Ellipse

This article is about the geometric figure. For other uses, see Ellipse (disambiguation).

For the syntactic omission of words, see Ellipsis (linguistics). For the punctuation mark (…), see Ellipsis. For the exercise machine, see Elliptical trainer.

Not to be confused with eclipse.

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, 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 ${\displaystyle e}$, a number ranging from ${\displaystyle e=0}$ (the limiting case of a circle) to ${\displaystyle e=1}$ (the limiting case of infinite elongation, no longer an ellipse but a parabola).

An ellipse (red) obtained as the intersection of a cone with an inclined plane.

Ellipse: notations

Ellipses: examples with increasing eccentricity

An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.

Analytically, the equation of a standard ellipse centered at the origin with width ${\displaystyle 2a}$ and height ${\displaystyle 2b}$ is:

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}$

Assuming ${\displaystyle a\geq b}$, the foci are ${\displaystyle (\pm c,0)}$ for ${\textstyle c={\sqrt {a^{2}-b^{2}}}}$. The standard parametric equation is:

${\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .}$

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, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:

${\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.}$

Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–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 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 sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.

The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics.

Definition as locus of points

 

Ellipse: definition by sum of distances to foci

 

Ellipse: definition by focus and circular directrix

An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:

Given two fixed points ${\displaystyle F_{1},F_{2}}$  called the foci and a distance ${\displaystyle 2a}$  which is greater than the distance between the foci, the ellipse is the set of points ${\displaystyle P}$  such that the sum of the distances ${\displaystyle |PF_{1}|,\ |PF_{2}|}$  is equal to ${\displaystyle 2a}$ :${\displaystyle E=\left\{P\in \mathbb {R} ^{2}\,\mid \,|PF_{2}|+|PF_{1}|=2a\right\}\ .}$ 

The midpoint ${\displaystyle C}$  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 ${\displaystyle V_{1},V_{2}}$ , which have distance ${\displaystyle a}$  to the center. The distance ${\displaystyle c}$  of the foci to the center is called the focal distance or linear eccentricity. The quotient ${\displaystyle e={\tfrac {c}{a}}}$  is the eccentricity.

The case ${\displaystyle F_{1}=F_{2}}$  yields a circle and is included as a special type of ellipse.

The equation ${\displaystyle |PF_{2}|+|PF_{1}|=2a}$  can be viewed in a different way (see figure):

If ${\displaystyle c_{2}}$  is the circle with center ${\displaystyle F_{2}}$  and radius ${\displaystyle 2a}$ , then the distance of a point ${\displaystyle P}$  to the circle ${\displaystyle c_{2}}$  equals the distance to the focus ${\displaystyle F_{1}}$ :

${\displaystyle |PF_{1}|=|Pc_{2}|.}$ 

${\displaystyle c_{2}}$  is called the circular directrix (related to focus ${\displaystyle F_{2}}$ ) of the ellipse.^[1][2] 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

 

Shape parameters:

• a: semi-major axis,
• b: semi-minor axis,
• c: linear eccentricity,
• p: semi-latus rectum (usually ${\displaystyle \ell }$ ).

Standard equation

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:

the foci are the points ${\displaystyle F_{1}=(c,\,0),\ F_{2}=(-c,\,0)}$ ,

the vertices are ${\displaystyle V_{1}=(a,\,0),\ V_{2}=(-a,\,0)}$ .

For an arbitrary point ${\displaystyle (x,y)}$  the distance to the focus ${\displaystyle (c,0)}$  is ${\textstyle {\sqrt {(x-c)^{2}+y^{2}}}}$  and to the other focus ${\textstyle {\sqrt {(x+c)^{2}+y^{2}}}}$ . Hence the point ${\displaystyle (x,\,y)}$  is on the ellipse whenever:

${\displaystyle {\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a\ .}$ 

Removing the radicals by suitable squarings and using ${\displaystyle b^{2}=a^{2}-c^{2}}$  (see diagram) produces the standard equation of the ellipse:^[3]

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,}$ 

or, solved for y:

${\displaystyle y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.}$ 

The width and height parameters ${\displaystyle a,\;b}$  are called the semi-major and semi-minor axes. The top and bottom points ${\displaystyle V_{3}=(0,\,b),\;V_{4}=(0,\,-b)}$  are the co-vertices. The distances from a point ${\displaystyle (x,\,y)}$  on the ellipse to the left and right foci are ${\displaystyle a+ex}$  and ${\displaystyle a-ex}$ .

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 ${\displaystyle a}$  and ${\displaystyle b}$ , respectively, i.e. ${\displaystyle a\geq b>0\ .}$ 

In principle, the canonical ellipse equation ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$  may have ${\displaystyle a<b}$  (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 ${\displaystyle x}$  and ${\displaystyle y}$  and the parameter names ${\displaystyle a}$  and ${\displaystyle b.}$ 

Linear eccentricity

This is the distance from the center to a focus: ${\displaystyle c={\sqrt {a^{2}-b^{2}}}}$ .

Eccentricity

The eccentricity can be expressed as:

${\displaystyle e={\frac {c}{a}}={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}},}$ 

assuming ${\displaystyle a>b.}$  An ellipse with equal axes (${\displaystyle a=b}$ ) 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 ${\displaystyle \ell }$ . A calculation shows:

${\displaystyle \ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).}$ ^[4]

The semi-latus rectum ${\displaystyle \ell }$  is equal to the radius of curvature at the vertices (see section curvature).

Tangent

An arbitrary line ${\displaystyle g}$  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 ${\displaystyle (x_{1},\,y_{1})}$  of the ellipse ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$  has the coordinate equation:

${\displaystyle {\frac {x_{1}}{a^{2}}}x+{\frac {y_{1}}{b^{2}}}y=1.}$ 

A vector parametric equation of the tangent is:

${\displaystyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}\;\!-y_{1}a^{2}\\\;\ \ \ x_{1}b^{2}\end{pmatrix}}\ }$  with ${\displaystyle \ s\in \mathbb {R} \ .}$ 

Proof: Let ${\displaystyle (x_{1},\,y_{1})}$  be a point on an ellipse and ${\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}}$  be the equation of any line ${\displaystyle g}$  containing ${\displaystyle (x_{1},\,y_{1})}$ . Inserting the line's equation into the ellipse equation and respecting ${\displaystyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1}$  yields:

${\displaystyle {\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\ .}$ 

There are then cases:

1. ${\displaystyle {\frac {x_{1}}{a^{2}}}u+{\frac {y_{1}}{b^{2}}}v=0.}$  Then line ${\displaystyle g}$  and the ellipse have only point ${\displaystyle (x_{1},\,y_{1})}$  in common, and ${\displaystyle g}$  is a tangent. The tangent direction has perpendicular vector ${\displaystyle {\begin{pmatrix}{\frac {x_{1}}{a^{2}}}&{\frac {y_{1}}{b^{2}}}\end{pmatrix}}}$ , so the tangent line has equation ${\textstyle {\frac {x_{1}}{a^{2}}}x+{\tfrac {y_{1}}{b^{2}}}y=k}$  for some ${\displaystyle k}$ . Because ${\displaystyle (x_{1},\,y_{1})}$  is on the tangent and the ellipse, one obtains ${\displaystyle k=1}$ .
2. ${\displaystyle {\frac {x_{1}}{a^{2}}}u+{\frac {y_{1}}{b^{2}}}v\neq 0.}$  Then line ${\displaystyle g}$  has a second point in common with the ellipse, and is a secant.

Using (1) one finds that ${\displaystyle {\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}}$  is a tangent vector at point ${\displaystyle (x_{1},\,y_{1})}$ , which proves the vector equation.

If ${\displaystyle (x_{1},y_{1})}$  and ${\displaystyle (u,v)}$  are two points of the ellipse such that ${\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0}$ , then the points lie on two conjugate diameters (see below). (If ${\displaystyle a=b}$ , the ellipse is a circle and "conjugate" means "orthogonal".)

Shifted ellipse

If the standard ellipse is shifted to have center ${\displaystyle \left(x_{\circ },\,y_{\circ }\right)}$ , its equation is

${\displaystyle {\frac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\frac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1\ .}$ 

The axes are still parallel to the x- and y-axes.

General ellipse

Main article: Matrix representation of conic sections

In analytic geometry, the ellipse is defined as a quadric: the set of points ${\displaystyle (X,\,Y)}$  of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation^[5][6]

${\displaystyle AX^{2}+BXY+CY^{2}+DX+EY+F=0}$ 

provided ${\displaystyle B^{2}-4AC<0.}$ 

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant

${\displaystyle \Delta ={\begin{vmatrix}A&{\frac {1}{2}}B&{\frac {1}{2}}D\\{\frac {1}{2}}B&C&{\frac {1}{2}}E\\{\frac {1}{2}}D&{\frac {1}{2}}E&F\end{vmatrix}}=\left(AC-{\tfrac {1}{4}}B^{2}\right)F+{\tfrac {1}{4}}BED-{\tfrac {1}{4}}CD^{2}-{\tfrac {1}{4}}AE^{2}.}$ 

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.^[7]: p.63

The general equation's coefficients can be obtained from known semi-major axis ${\displaystyle a}$ , semi-minor axis ${\displaystyle b}$ , center coordinates ${\displaystyle \left(x_{\circ },\,y_{\circ }\right)}$ , and rotation angle ${\displaystyle \theta }$  (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:

${\displaystyle {\begin{aligned}A&=a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta \\[3mu]B&=2\left(b^{2}-a^{2}\right)\sin \theta \cos \theta \\[3mu]C&=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta \\[3mu]D&=-2Ax_{\circ }-By_{\circ }\\[3mu]E&=-Bx_{\circ }-2Cy_{\circ }\\[3mu]F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}}$ 

These expressions can be derived from the canonical equation

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$ 

by an affine transformation of the coordinates ${\displaystyle (x,\,y)}$ :

${\displaystyle {\begin{aligned}x&=\left(X-x_{\circ }\right)\cos \theta +\left(Y-y_{\circ }\right)\sin \theta \\y&=-\left(X-x_{\circ }\right)\sin \theta +\left(Y-y_{\circ }\right)\cos \theta .\end{aligned}}}$ 

Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:^{[citation needed]}

${\displaystyle {\begin{aligned}a,b&={\frac {-{\sqrt {2{\big (}AE^{2}+CD^{2}-BDE+(B^{2}-4AC)F{\big )}{\big (}(A+C)\pm {\sqrt {(A-C)^{2}+B^{2}}}{\big )}}}}{B^{2}-4AC}}\\x_{\circ }&={\frac {2CD-BE}{B^{2}-4AC}}\\[5mu]y_{\circ }&={\frac {2AE-BD}{B^{2}-4AC}}\\[5mu]\theta &={\begin{cases}\operatorname {arccot} {\dfrac {C-A-{\sqrt {(A-C)^{2}+B^{2}}}}{B}}&{\text{for }}B\neq 0\\[5mu]0&{\text{for }}B=0,\ A<C\\[10mu]90^{\circ }&{\text{for }}B=0,\ A>C\\\end{cases}}\end{aligned}}}$ 

Parametric representation

 

The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire

 

Ellipse points calculated by the rational representation with equal spaced parameters (${\displaystyle \Delta u=0.2}$ ).

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$  is:

${\displaystyle (x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \ .}$ 

The parameter t (called the eccentric anomaly in astronomy) is not the angle of ${\displaystyle (x(t),y(t))}$  with the x-axis, but has a geometric meaning due to Philippe de La Hire (see Drawing ellipses below).^[8]

Rational representation

With the substitution ${\textstyle u=\tan \left({\frac {t}{2}}\right)}$  and trigonometric formulae one obtains

${\displaystyle \cos t={\frac {1-u^{2}}{1+u^{2}}}\ ,\quad \sin t={\frac {2u}{1+u^{2}}}}$ 

and the rational parametric equation of an ellipse

${\displaystyle {\begin{aligned}x(u)&=a{\frac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)&=b{\frac {2u}{1+u^{2}}}\end{aligned}}\;,\quad -\infty <u<\infty \;,}$ 

which covers any point of the ellipse ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$  except the left vertex ${\displaystyle (-a,\,0)}$ .

For ${\displaystyle u\in [0,\,1],}$  this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing ${\displaystyle u.}$  The left vertex is the limit ${\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.}$ 

Alternately, if the parameter ${\displaystyle [u:v]}$  is considered to be a point on the real projective line ${\textstyle \mathbf {P} (\mathbf {R} )}$ , then the corresponding rational parametrization is

${\displaystyle [u:v]\mapsto \left(a{\frac {v^{2}-u^{2}}{v^{2}+u^{2}}},b{\frac {2uv}{v^{2}+u^{2}}}\right).}$ 

Then ${\textstyle [1:0]\mapsto (-a,\,0).}$ 

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 ${\displaystyle m}$  of the tangent at a point of the ellipse can be obtained from the derivative of the standard representation ${\displaystyle {\vec {x}}(t)=(a\cos t,\,b\sin t)^{\mathsf {T}}}$ :

${\displaystyle {\vec {x}}'(t)=(-a\sin t,\,b\cos t)^{\mathsf {T}}\quad \rightarrow \quad m=-{\frac {b}{a}}\cot t\quad \rightarrow \quad \cot t=-{\frac {ma}{b}}.}$ 

With help of trigonometric formulae one obtains:

${\displaystyle \cos t={\frac {\cot t}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {-ma}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\ ,\quad \quad \sin t={\frac {1}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {b}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}.}$ 

Replacing ${\displaystyle \cos t}$  and ${\displaystyle \sin t}$  of the standard representation yields:

${\displaystyle {\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}},\;{\frac {b^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\right),\,m\in \mathbb {R} .}$ 

Here ${\displaystyle m}$  is the slope of the tangent at the corresponding ellipse point, ${\displaystyle {\vec {c}}_{+}}$  is the upper and ${\displaystyle {\vec {c}}_{-}}$  the lower half of the ellipse. The vertices${\displaystyle (\pm a,\,0)}$ , having vertical tangents, are not covered by the representation.

The equation of the tangent at point ${\displaystyle {\vec {c}}_{\pm }(m)}$  has the form ${\displaystyle y=mx+n}$ . The still unknown ${\displaystyle n}$  can be determined by inserting the coordinates of the corresponding ellipse point ${\displaystyle {\vec {c}}_{\pm }(m)}$ :

${\displaystyle y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}\;.}$ 

This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

General ellipse

 

Ellipse as an affine image of the unit circle

Another definition of an ellipse uses affine transformations:

Any ellipse is an affine image of the unit circle with equation ${\displaystyle x^{2}+y^{2}=1}$ .

Parametric representation

An affine transformation of the Euclidean plane has the form ${\displaystyle {\vec {x}}\mapsto {\vec {f}}\!_{0}+A{\vec {x}}}$ , where ${\displaystyle A}$  is a regular matrix (with non-zero determinant) and ${\displaystyle {\vec {f}}\!_{0}}$  is an arbitrary vector. If ${\displaystyle {\vec {f}}\!_{1},{\vec {f}}\!_{2}}$  are the column vectors of the matrix ${\displaystyle A}$ , the unit circle ${\displaystyle (\cos(t),\sin(t))}$ , ${\displaystyle 0\leq t\leq 2\pi }$ , is mapped onto the ellipse:

${\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}\!_{0}+{\vec {f}}\!_{1}\cos t+{\vec {f}}\!_{2}\sin t\ .}$ 

Here ${\displaystyle {\vec {f}}\!_{0}}$  is the center and ${\displaystyle {\vec {f}}\!_{1},\;{\vec {f}}\!_{2}}$  are the directions of two conjugate diameters, in general not perpendicular.

Vertices

The four vertices of the ellipse are ${\displaystyle {\vec {p}}(t_{0}),\;{\vec {p}}\left(t_{0}\pm {\tfrac {\pi }{2}}\right),\;{\vec {p}}\left(t_{0}+\pi \right)}$ , for a parameter ${\displaystyle t=t_{0}}$  defined by:

${\displaystyle \cot(2t_{0})={\frac {{\vec {f}}\!_{1}^{\,2}-{\vec {f}}\!_{2}^{\,2}}{2{\vec {f}}\!_{1}\cdot {\vec {f}}\!_{2}}}.}$ 

(If ${\displaystyle {\vec {f}}\!_{1}\cdot {\vec {f}}\!_{2}=0}$ , then ${\displaystyle t_{0}=0}$ .) This is derived as follows. The tangent vector at point ${\displaystyle {\vec {p}}(t)}$  is:

${\displaystyle {\vec {p}}\,'(t)=-{\vec {f}}\!_{1}\sin t+{\vec {f}}\!_{2}\cos t\ .}$ 

At a vertex parameter ${\displaystyle t=t_{0}}$ , the tangent is perpendicular to the major/minor axes, so:

${\displaystyle 0={\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}\!_{0}\right)=\left(-{\vec {f}}\!_{1}\sin t+{\vec {f}}\!_{2}\cos t\right)\cdot \left({\vec {f}}\!_{1}\cos t+{\vec {f}}\!_{2}\sin t\right).}$ 

Expanding and applying the identities ${\displaystyle \;\cos ^{2}t-\sin ^{2}t=\cos 2t,\ \ 2\sin t\cos t=\sin 2t\;}$  gives the equation for ${\displaystyle t=t_{0}\;.}$ 

Area

From Apollonios theorem (see below) one obtains:
The area of an ellipse ${\displaystyle \;{\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t\;}$  is

${\displaystyle A=\pi |\det({\vec {f}}_{1},{\vec {f}}_{2})|\ .}$ 

Semiaxes

With the abbreviations ${\displaystyle \;M={\vec {f}}_{1}^{2}+{\vec {f}}_{2}^{2},\ N=\left|\det({\vec {f}}_{1},{\vec {f}}_{2})\right|}$  the statements of Apollonios's theorem can be written as:

${\displaystyle a^{2}+b^{2}=M,\quad ab=N\ .}$ 

Solving this nonlinear system for ${\displaystyle a,b}$  yields the semiaxes:

${\displaystyle a={\frac {1}{2}}({\sqrt {M+2N}}+{\sqrt {M-2N}})}$ 

${\displaystyle b={\frac {1}{2}}({\sqrt {M+2N}}-{\sqrt {M-2N}})\ .}$ 

Implicit representation

Solving the parametric representation for ${\displaystyle \;\cos t,\sin t\;}$  by Cramer's rule and using ${\displaystyle \;\cos ^{2}t+\sin ^{2}t-1=0\;}$ , one obtains the implicit representation

${\displaystyle \det({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2})^{2}+\det({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0})^{2}-\det({\vec {f}}\!_{1},{\vec {f}}\!_{2})^{2}=0}$ .

Conversely: If the equation

${\displaystyle x^{2}+2cxy+d^{2}y^{2}-e^{2}=0\ ,}$  with ${\displaystyle \;d^{2}-c^{2}>0\;,}$ 

of an ellipse centered at the origin is given, then the two vectors

${\displaystyle {\vec {f}}_{1}={e \choose 0},\quad {\vec {f}}_{2}={\frac {e}{\sqrt {d^{2}-c^{2}}}}{-c \choose 1}\ }$ 

point to two conjugate points and the tools developed above are applicable.

Example: For the ellipse with equation ${\displaystyle \;x^{2}+2xy+3y^{2}-1=0\;}$  the vectors are

${\displaystyle {\vec {f}}_{1}={1 \choose 0},\quad {\vec {f}}_{2}={\frac {1}{\sqrt {2}}}{-1 \choose 1}}$ .

 

Whirls: nested, scaled and rotated ellipses. The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.

Rotated Standard ellipse

For ${\displaystyle {\vec {f}}_{0}={0 \choose 0},\;{\vec {f}}_{1}=a{\cos \theta \choose \sin \theta },\;{\vec {f}}_{2}=b{-\sin \theta \choose \;\cos \theta }}$  one obtains a parametric representation of the standard ellipse rotated by angle ${\displaystyle \theta }$ :

${\displaystyle x=x_{\theta }(t)=a\cos \theta \cos t-b\sin \theta \sin t\ ,}$ 

${\displaystyle y=y_{\theta }(t)=a\sin \theta \cos t+b\cos \theta \sin t\ .}$ 

Ellipse in space

The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows ${\displaystyle {\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}}$  to be vectors in space.

Polar forms

Polar form relative to center

 

Polar coordinates centered at the center.

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate ${\displaystyle \theta }$  measured from the major axis, the ellipse's equation is^{[7]: p. 75}

${\displaystyle r(\theta )={\frac {ab}{\sqrt {(b\cos \theta )^{2}+(a\sin \theta )^{2}}}}={\frac {b}{\sqrt {1-(e\cos \theta )^{2}}}}}$ 

where ${\displaystyle e}$  is the eccentricity, not Euler's number

Polar form relative to focus

 

Polar coordinates centered at focus.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate ${\displaystyle \theta =0}$  still measured from the major axis, the ellipse's equation is

${\displaystyle r(\theta )={\frac {a(1-e^{2})}{1\pm e\cos \theta }}}$ 

where the sign in the denominator is negative if the reference direction ${\displaystyle \theta =0}$  points towards the center (as illustrated on the right), and positive if that direction points away from the center.

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate ${\displaystyle \phi }$ , the polar form is

${\displaystyle r(\theta )={\frac {a(1-e^{2})}{1-e\cos(\theta -\phi )}}.}$ 

The angle ${\displaystyle \theta }$  in these formulas is called the true anomaly of the point. The numerator of these formulas is the semi-latus rectum ${\displaystyle \ell =a(1-e^{2})}$ .

Eccentricity and the directrix property

 

Ellipse: directrix property

Each of the two lines parallel to the minor axis, and at a distance of ${\textstyle d={\frac {a^{2}}{c}}={\frac {a}{e}}}$  from it, is called a directrix of the ellipse (see diagram).

For an arbitrary point ${\displaystyle P}$  of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:

${\displaystyle {\frac {\left|PF_{1}\right|}{\left|Pl_{1}\right|}}={\frac {\left|PF_{2}\right|}{\left|Pl_{2}\right|}}=e={\frac {c}{a}}\ .}$ 

The proof for the pair ${\displaystyle F_{1},l_{1}}$  follows from the fact that ${\displaystyle \left|PF_{1}\right|^{2}=(x-c)^{2}+y^{2},\ \left|Pl_{1}\right|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}}$  and ${\displaystyle y^{2}=b^{2}-{\tfrac {b^{2}}{a^{2}}}x^{2}}$  satisfy the equation

${\displaystyle \left|PF_{1}\right|^{2}-{\frac {c^{2}}{a^{2}}}\left|Pl_{1}\right|^{2}=0\ .}$ 

The second case is proven analogously.

The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):

For any point ${\displaystyle F}$  (focus), any line ${\displaystyle l}$  (directrix) not through ${\displaystyle F}$ , and any real number ${\displaystyle e}$  with ${\displaystyle 0<e<1,}$  the ellipse is the locus of points for which the quotient of the distances to the point and to the line is ${\displaystyle e,}$  that is:

${\displaystyle E=\left\{P\ \left|\ {\frac {|PF|}{|Pl|}}=e\right.\right\}.}$ 

The extension to ${\displaystyle e=0}$ , which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the line at infinity in the projective plane.

(The choice ${\displaystyle e=1}$  yields a parabola, and if ${\displaystyle e>1}$ , a hyperbola.)

 

Pencil of conics with a common vertex and common semi-latus rectum

Proof

Let ${\displaystyle F=(f,\,0),\ e>0}$ , and assume ${\displaystyle (0,\,0)}$  is a point on the curve. The directrix ${\displaystyle l}$  has equation ${\displaystyle x=-{\tfrac {f}{e}}}$ . With ${\displaystyle P=(x,\,y)}$ , the relation ${\displaystyle |PF|^{2}=e^{2}|Pl|^{2}}$  produces the equations

${\displaystyle (x-f)^{2}+y^{2}=e^{2}\left(x+{\frac {f}{e}}\right)^{2}=(ex+f)^{2}}$  and ${\displaystyle x^{2}\left(e^{2}-1\right)+2xf(1+e)-y^{2}=0.}$ 

The substitution ${\displaystyle p=f(1+e)}$  yields

${\displaystyle x^{2}\left(e^{2}-1\right)+2px-y^{2}=0.}$ 

This is the equation of an ellipse (${\displaystyle e<1}$ ), or a parabola (${\displaystyle e=1}$ ), or a hyperbola (${\displaystyle e>1}$ ). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).

If ${\displaystyle e<1}$ , introduce new parameters ${\displaystyle a,\,b}$  so that ${\displaystyle 1-e^{2}={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}}$ , and then the equation above becomes

${\displaystyle {\frac {(x-a)^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\ ,}$ 

which is the equation of an ellipse with center ${\displaystyle (a,\,0)}$ , the x-axis as major axis, and the major/minor semi axis ${\displaystyle a,\,b}$ .

 

Construction of a directrix

Construction of a directrix

Because of ${\displaystyle c\cdot {\tfrac {a^{2}}{c}}=a^{2}}$  point ${\displaystyle L_{1}}$  of directrix ${\displaystyle l_{1}}$  (see diagram) and focus ${\displaystyle F_{1}}$  are inverse with respect to the circle inversion at circle ${\displaystyle x^{2}+y^{2}=a^{2}}$  (in diagram green). Hence ${\displaystyle L_{1}}$  can be constructed as shown in the diagram. Directrix ${\displaystyle l_{1}}$  is the perpendicular to the main axis at point ${\displaystyle L_{1}}$ .

General ellipse

If the focus is ${\displaystyle F=\left(f_{1},\,f_{2}\right)}$  and the directrix ${\displaystyle ux+vy+w=0}$ , one obtains the equation

${\displaystyle \left(x-f_{1}\right)^{2}+\left(y-f_{2}\right)^{2}=e^{2}{\frac {\left(ux+vy+w\right)^{2}}{u^{2}+v^{2}}}\ .}$ 

(The right side of the equation uses the Hesse normal form of a line to calculate the distance ${\displaystyle |Pl|}$ .)

Focus-to-focus reflection property

 

Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.

 

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 ${\displaystyle P}$  bisects the angle between the lines ${\displaystyle {\overline {PF_{1}}},\,{\overline {PF_{2}}}}$ .

Proof

Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too.

Let ${\displaystyle L}$  be the point on the line ${\displaystyle {\overline {PF_{2}}}}$  with the distance ${\displaystyle 2a}$  to the focus ${\displaystyle F_{2}}$ , ${\displaystyle a}$  is the semi-major axis of the ellipse. Let line ${\displaystyle w}$  be the bisector of the supplementary angle to the angle between the lines ${\displaystyle {\overline {PF_{1}}},\,{\overline {PF_{2}}}}$ . In order to prove that ${\displaystyle w}$  is the tangent line at point ${\displaystyle P}$ , one checks that any point ${\displaystyle Q}$  on line ${\displaystyle w}$  which is different from ${\displaystyle P}$  cannot be on the ellipse. Hence ${\displaystyle w}$  has only point ${\displaystyle P}$  in common with the ellipse and is, therefore, the tangent at point ${\displaystyle P}$ .

From the diagram and the triangle inequality one recognizes that ${\displaystyle 2a=\left|LF_{2}\right|<\left|QF_{2}\right|+\left|QL\right|=\left|QF_{2}\right|+\left|QF_{1}\right|}$  holds, which means: ${\displaystyle \left|QF_{2}\right|+\left|QF_{1}\right|>2a}$  . The equality ${\displaystyle \left|QL\right|=\left|QF_{1}\right|}$  is true from the Angle bisector theorem because ${\displaystyle {\frac {\left|PL\right|}{\left|PF_{1}\right|}}={\frac {\left|QL\right|}{\left|QF_{1}\right|}}}$  and ${\displaystyle \left|PL\right|=\left|PF_{1}\right|}$  . But if ${\displaystyle Q}$  is a point of the ellipse, the sum should be ${\displaystyle 2a}$ .

Application

The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).

Conjugate diameters

Definition of conjugate diameters

 

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 article: 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 ${\displaystyle d_{1},\,d_{2}}$  of an ellipse are conjugate if the midpoints of chords parallel to ${\displaystyle d_{1}}$  lie on ${\displaystyle d_{2}\ .}$ 

From the diagram one finds:

Two diameters ${\displaystyle {\overline {P_{1}Q_{1}}},\,{\overline {P_{2}Q_{2}}}}$  of an ellipse are conjugate whenever the tangents at ${\displaystyle P_{1}}$  and ${\displaystyle Q_{1}}$  are parallel to ${\displaystyle {\overline {P_{2}Q_{2}}}}$ .

Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.

In the parametric equation for a general ellipse given above,

${\displaystyle {\vec {x}}={\vec {p}}(t)={\vec {f}}\!_{0}+{\vec {f}}\!_{1}\cos t+{\vec {f}}\!_{2}\sin t,}$ 

any pair of points ${\displaystyle {\vec {p}}(t),\ {\vec {p}}(t+\pi )}$  belong to a diameter, and the pair ${\displaystyle {\vec {p}}\left(t+{\tfrac {\pi }{2}}\right),\ {\vec {p}}\left(t-{\tfrac {\pi }{2}}\right)}$  belong to its conjugate diameter.

For the common parametric representation ${\displaystyle (a\cos t,b\sin t)}$  of the ellipse with equation ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$  one gets: The points

${\displaystyle (x_{1},y_{1})=(\pm a\cos t,\pm b\sin t)\quad }$  (signs: (+,+) or (-,-) )

${\displaystyle (x_{2},y_{2})=({\color {red}{\mp }}a\sin t,\pm b\cos t)\quad }$  (signs: (-,+) or (+,-) )

are conjugate and

${\displaystyle {\frac {x_{1}x_{2}}{a^{2}}}+{\frac {y_{1}y_{2}}{b^{2}}}=0\ .}$ 

In case of a circle the last equation collapses to ${\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\ .}$ 

Theorem of Apollonios on conjugate diameters

 

Theorem of Apollonios

 

For the alternative area formula

For an ellipse with semi-axes ${\displaystyle a,\,b}$  the following is true:^[9][10]

Let ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  be halves of two conjugate diameters (see diagram) then

1. ${\displaystyle c_{1}^{2}+c_{2}^{2}=a^{2}+b^{2}}$ .
2. The triangle ${\displaystyle O,P_{1},P_{2}}$  with sides ${\displaystyle c_{1},\,c_{2}}$  (see diagram) has the constant area ${\textstyle A_{\Delta }={\frac {1}{2}}ab}$ , which can be expressed by ${\displaystyle A_{\Delta }={\tfrac {1}{2}}c_{2}d_{1}={\tfrac {1}{2}}c_{1}c_{2}\sin \alpha }$ , too. ${\displaystyle d_{1}}$  is the altitude of point ${\displaystyle P_{1}}$

March 24, 2023