}\ .</math>
The diagram shows an easy way to find the centers of curvature <math>C_1 = \left(a - \tfrac{b^2}{a}, 0\right),\, C_3 = \left(0, b - \tfrac{a^2}{b}\right)</math> at vertex <math>V_1</math> and co-vertex <math>V_3</math>, respectively:
# mark the auxiliary point <math>H = (a,\, b)</math> and draw the line segment <math>V_1 V_3\ ,</math>
# draw the line through <math>H</math>, which is perpendicular to the line <math>V_1 V_3\ ,</math>
# 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]]s.
=== Steiner generation ===
[[File:Ellipse-steiner-e.svg|250px|thumb|Ellipse: Steiner generation]]
[[File:Ellipse construction - parallelogram method.gif|200px|thumb|Ellipse: Steiner generation]]
The following method to construct single points of an ellipse relies on the [[Steiner conic|Steiner generation of a conic section]]:
: Given two [[pencil (mathematics)|pencils]] <math>B(U),\, B(V)</math> of lines at two points <math>U,\, V</math> (all lines containing <math>U</math> and <math>V</math>, respectively) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>, then the intersection points of corresponding lines form a non-degenerate projective conic section.
For the generation of points of the ellipse <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math> one uses the pencils at the vertices <math>V_1,\, V_2</math>. Let <math>P = (0,\, b)</math> be an upper co-vertex of the ellipse and <math>A = (-a,\, 2b),\, B = (a,\,2b)</math>.
<math>P</math> is the center of the rectangle <math>V_1,\, V_2,\, B,\, A</math>. The side  <math>\overline{AB}</math> of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal <math>AV_2</math> as direction onto the line segment  <math>\overline{V_1B}</math> 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 <math>V_1</math> and <math>V_2</math> needed. The intersection points of any two related lines   <math>V_1 B_i</math> and <math>V_2 A_i</math> are points of the uniquely defined ellipse. With help of the points <math>C_1,\, \dotsc</math> 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 ===
[[File:Ellipse as hypotrochoid.gif|right|300px|thumb|An ellipse (in red) as a special case of the [[hypotrochoid]] with&nbsp;''R''&nbsp;=&nbsp;2''r'']]
The ellipse is a special case of the [[hypotrochoid]] when <math>R = 2r</math>, as shown in the adjacent image. The special case of a moving circle with radius <math>r</math> inside a circle with radius <math>R = 2r</math> is called a [[Tusi couple]].
== Inscribed angles and three-point form ==
=== Circles ===
[[File:Inscribe-a-c.svg|thumb|Circle: inscribed angle theorem]]
A circle with equation <math>\left(x - x_\circ\right)^2 + \left(y - y_\circ\right)^2 = r^2</math> is uniquely determined by three points <math>\left(x_1, y_1\right),\; \left(x_2,\,y_2\right),\; \left(x_3,\, y_3\right)</math> not on a line. A simple way to determine the parameters <math>x_\circ,y_\circ,r</math> uses the ''[[inscribed angle theorem]]'' for circles:
: For four points <math>P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\,</math> (see diagram) the following statement is true:
: The four points are on a circle if and only if the angles at <math>P_3</math> and <math>P_4</math> 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 <math>y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2,</math>  one uses the quotient: <math display="block">\frac{1 + m_1 m_2}{m_2 - m_1} = \cot\theta\ .</math>
====Inscribed angle theorem for circles====
For four points  <math>P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\,</math> 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 <math>P_3</math> and <math>P_4</math> are equal. In terms of the angle measurement above, this means: <math display="block">
\frac{(x_4 - x_1)(x_4 - x_2) + (y_4 - y_1)(y_4 - y_2)}
{(y_4 - y_1)(x_4 - x_2) - (y_4 - y_2)(x_4 - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)}
{(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}.
</math>
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-collinear points <math>P_i = \left(x_i,\, y_i\right)</math>: <math display="block">
\frac{({\color{red}x} - x_1)({\color{red}x} - x_2) + ({\color{red}y} - y_1)({\color{red}y} - y_2)}
{({\color{red}y} - y_1)({\color{red}x} - x_2) - ({\color{red}y} - y_2)({\color{red}x} - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)}
{(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}.
</math>
For example, for <math>P_1 = (2,\, 0),\; P_2 = (0,\, 1),\; P_3 = (0,\,0)</math> the three-point equation is:
: <math>\frac{(x - 2)x + y(y - 1)}{yx - (y - 1)(x - 2)} = 0</math>, which can be rearranged to <math>(x - 1)^2 + \left(y - \tfrac{1}{2}\right)^2 = \tfrac{5}{4}\ .</math>
Using vectors, [[dot product]]s and [[determinant]]s this formula can be arranged more clearly, letting <math>\vec x = (x,\, y)</math>:
<math display="block">
\frac{\left({\color{red}\vec x} - \vec x_1\right) \cdot \left({\color{red}\vec x} - \vec x_2\right)}
{\det\left({\color{red}\vec x} - \vec x_1,{\color{red}\vec x} - \vec x_2\right)} =
\frac{\left(\vec x_3 - \vec x_1\right) \cdot \left(\vec x_3 - \vec x_2\right)}
{\det\left(\vec x_3 - \vec x_1, \vec x_3 - \vec x_2\right)}.
</math>
The center of the circle <math>\left(x_\circ,\, y_\circ\right)</math> satisfies:
<math display="block">\begin{bmatrix}
1 & \dfrac{y_1 - y_2}{x_1 - x_2} \\[2ex]
\dfrac{x_1 - x_3}{y_1 - y_3} & 1
\end{bmatrix}
\begin{bmatrix} x_\circ \\[1ex] y_\circ \end{bmatrix}
=
\begin{bmatrix}
\dfrac{x_1^2 - x_2^2 + y_1^2 - y_2^2}{2(x_1 - x_2)} \\[2ex]
\dfrac{y_1^2 - y_3^2 + x_1^2 - x_3^2}{2(y_1 - y_3)}
\end{bmatrix}.
</math>
The radius is the distance between any of the three points and the center.
<math display="block">
r = \sqrt{\left(x_1 - x_\circ\right)^2 + \left(y_1 - y_\circ\right)^2}
= \sqrt{\left(x_2 - x_\circ\right)^2 + \left(y_2 - y_\circ\right)^2}
= \sqrt{\left(x_3 - x_\circ\right)^2 + \left(y_3 - y_\circ\right)^2}.
</math>
=== Ellipses ===
This section considers the family of ellipses defined by equations <math>\tfrac{\left(x - x_\circ\right)^2}{a^2} +  \tfrac{\left(y - y_\circ\right)^2}{b^2} = 1</math> with a ''fixed'' eccentricity <math>e</math>. It is convenient to use the parameter:
<math display="block">{\color{blue}q} = \frac{a^2}{b^2} = \frac{1}{1 - e^2},</math>
and to write the ellipse equation as:
<math display="block">\left(x - x_\circ\right)^2 + {\color{blue}q}\, \left(y - y_\circ\right)^2 = a^2,</math>
where ''q'' is fixed and <math>x_\circ,\, y_\circ,\, a</math> vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if <math>q < 1</math>, the major axis is parallel to the ''x''-axis; if <math>q > 1</math>, it is parallel to the ''y''-axis.)
[[File:Inscribe-a-e.svg|thumb|Inscribed angle theorem for an ellipse]]
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'':<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf E. Hartmann: Lecture Note '<nowiki/>'''Planar Circle Geometries'''', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 55]</ref><ref>W. Benz, ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973)</ref>
: In order to measure an angle between two lines with equations <math>y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2</math> one uses the quotient: <math display="block">\frac{1 + {\color{blue}q}\; m_1 m_2}{m_2 - m_1}\ .</math>
====Inscribed angle theorem for ellipses====
: Given four points <math>P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4</math>, no three of them on a line (see diagram).
: The four points are on an ellipse with equation <math>(x - x_\circ)^2 + {\color{blue}q}\, (y - y_\circ)^2 = a^2</math> if and only if the angles at <math>P_3</math> and <math>P_4</math> are equal in the sense of the measurement above—that is, if <math display="block">
\frac{(x_4 - x_1)(x_4 - x_2) + {\color{blue}q}\;(y_4 - y_1)(y_4 - y_2)}
{(y_4 - y_1)(x_4 - x_2) - (y_4 - y_2)(x_4 - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + {\color{blue}q}\;(y_3 - y_1)(y_3 - y_2)}
{(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}\ .
</math>
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-collinear points <math>P_i = \left(x_i,\, y_i\right)</math>: <math display="block">
\frac{({\color{red}x} - x_1)({\color{red}x} - x_2) + {\color{blue}q}\;({\color{red}y} - y_1)({\color{red}y} - y_2)}
{({\color{red}y} - y_1)({\color{red}x} - x_2) - ({\color{red}y} - y_2)({\color{red}x} - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + {\color{blue}q}\;(y_3 - y_1)(y_3 - y_2)}
{(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}\ .
</math>
For example, for <math>P_1 = (2,\, 0),\; P_2 = (0,\,1),\; P_3 = (0,\, 0)</math> and <math>q = 4</math> one obtains the three-point form
: <math>\frac{(x - 2)x + 4y(y - 1)}{yx - (y - 1)(x - 2)} = 0</math> and after conversion <math>\frac{(x - 1)^2}{2} + \frac{\left(y - \frac{1}{2}\right)^2}{\frac{1}{2}} = 1.</math>
Analogously to the circle case, the equation can be written more clearly using vectors:
<math display="block">
\frac{\left({\color{red}\vec x} - \vec x_1\right)*\left({\color{red}\vec x} - \vec x_2\right)}
{\det\left({\color{red}\vec x} - \vec x_1,{\color{red}\vec x} - \vec x_2\right)}
= \frac{\left(\vec x_3 - \vec x_1\right)*\left(\vec x_3 - \vec x_2\right)}
{\det\left(\vec x_3 - \vec x_1, \vec x_3 - \vec x_2\right)},
</math>
where <math>*</math> is the modified [[dot product]] <math>\vec u*\vec v = u_x v_x + {\color{blue}q}\,u_y v_y.</math>
== Pole-polar relation ==
[[File:Ellipse-pol.svg|250px|thumb|Ellipse: pole-polar relation]]
Any ellipse can be described in a suitable coordinate system by an equation <math>\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1</math>. The equation of the tangent at a point <math>P_1 = \left(x_1,\, y_1\right)</math> of the ellipse is <math>\tfrac{x_1x}{a^2} + \tfrac{y_1y}{b^2} = 1.</math> If one allows point <math>P_1 = \left(x_1,\, y_1\right)</math> to be an arbitrary point different from the origin, then
: point <math>P_1 = \left(x_1,\, y_1\right) \neq (0,\, 0)</math> is mapped onto the line <math>\tfrac{x_1 x}{a^2} + \tfrac{y_1 y}{b^2} = 1</math>, not through the center of the ellipse.
This relation between points and lines is a [[bijection]].
The [[inverse function]] maps
* line <math>y = mx + d,\ d \ne 0</math> onto the point <math>\left(-\tfrac{ma^2}{d},\, \tfrac{b^2}{d}\right)</math> and
* line <math>x = c,\ c \ne 0</math> onto the point <math>\left(\tfrac{a^2}{c},\, 0\right).</math>
Such a relation between points and lines generated by a conic is called ''[[Pole and polar|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: {{nowrap|<math>P_1,\, p_1</math>).}}
* For a pole <math>P</math> ''outside'' the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing <math>P</math> (see diagram: {{nowrap|<math>P_2,\, p_2</math>).}}
* For a point ''within'' the ellipse, the polar has no point with the ellipse in common (see diagram: {{nowrap|<math>F_1,\, l_1</math>).}}
# The intersection point of two polars is the pole of the line through their poles.
# The foci <math>(c,\, 0)</math> and <math>(-c,\, 0)</math>, respectively, and the directrices <math>x = \tfrac{a^2}{c}</math> and <math>x = -\tfrac{a^2}{c}</math>, respectively, belong to pairs of pole and polar. Because they are even polar pairs with respect to the circle <math>x^2+y^2=a^2</math>, 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
{{NumBlk||<math display="block">\frac{x^2}{a^2}+\frac{y^2}{b^2}= 1 </math>|{{EquationRef|1}}}}
except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.({{EquationNote|1}}) will be given.
=== Area ===
The [[area]] <math>A_\text{ellipse}</math> enclosed by an ellipse is:
{{NumBlk||<math display="block">A_\text{ellipse} = \pi ab</math>|{{EquationRef|2}}}}
where <math>a</math> and <math>b</math> are the lengths of the semi-major and semi-minor axes, respectively.  The area formula <math>\pi a b</math> is intuitive: start with a circle of radius <math>b</math> (so its area is <math>\pi b^2</math>) and stretch it by a factor <math>a/b</math> to make an ellipse. This scales the area by the same factor: <math>\pi b^2(a/b) = \pi a b.</math><ref>{{Cite book|last=Archimedes.|url=https://www.worldcat.org/oclc/48876646|title=The works of Archimedes|date=1897|publisher=Dover Publications|others=Heath, Thomas Little, Sir, 1861-1940. | isbn=0-486-42084-1 | location=Mineola, N.Y.|pages=115|oclc=48876646}}</ref> However, using the same approach for the circumference would be fallacious – compare the [[Integral|integrals]] <math display="inline">\int f(x)\, dx</math> and <math display="inline"> \int \sqrt{1+f'^2(x)}\, dx</math>. It is also easy to rigorously prove the area formula using integration as follows.  Equation ({{EquationNote|1}}) can be rewritten as <math display="inline">y(x) = b \sqrt{1 - x^2 / a^2}.</math>  For <math>x\in[-a,a],</math> this curve is the top half of the ellipse.  So twice the integral of <math>y(x)</math> over the interval <math>[-a,a]</math> will be the area of the ellipse:
<math display="block">\begin{align}
A_\text{ellipse} &= \int_{-a}^a 2b\sqrt{1 - \frac{x^2}{a^2}}\,dx\\
&= \frac ba \int_{-a}^a 2\sqrt{a^2 - x^2}\,dx.
\end{align}</math>
The second integral is the area of a circle of radius <math>a,</math> that is, <math>\pi a^2.</math> So
<math display="block">A_\text{ellipse} = \frac{b}{a}\pi a^2 = \pi ab.</math>
An ellipse defined implicitly by <math>Ax^2+ Bxy + Cy^2 = 1 </math> has area <math>2\pi / \sqrt{4AC - B^2}.</math>
The area can also be expressed in terms of eccentricity and the length of the semi-major axis as <math>a^2\pi\sqrt{1-e^2}</math> (obtained by solving for [[flattening]], then computing the semi-minor axis).
[[Image:tiltedEllipse2.jpg|thumb|The area enclosed by a tilted ellipse is <math>\pi\; y_\text{int}\, x_\text{max}</math>.]]
So far we have dealt with ''erect'' ellipses, whose major and minor axes are parallel to the <math>x</math> and <math>y</math> 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
{{NumBlk||<math display="block">A_\text{ellipse} = \pi\; y_\text{int}\, x_\text{max} = \pi\; x_\text{int}\, y_\text{max}</math>|{{EquationRef|3}}}}
where <math>y_{\text{int}}</math>, <math>x_{\text{int}}</math> are intercepts and <math>x_{\text{max}}</math>, <math>y_{\text{max}}</math> are maximum values. It follows directly from [[Ellipse#Theorem of Apollonios on conjugate diameters|Apollonios's theorem]].
===Circumference===
{{further|Meridian arc#Quarter meridian}}
[[File:Ellipses same circumference.png|thumb|Ellipses with same circumference]]
The circumference <math>C</math> of an ellipse is:
<math display="block">C \,=\, 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta \,=\, 4 a \,E(e)</math>
where again <math>a</math> is the length of the semi-major axis, <math display="inline">e=\sqrt{1 - b^2/a^2}</math> is the eccentricity, and the function <math>E</math> is the [[complete elliptic integral of the second kind]],
<math display="block">E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</math>
which is in general not an [[elementary function]].
The circumference of the ellipse may be evaluated in terms of <math>E(e)</math> using [[arithmetic-geometric mean|Gauss's arithmetic-geometric mean]];<ref>{{dlmf|first=B. C.|last=Carlson|id=19.8.E6|title=Elliptic Integrals}}</ref> this is a quadratically converging iterative method (see [[Elliptic integral#Computation|here]] for details).
The exact [[infinite series]] is:
<math display="block">\begin{align}
\frac C{2\pi a} &= 1 - \left(\frac{1}{2}\right)^2e^2 - \left(\frac{1\cdot 3}{2\cdot 4}\right)^2\frac{e^4}{3} - \left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^2\frac{e^6}{5} - \cdots \\
&= 1 - \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{e^{2n}}{2n-1} \\
&= -\sum_{n=0}^\infty \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{e^{2n}}{2n-1},
\end{align}
</math>
where <math>n!!</math> is the [[double factorial]] (extended to negative odd integers in the usual way, giving <math>(-1)!! = 1</math> and <math>(-3)!! = -1</math>).
This series converges, but by expanding in terms of <math>h = (a-b)^2 / (a+b)^2,</math> [[James Ivory (mathematician)|James Ivory]],<ref>{{cite journal
|last = Ivory
|first = J.
|title = A new series for the rectification of the ellipsis
|author-link = James Ivory (mathematician)
|journal = Transactions of the Royal Society of Edinburgh
|year = 1798
|volume = 4
|issue = 2
|pages = 177{{ndash}}190
|url =https://books.google.com/books?id=FaUaqZZYYPAC&pg=PA177
|doi=10.1017/s0080456800030817
|s2cid = 251572677
}}</ref> [[Friedrich Wilhelm Bessel|Bessel]]<ref>{{cite journal
|ref = {{harvid|Bessel|1825}}
|last = Bessel
|first = F. W.
|title = The calculation of longitude and latitude from geodesic measurements (1825)
|author-link = Friedrich Bessel
|journal = [[Astron. Nachr.]]
|year = 2010
|volume = 331
|number = 8
|pages = 852{{ndash}}861
|arxiv = 0908.1824
|doi = 10.1002/asna.201011352
|bibcode = 2010AN....331..852K
|s2cid = 118760590
}} English translation of {{cite journal
|first1= F. W. | last1=Bessel
| doi=10.1002/asna.18260041601
| year=1825
| bibcode=1825AN......4..241B
|title=Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermesssungen
| journal=[[Astron. Nachr.]] |volume=4 |issue=16 |pages =241{{ndash}}254
|arxiv=0908.1823 | s2cid=118630614 | lang=de
}}</ref> and [[Ernst Kummer|Kummer]]<ref name=Linderholm95>{{cite journal
|title=An Overlooked Series for the Elliptic Perimeter
|first1=Carl E. |last1=Linderholm   |first2=Arthur C. |last2=Segal
|journal=Mathematics Magazine |volume=68 |issue=3 |pages=216–220
|date=June 1995
|doi=10.1080/0025570X.1995.11996318
}} which cites to {{cite journal
|last=Kummer |first=Ernst Eduard |author-link=Ernst Kummer
|title=Uber die Hypergeometrische Reihe |lang=de
|trans-title=About the hypergeometric series
|journal=[[Journal für die Reine und Angewandte Mathematik]]
|volume=15 |issue=1, 2 |year=1836 |pages=39–83, 127–172
|doi=10.1515/crll.1836.15.39
|url=https://archive.org/details/sim_journal-fuer-die-reine-und-angewandte-mathematik_1836_15
}}</ref> derived a series that converges much more rapidly.  It is most concisely written in terms of the [[Binomial coefficient#Binomial coefficient with n = 1/2|binomial coefficient with <math>n = 1/2</math>]]:
<math display="block">\begin{align}
\frac{C}{\pi(a+b)}
&= \sum_{n=0}^\infty {\frac 12 \choose n}^2 h^n \\
&= \sum_{n=0}^\infty \left(\frac{(2n-3)!!}{(2n)!!}\right)^2 h^n \\
&= \sum_{n=0}^\infty \left(\frac{(2n-3)!!}{2^n n!}\right)^2 h^n \\
&= \sum_{n=0}^\infty \left(\frac{1}{(2n-1)4^n}\binom{2n}{n}\right)^2 h^n \\
&= 1 + \frac{h}{4} + \frac{h^2}{64} + \frac{h^3}{256} + \frac{25\,h^4}{16384} + \frac{49\,h^5}{65536} + \frac{441\,h^6}{2^{20}} + \frac{1089\,h^7}{2^{22}} + \cdots.
\end{align}</math>
The coefficients are slightly smaller (by a factor of <math>2n-1</math>), but also <math>e^4/16 \le h \le e^4</math> is numerically much smaller than <math>e</math> except at <math>h = e = 0</math> and <math>h = e = 1</math>.  For eccentricities less than 0.5 {{nobr|(<math>h < 0.005</math>),}} the error is at the limits of [[double-precision floating-point]] after the <math>h^4</math> term.<ref name=Cook23>{{cite web
|title=Comparing approximations for ellipse perimeter
|date=28 May 2023
|first=John D. |last=Cook
|website=John D. Cook Consulting blog
|url=https://www.johndcook.com/blog/2023/05/28/approximate-ellipse-perimeter/
|access-date=2024-09-16
}}</ref>
[[Srinivasa Ramanujan]] gave two close [[approximations]] for the circumference in §16 of "Modular Equations and Approximations to <math>\pi</math>";<ref>{{cite journal
|last=Ramanujan |first=Srinivasa |author-link=Srinivasa Ramanujan
|title=Modular Equations and Approximations to ''π''
|journal = Quart. J. Pure App. Math.
|volume = 45
|pages = 350{{ndash}}372
|year = 1914
|url = http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf#page=24
|isbn = 978-0-8218-2076-6
}}</ref> they are
<math display="block">\frac C\pi \approx 3(a + b) - \sqrt{(3a + b)(a + 3b)}  = 3(a + b) - \sqrt{3(a+b)^2 + 4ab}</math>
and
<math display="block">\frac C{\pi(a+b)} \approx 1+\frac{3h}{10+\sqrt{4-3h}},</math>
where <math>h</math> takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order <math>h^3</math> and <math>h^5,</math> respectively.<ref name=Villarino>{{cite arxiv |eprint=math.CA/0506384 |title=Ramanujan’s Perimeter of an Ellipse |first=Mark B. |last=Villarino |date=20 June 2005 |quote=We present a detailed analysis of Ramanujan’s most accurate approximation to the perimeter of an ellipse.}}  In particular, the second equation underestimates the circumference by <math>\pi(a+b)h^5\theta(h),</math> where <math>22.888\cdot 10^{-6} < 3\cdot 2^{-17} < \theta(h) \le 4\left(1 - \frac{7\pi}{22}\right) < 1.60935\cdot10^{-3}</math> is an increasing function of <math>0 \le h \le 1.</math></ref><ref>{{cite web
|title=Error in Ramanujan’s approximation for ellipse perimeter
|date=22 September 2024
|first=John D. |last=Cook
|website=John D. Cook Consulting blog
|url=https://www.johndcook.com/blog/2024/09/22/ellipse-perimeter-approx/
|access-date=2024-12-01
|quote=the relative error when {{math|''b'' {{=}} 1}} and {{mvar|a}} varies ... is bound by {{math|4/''π'' − 14/11 {{=}} 0.00051227…}}.
}}</ref>  This is because the second formula's infinite series expansion matches Ivory's formula up to the <math>h^4</math> term.{{r|Villarino|p=3}}
===Arc length===
{{further|Meridian arc#Calculation}}
More generally, the [[arc length]] of a portion of the circumference, as a function of the angle subtended (or {{nobr|{{mvar|x}} 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
<math display="block"> y = b\ \sqrt{ 1-\frac{x^{2}}{a^{2}}\ } ~.</math>
Then the arc length <math>s</math> from <math>\ x_{1}\ </math> to <math>\ x_{2}\ </math> is:
<math display="block">s = -b\int_{\arccos \frac{x_1}{a}}^{\arccos \frac{x_2}{a}} \sqrt{\ 1 + \left( \tfrac{a^2}{b^2} - 1 \right)\ \sin^2 z ~} \; dz ~.</math>
This is equivalent to
<math display="block"> s = b\ \left[ \; E\left(z \;\Biggl|\; 1 - \frac{a^2}{b^2} \right) \; \right]^{\arccos \frac{x_1}{a}}_{z\ =\ \arccos \frac{x_2}{a}} </math>
where <math>E(z \mid m)</math> is the incomplete elliptic integral of the second kind with parameter <math>m=k^{2}.</math>
Some lower and upper bounds on the circumference of the canonical ellipse <math>\ x^2/a^2 + y^2/b^2 = 1\ </math> with <math>\ a \geq b\ </math> are<ref>{{cite journal |last1=Jameson |first1=G.J.O. |year=2014 |title=Inequalities for the perimeter of an ellipse |journal= Mathematical Gazette |volume=98 |issue=542 |pages=227–234 |doi=10.1017/S002555720000125X |s2cid=125063457}}</ref>
<math display="block">\begin{align}
2\pi b &\le C \le 2\pi a\ , \\
\pi (a+b) &\le C \le 4(a+b)\ , \\
4\sqrt{a^2+b^2\ } &\le C \le \sqrt{2\ } \pi \sqrt{a^2 + b^2\ } ~.
\end{align}</math>
Here the upper bound <math>\ 2\pi a\ </math> is the circumference of a [[circumscribed circle|circumscribed]] [[concentric circle]] passing through the endpoints of the ellipse's major axis, and the lower bound <math>4\sqrt{a^2+b^2}</math> is the perimeter of an [[inscribed figure|inscribed]] [[rhombus]] with [[vertex (geometry)|vertices]] at the endpoints of the major and the minor axes.
Given an ellipse whose axes are drawn, we can construct the endpoints of a particular elliptic arc whose length is one eighth of the ellipse's circumference using only [[Straightedge and compass construction|straightedge and compass]] in a finite number of steps; for some specific shapes of ellipses, such as when the axes have a length ratio of {{tmath|\sqrt2 : 1}}, it is additionally possible to construct the endpoints of a particular arc whose length is one twelfth of the circumference.<ref>{{Cite book |last1=Prasolov |first1=V. |last2=Solovyev|first2=Y.|title=Elliptic Functions and Elliptic Integrals|publisher=American Mathematical Society |year=1997 |isbn=0-8218-0587-8|page=58—60}}</ref> (The vertices and co-vertices are already endpoints of arcs whose length is one half or one quarter of the ellipse's circumference.) However, the general theory of straightedge-and-compass elliptic division appears to be unknown, unlike in [[Constructible polygon|the case of the circle]] and [[Lemniscate elliptic functions|the lemniscate]]. The division in special cases has been investigated by [[Adrien-Marie Legendre|Legendre]] in his classical treatise.<ref>Legendre's ''Traité des fonctions elliptiques et des intégrales eulériennes''</ref>
=== Curvature ===
The [[curvature]] is given by:
<math display="block">\kappa = \frac{1}{a^2 b^2}\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right)^{-\frac{3}{2}}\ ,</math>
and the [[Radius of curvature#Ellipses|radius of curvature]], ρ = 1/κ, at point <math>(x,y)</math>:
<math display="block">\rho = a^2 b^2 \left(\frac{x^{2}}{a^4} + \frac{y^{2}}{b^4}\right)^\frac{3}{2} = \frac{1}{a^4 b^4} \sqrt{\left(a^4 y^{2} + b^4 x^{2}\right)^3} \ .</math>The radius of curvature of an ellipse, as a function of angle ''{{mvar|θ}}'' from the center, is:
<math display="block">R(\theta)=\frac{a^2}{b}\biggl(\frac{1-e^2(2-e^2)(\cos\theta)^2)}{1-e^2(\cos\theta)^2}\biggr)^{3/2}\,,</math>where e is the eccentricity.
Radius of curvature at the two ''vertices'' <math>(\pm a,0)</math> and the centers of curvature:
<math display="block">\rho_0 = \frac{b^2}{a}=p\ , \qquad \left(\pm\frac{c^2}{a}\,\bigg|\,0\right)\ .</math>
Radius of curvature at the two ''co-vertices'' <math>(0,\pm b)</math> and the centers of curvature:
<math display="block">\rho_1 = \frac{a^2}{b}\ , \qquad \left(0\,\bigg|\,\pm\frac{c^2}{b}\right)\ .</math>The locus of all the centers of curvature is called an [[Evolute#Evolute of an ellipse|evolute]]. In the case of an ellipse, the evolute is an [[astroid]].
== In triangle geometry ==
Ellipses appear in triangle geometry as
# [[Steiner ellipse]]: ellipse through the vertices of the triangle with center at the centroid,
# [[inellipse]]s: ellipses which touch the sides of a triangle.  Special cases are the [[Steiner inellipse]] and the [[Mandart inellipse]].
== As plane sections of quadrics ==
Ellipses appear as plane sections of the following [[quadric]]s:
* [[Ellipsoid]]
* Elliptic cone
* Elliptic [[cylinder]]
* [[Hyperboloid of one sheet]]
* [[Hyperboloid of two sheets]]
<gallery>
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
</gallery>
== Applications ==
===Physics===
==== Elliptical reflectors and acoustics ====
{{See also|Fresnel zone}}
[[File: "Wave pattern of a little droplet dropped into mercury in one focus of the ellipse " by Weber Bros..jpg|thumb|Wave pattern of a little droplet dropped into mercury in the foci of the ellipse]]
If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after [[reflection (physics)|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]], 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]]), 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]] along a line of the paper; such mirrors are used in some [[image scanner|document scanner]]s.
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 [[cupola|vaulted roof]] shaped as a section of a prolate spheroid. Such a room is called a ''[[whisper chamber]]''. 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]] (where [[John Quincy Adams]] is said to have used this property for eavesdropping on political matters); the [[Mormon Tabernacle]] at [[Temple Square]] in [[Salt Lake City]], [[Utah]]; at an exhibit on sound at the [[Museum of Science and Industry (Chicago)|Museum of Science and Industry]] in [[Chicago]]; in front of the [[University of Illinois at Urbana–Champaign]] Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the [[Alhambra]].
==== Planetary orbits ====
{{Main|Elliptic orbit}}
In the 17th century, [[Johannes Kepler]] discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his [[Kepler's laws of planetary motion|first law of planetary motion]]. Later, [[Isaac Newton]] explained this as a corollary of his [[Newton's law of universal gravitation|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 [[Similarity (geometry)|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]] and [[quantum mechanics|quantum effects]], which become significant when the particles are moving at high speed.)
For [[elliptical orbit]]s, useful relations involving the eccentricity <math>e</math> are:
<math display="block">\begin{align}
e &= \frac{r_a - r_p}{r_a + r_p} = \frac{r_a - r_p}{2a} \\
r_a &= (1 + e)a \\
r_p &= (1 - e)a
\end{align}</math>
where
* <math>r_a</math> is the radius at [[apoapsis]], i.e., the farthest distance of the orbit to the [[barycenter]] of the system, which is a [[Focus (geometry)|focus]] of the ellipse
* <math>r_p</math> is the radius at [[periapsis]], the closest distance
* <math>a</math> is the length of the [[semi-major axis]]
Also, in terms of <math>r_a</math> and <math>r_p</math>, the semi-major axis <math>a</math> is their [[arithmetic mean]], the semi-minor axis <math>b</math> is their [[geometric mean]], and the [[conic section#Features|semi-latus rectum]] <math>\ell</math> is their [[harmonic mean]]. In other words,
<math display="block">\begin{align}
a &= \frac{r_a + r_p}{2} \\[2pt]
b &= \sqrt{r_a r_p} \\[2pt]
\ell &= \frac{2}{\frac{1}{r_a} + \frac{1}{r_p}} = \frac{2r_ar_p}{r_a + r_p}.
\end{align}</math>
==== Harmonic oscillators ====
The general solution for a [[harmonic oscillator]] in two or more [[dimension]]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 (mechanics)|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 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]] display is an ellipse, rather than a straight line, the two signals are out of phase.
==== Elliptical gears ====
Two [[non-circular gear]]s 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]] or [[toothed belt|timing belt]], or in the case of a bicycle the main [[chainwheel#ovoid chainwheels|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]] or [[torque]] 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]].
Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.<ref>David Drew.
"Elliptical Gears".
[http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Drew/Emat6890/Elliptical%20Gears.htm]
</ref>
An example gear application would be a device that winds thread onto a conical [[bobbin]] on a [[Spinning (textiles)|spinning]] machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.<ref>{{cite book |first=George B. |last=Grant |title=A treatise on gear wheels |url=https://books.google.com/books?id=fPoOAAAAYAAJ&pg=PA72 |year=1906 |publisher=Philadelphia Gear Works |page=72}}</ref>
==== Optics ====
* In a material that is optically [[anisotropic]] ([[birefringent]]), the [[refractive index]] 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-[[laser pumping|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).<ref>[http://www.rp-photonics.com/lamp_pumped_lasers.html Encyclopedia of Laser Physics and Technology - lamp-pumped lasers, arc lamps, flash lamps, high-power, Nd:YAG laser<!-- Bot generated title -->]</ref>
* In laser-plasma produced [[Extreme ultraviolet|EUV]] light sources used in microchip [[Extreme ultraviolet lithography|lithography]], 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.<ref>{{cite web |url=http://www.cymer.com/plasma_chamber_detail/ |title=Cymer - EUV Plasma Chamber Detail Category Home Page |access-date=2013-06-20 |url-status=dead |archive-url=https://web.archive.org/web/20130517100847/http://www.cymer.com/plasma_chamber_detail |archive-date=2013-05-17 }}</ref>
===Statistics and finance===
In [[statistics]], a bivariate [[random vector]] <math>(X, Y)</math> is [[elliptical distribution|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]]. 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.<ref>{{cite journal |author=Chamberlain, G. |title=A characterization of the distributions that imply mean—Variance utility functions |journal=[[Journal of Economic Theory]] |volume=29 |issue=1 |pages=185–201 |date=February 1983 |doi=10.1016/0022-0531(83)90129-1 }}</ref><ref>{{cite journal |author1=Owen, J. |author2=Rabinovitch, R. |title=On the class of elliptical distributions and their applications to the theory of portfolio choice |journal=[[Journal of Finance]] |volume=38 |issue= 3|pages=745–752 |date=June 1983 |jstor=2328079 |doi=10.1111/j.1540-6261.1983.tb02499.x}}</ref>
=== 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.<ref>{{cite journal |author=Pitteway, M.L.V. |title=Algorithm for drawing ellipses or hyperbolae with a digital plotter |journal=The Computer Journal |volume=10 |issue=3 |pages=282–9 |year=1967 |doi=10.1093/comjnl/10.3.282 |doi-access=free }}</ref> Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.<ref>{{cite journal |author=Van Aken, J.R. |title=An Efficient Ellipse-Drawing Algorithm |journal=IEEE Computer Graphics and Applications |volume=4 |issue=9 |pages=24–35 |date=September 1984 |doi=10.1109/MCG.1984.275994 |s2cid=18995215 }}</ref>
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.<ref>{{cite journal |author=Smith, L.B. |title=Drawing ellipses, hyperbolae or parabolae with a fixed number of points |journal=The Computer Journal |volume=14 |issue=1 |pages=81–86 |year=1971 |doi=10.1093/comjnl/14.1.81 |doi-access=free }}</ref> 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 curve]]s may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an [[affine transformation]] of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent [[Bézier curve]]s 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 ==
{{Portal|Solar System|Science|Mathematics|Astronomy|Biography|Technology}}
{{Div col|colwidth=30em}}
* [[Cartesian oval]], a generalization of the ellipse
* [[Circumconic and inconic]]
* [[Distance of closest approach of ellipses]]
* [[Ellipse fitting]]
* [[Elliptic coordinates]], an orthogonal coordinate system based on families of ellipses and [[hyperbola]]e
* [[Elliptic partial differential equation]]
* [[Elliptical distribution]], in statistics
* [[Elliptical dome]]
* [[Geodesics on an ellipsoid]]
* [[Great ellipse]]
* [[Kepler's laws of planetary motion]]
* [[n-ellipse|''n''-ellipse]], a generalization of the ellipse for ''n'' foci
* [[Oval]]
* [[Spheroid]], 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 anomaly|True]], [[eccentric anomaly|eccentric]], and [[mean anomaly]]
{{Div col end}}
==Notes==
{{Reflist|30em}}
== References ==
* {{cite book |first=W.H. |last=Besant | title=Conic Sections |chapter=Chapter III. The Ellipse
| publisher=George Bell and Sons| location=London | year=1907
|chapter-url=https://books.google.com/books?id=TRJLAAAAYAAJ&pg=PA50 |page=50 }}
* {{cite book |author=Coxeter, H.S.M. |title=Introduction to Geometry |url=https://archive.org/details/introductiontoge0002coxe |url-access=registration |publisher=Wiley |location=New York |year=1969 |pages=[https://archive.org/details/introductiontoge0002coxe/page/115 115–9] |edition=2nd}}
* {{citation|first=Bruce E.|last=Meserve|title=Fundamental Concepts of Geometry|year=1983|orig-year=1959|publisher=Dover Publications | isbn=978-0-486-63415-9}}
* {{cite book |author1=Miller, Charles D. |author2=Lial, Margaret L. |author3=Schneider, David I. |title=Fundamentals of College Algebra |publisher=Scott Foresman/Little |year=1990 |isbn=978-0-673-38638-0 |page=[https://archive.org/details/fundamentalsofco0000mill_g1q3/page/381 381] |edition=3rd |url=https://archive.org/details/fundamentalsofco0000mill_g1q3/page/381 }}
* {{ citation | last1 = Protter | first1 = Murray H. | last2 = Morrey | first2 = Charles B. Jr. | title = College Calculus with Analytic Geometry | edition = 2nd | location = Reading | publisher = [[Addison-Wesley]] | year = 1970 | lccn = 76087042 }}
== External links ==
* {{Wikiquote-inline}}
* {{Commons category-inline|Ellipses}}
* {{PlanetMath |urlname= ellipse|title= ellipse}}
* {{MathWorld |id=Ellipse |title=Ellipse}}
* {{MathWorld |id=Hypotrochoid|title=Ellipse as special case of hypotrochoid}}
* [https://web.archive.org/web/20070715063900/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=203 Apollonius' Derivation of the Ellipse] at Convergence
* [http://faculty.evansville.edu/ck6/ellipse.pdf ''The Shape and History of The Ellipse in Washington, D.C.''] by [[Clark Kimberling]]
* [http://www.fxsolver.com/solve/share/ON58ARMtP65D1khWt1uwUA==/ Ellipse circumference calculator]
* [http://www.mathopenref.com/tocs/ellipsetoc.html Collection of animated ellipse demonstrations]
* {{springer| title=Ellipse | id=Ellipse&oldid=11394 | last=Ivanov | first=A.B. }}
* [https://commons.wikimedia.org/wiki/File:01-Ellipsenzirkel-van_Schooten-3.svg#{{int:filedesc}} Trammel according Frans van Schooten]
* {{YouTube|id=5nW3nJhBHL0|title="Why is there no equation for the perimeter of an ellipse‽"}} by [[Matt Parker]]
{{Authority control}}
[[Category:Conic sections]]
[[Category:Plane curves]]
[[Category:Elementary shapes]]
[[Category:Algebraic curves]]