Ellipse

In Cartesian coordinates

Standard equation

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

• the foci are the points ${\displaystyle F_1=(c,0),F_2=(-c,0)}$,
• 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 $\sqrt{(x-c{)}^2+y^2}$ and to the other focus $\sqrt{(x+c{)}^2+y^2}$. 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: $${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,}$$ or, solved for y: $${\displaystyle y=\pm \frac{b}{a}\sqrt{\phantom{\left(a^2-x^2\right)}{a}^2-x^2}=\pm \sqrt{\left(a^2-x^2\right)\left(1-e^2\right)}.}$$

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. ${\displaystyle a\ge b>0.}$

In principle, the canonical ellipse equation ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ 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: $${\displaystyle \ell =\frac{b^2}{a}=a\left(1-e^2\right).}$$

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 ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ 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: $${\displaystyle \overrightarrow{x}=\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)+s\left(\begin{array}{r}-y_1{a}^2\\ x_1{b}^2\end{array}\right),s\in \mathbb{R}.}$$

Proof: Let ${\displaystyle (x_1,y_1)}$ be a point on an ellipse and $\overrightarrow{x}=\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)+s\left(\begin{array}{c}u\\ v\end{array}\right)$ 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 $\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1$ yields: $${\displaystyle \frac{{\left(x_1+su\right)}^2}{a^2}+\frac{{\left(y_1+sv\right)}^2}{b^2}=1⟹2s\left(\frac{x_{1}u}{a^2}+\frac{y_{1}v}{b^2}\right)+s^2\left(\frac{u^2}{a^2}+\frac{v^2}{b^2}\right)=0.}$$ 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 ${\displaystyle \left(\begin{array}{cc}\frac{x_1}{a^2}& \frac{y_1}{b^2}\end{array}\right)}$, so the tangent line has equation $\frac{x_1}{a^2}x+\frac{y_1}{b^2}y=k$ for some ${\displaystyle k}$. Because ${\displaystyle (x_1,y_1)}$ is on the tangent and the ellipse, one obtains ${\displaystyle k=1}$.
2. ${\displaystyle \frac{x_1}{a^2}u+\frac{y_1}{b^2}v\ne 0.}$ Then line ${\displaystyle g}$ has a second point in common with the ellipse, and is a secant.

Using (1) one finds that ${\displaystyle \left(\begin{array}{cc}-y_1{a}^2& x_1{b}^2\end{array}\right)}$ 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 $\frac{x_{1}u}{a^2}+\frac{y_{1}v}{b^2}=0$, 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

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 $${\displaystyle A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0}$$ provided ${\displaystyle B^2-4AC<0.}$

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant $${\displaystyle \mathrm{\Delta }=\left|\begin{array}{ccc}A& \frac{1}{2}B& \frac{1}{2}D\\ \frac{1}{2}B& C& \frac{1}{2}E\\ \frac{1}{2}D& \frac{1}{2}E& F\end{array}\right|=ACF+\frac{1}{4}BDE-\frac{1}{4}(A{E}^2+C{D}^2+F{B}^2).}$$

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.

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: $${\displaystyle \begin{array}{cccc}A& =a^2{\sin }^2\theta +b^2{\cos }^2\theta & B& =2\left(b^2-a^2\right)\sin \theta \cos \theta \\ C& =a^2{\cos }^2\theta +b^2{\sin }^2\theta & D& =-2A{x}_{\circ }-B{y}_{\circ }\\ E& =-B{x}_{\circ }-2C{y}_{\circ }& F& =A{x}_{\circ }^2+B{x}_{\circ }{y}_{\circ }+C{y}_{\circ }^2-a^2{b}^2.\end{array}}$$

These expressions can be derived from the canonical equation $${\displaystyle \frac{X^2}{a^2}+\frac{Y^2}{b^2}=1}$$ by a Euclidean transformation of the coordinates ${\displaystyle (X,Y)}$: $${\displaystyle \begin{array}{cc}X& =\left(x-x_{\circ }\right)\cos \theta +\left(y-y_{\circ }\right)\sin \theta ,\\ Y& =-\left(x-x_{\circ }\right)\sin \theta +\left(y-y_{\circ }\right)\cos \theta .\end{array}}$$

Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:

$${\displaystyle \begin{array}{cc}a,b& =\frac{-\sqrt{2(A{E}^2+C{D}^2-BDE+(B^2-4AC)F)((A+C)\pm \sqrt{(A-C{)}^2+B^2})}}{B^2-4AC},\\ x_{\circ }& =\frac{2CD-BE}{B^2-4AC},\\ y_{\circ }& =\frac{2AE-BD}{B^2-4AC},\\ \theta & =\frac{1}{2}\mathrm{atan2}(-B,C-A),\end{array}}$$

where atan2 is the 2-argument arctangent function.

Parametric representation

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ is: $${\displaystyle (x,y)=(a\cos t,b\sin t),0\le t<2\pi .}$$

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).

Rational representation

With the substitution $u=\tan \left(\frac{t}{2}\right)$ and trigonometric formulae one obtains $${\displaystyle \cos t=\frac{1-u^2}{1+u^2},\sin t=\frac{2u}{1+u^2}}$$

and the rational parametric equation of an ellipse $${\displaystyle \{\begin{array}{l}x(u)=a{\displaystyle \frac{1-u^2}{1+u^2}}\\ y(u)=b{\displaystyle \frac{2u}{1+u^2}}\\ -\mathrm{\infty }<u<\mathrm{\infty }\end{array}}$$

which covers any point of the ellipse ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ 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 $\underset{u\to \pm \mathrm{\infty }}{lim}(x(u),y(u))=(-a,0).$

Alternately, if the parameter ${\displaystyle [u:v]}$ is considered to be a point on the real projective line $\mathbf{P}(\mathbf{R})$, 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 $[1:0]\mapsto (-a,0).$

Rational representations of conic sections are commonly used in computer-aided design (see Bézier 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 ${\displaystyle \overrightarrow{x}(t)=(a\cos t,b\sin t)}$: $${\displaystyle {\overrightarrow{x}}^{\prime }(t)=(-a\sin t,b\cos t)\to m=-\frac{b}{a}\cot t\to \cot t=-\frac{ma}{b}.}$$

With help of trigonometric formulae one obtains: $${\displaystyle \cos t=\frac{\cot t}{\pm \sqrt{1+{\cot }^{2}t}}=\frac{-ma}{\pm \sqrt{m^2{a}^2+b^2}},\sin t=\frac{1}{\pm \sqrt{1+{\cot }^{2}t}}=\frac{b}{\pm \sqrt{m^2{a}^2+b^2}}.}$$

Replacing ${\displaystyle \cos t}$ and ${\displaystyle \sin t}$ of the standard representation yields: $${\displaystyle {\overrightarrow{c}}_{\pm }(m)=\left(-\frac{m{a}^2}{\pm \sqrt{m^2{a}^2+b^2}},\frac{b^2}{\pm \sqrt{m^2{a}^2+b^2}}\right),m\in \mathbb{R}.}$$

Here ${\displaystyle m}$ is the slope of the tangent at the corresponding ellipse point, ${\displaystyle {\overrightarrow{c}}_{+}}$ is the upper and ${\displaystyle {\overrightarrow{c}}_{-}}$ 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 ${\displaystyle {\overrightarrow{c}}_{\pm }(m)}$ has the form ${\displaystyle y=mx+n}$. The still unknown ${\displaystyle n}$ can be determined by inserting the coordinates of the corresponding ellipse point ${\displaystyle {\overrightarrow{c}}_{\pm }(m)}$: $${\displaystyle y=mx\pm \sqrt{m^2{a}^2+b^2}.}$$

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

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 ${\displaystyle \overrightarrow{x}\mapsto \overrightarrow{f}{}_0+A\overrightarrow{x}}$, where ${\displaystyle A}$ is a regular matrix (with non-zero determinant) and ${\displaystyle \overrightarrow{f}{}_0}$ is an arbitrary vector. If ${\displaystyle \overrightarrow{f}{}_1,\overrightarrow{f}{}_2}$ are the column vectors of the matrix ${\displaystyle A}$, the unit circle ${\displaystyle (\cos (t),\sin (t))}$, ${\displaystyle 0\le t\le 2\pi }$, is mapped onto the ellipse: $${\displaystyle \overrightarrow{x}=\overrightarrow{p}(t)=\overrightarrow{f}{}_0+\overrightarrow{f}{}_1\cos t+\overrightarrow{f}{}_2\sin t.}$$

Here ${\displaystyle \overrightarrow{f}{}_0}$ is the center and ${\displaystyle \overrightarrow{f}{}_1,\overrightarrow{f}{}_2}$ are the directions of two conjugate diameters, in general not perpendicular.

Vertices

The four vertices of the ellipse are ${\displaystyle \overrightarrow{p}(t_0),\overrightarrow{p}\left(t_0\pm \frac{\pi }{2}\right),\overrightarrow{p}\left(t_0+\pi \right)}$, for a parameter ${\displaystyle t=t_0}$ defined by: $${\displaystyle \cot (2t_0)=\frac{\overrightarrow{f}{}_1^2-\overrightarrow{f}{}_2^2}{2\overrightarrow{f}{}_1\cdot \overrightarrow{f}{}_2}.}$$

(If ${\displaystyle \overrightarrow{f}{}_1\cdot \overrightarrow{f}{}_2=0}$, then ${\displaystyle t_0=0}$.) This is derived as follows. The tangent vector at point ${\displaystyle \overrightarrow{p}(t)}$ is: $${\displaystyle \overrightarrow{p}{}^{\prime }(t)=-\overrightarrow{f}{}_1\sin t+\overrightarrow{f}{}_2\cos t.}$$

At a vertex parameter ${\displaystyle t=t_0}$, the tangent is perpendicular to the major/minor axes, so: $${\displaystyle 0={\overrightarrow{p}}^{\prime }(t)\cdot \left(\overrightarrow{p}(t)-\overrightarrow{f}{}_0\right)=\left(-\overrightarrow{f}{}_1\sin t+\overrightarrow{f}{}_2\cos t\right)\cdot \left(\overrightarrow{f}{}_1\cos t+\overrightarrow{f}{}_2\sin t\right).}$$

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 ${\displaystyle \overrightarrow{x}={\overrightarrow{f}}_0+{\overrightarrow{f}}_1\cos t+{\overrightarrow{f}}_2\sin t}$ is $${\displaystyle A=\pi \left|det({\overrightarrow{f}}_1,{\overrightarrow{f}}_2)\right|.}$$

Semiaxes

With the abbreviations ${\displaystyle M={\overrightarrow{f}}_1^2+{\overrightarrow{f}}_2^2,N=\left|det({\overrightarrow{f}}_1,{\overrightarrow{f}}_2)\right|}$ the statements of Apollonios's theorem can be written as: $${\displaystyle a^2+b^2=M,ab=N.}$$ Solving this nonlinear system for ${\displaystyle a,b}$ yields the semiaxes: $${\displaystyle \begin{array}{cc}a& =\frac{1}{2}(\sqrt{M+2N}+\sqrt{M-2N})\\ b& =\frac{1}{2}(\sqrt{M+2N}-\sqrt{M-2N}).\end{array}}$$

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 $${\displaystyle det{\left(\overrightarrow{x}-\overrightarrow{f}{}_0,\overrightarrow{f}{}_2\right)}^2+det{\left(\overrightarrow{f}{}_1,\overrightarrow{x}-\overrightarrow{f}{}_0\right)}^2-det{\left(\overrightarrow{f}{}_1,\overrightarrow{f}{}_2\right)}^2=0.}$$

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 $${\displaystyle {\overrightarrow{f}}_1=(\genfrac{}{}{0ex}{}{e}{0}),{\overrightarrow{f}}_2=\frac{e}{\sqrt{d^2-c^2}}(\genfrac{}{}{0ex}{}{-c}{1})}$$ 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 $${\displaystyle {\overrightarrow{f}}_1=(\genfrac{}{}{0ex}{}{1}{0}),{\overrightarrow{f}}_2=\frac{1}{\sqrt{2}}(\genfrac{}{}{0ex}{}{-1}{1}).}$$

Rotated standard ellipse

For ${\displaystyle {\overrightarrow{f}}_0=(\genfrac{}{}{0ex}{}{0}{0}),{\overrightarrow{f}}_1=a(\genfrac{}{}{0ex}{}{\cos \theta }{\sin \theta }),{\overrightarrow{f}}_2=b(\genfrac{}{}{0ex}{}{-\sin \theta }{\cos \theta })}$ one obtains a parametric representation of the standard ellipse rotated by angle ${\displaystyle \theta }$: $${\displaystyle \begin{array}{cc}x& =x_{\theta }(t)=a\cos \theta \cos t-b\sin \theta \sin t,\\ y& =y_{\theta }(t)=a\sin \theta \cos t+b\cos \theta \sin t.\end{array}}$$

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 ${\displaystyle \overrightarrow{f}{}_0,\overrightarrow{f}{}_1,\overrightarrow{f}{}_2}$ to be vectors in space.

Polar forms

Polar form relative to 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 $${\displaystyle r(\theta )=\frac{ab}{\sqrt{(b\cos \theta {)}^2+(a\sin \theta {)}^2}}=\frac{b}{\sqrt{1-(e\cos \theta {)}^2}}}$$ where ${\displaystyle e}$ is the eccentricity (not Euler's number).

Polar form relative to focus

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.

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

Eccentricity and the directrix property

Each of the two lines parallel to the minor axis, and at a distance of $d=\frac{a^2}{c}=\frac{a}{e}$ 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: $${\displaystyle \frac{\left|P{F}_1\right|}{\left|P{l}_1\right|}=\frac{\left|P{F}_2\right|}{\left|P{l}_2\right|}=e=\frac{c}{a}.}$$

The proof for the pair ${\displaystyle F_1,l_1}$ follows from the fact that ${\left|P{F}_1\right|}^2=(x-c{)}^2+y^2,{\left|P{l}_1\right|}^2={\left(x-\frac{a^2}{c}\right)}^2$ and ${\displaystyle y^2=b^2-\frac{b^2}{a^2}{x}^2}$ satisfy the equation $${\displaystyle {\left|P{F}_1\right|}^2-\frac{c^2}{a^2}{\left|P{l}_1\right|}^2=0.}$$

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

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.)

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 ${\displaystyle x=-\frac{f}{e}}$. 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 ${\displaystyle 1-e^2=\frac{b^2}{a^2},\text{ and }p=\frac{b^2}{a}}$, 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

Because of ${\displaystyle c\cdot \frac{a^2}{c}=a^2}$ 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

An ellipse possesses the following property:

The normal at a point ${\displaystyle P}$ bisects the angle between the lines ${\displaystyle \overline{P{F}_1},\overline{P{F}_2}}$.

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 ${\displaystyle L}$ be the point on the line ${\displaystyle \overline{P{F}_2}}$ with distance ${\displaystyle 2a}$ to the focus ${\displaystyle F_2}$, where ${\displaystyle a}$ is the semi-major axis of the ellipse. Let line ${\displaystyle w}$ be the external angle bisector of the lines ${\displaystyle \overline{P{F}_1}}$ and ${\displaystyle \overline{P{F}_2}.}$ Take any other point ${\displaystyle Q}$ on ${\displaystyle w.}$ By the triangle inequality and the angle bisector theorem, ${\displaystyle 2a=\left|L{F}_2\right|<}$${\displaystyle \left|Q{F}_2\right|+\left|QL\right|=}$${\displaystyle \left|Q{F}_2\right|+\left|Q{F}_1\right|,}$ so ${\displaystyle Q}$ must be outside the ellipse. As this is true for every choice of ${\displaystyle Q,}$ ${\displaystyle w}$ only intersects the ellipse at the single point ${\displaystyle P}$ 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). It also serves to formulate Alhazen's problem of reflection on a circle tangent to the ellipse.

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

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, $${\displaystyle \overrightarrow{x}=\overrightarrow{p}(t)=\overrightarrow{f}{}_0+\overrightarrow{f}{}_1\cos t+\overrightarrow{f}{}_2\sin t,}$$

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

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

${\displaystyle (x_1,y_1)=(\pm a\cos t,\pm b\sin t)}$ (signs: (+,+) or (−,−) )

${\displaystyle (x_2,y_2)=(\mp a\sin t,\pm b\cos t)}$ (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

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

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 $A_{\mathrm{\Delta }}=\frac{1}{2}ab$, which can be expressed by ${\displaystyle A_{\mathrm{\Delta }}=\frac{1}{2}{c}_2{d}_1=\frac{1}{2}{c}_1{c}_2\sin \alpha }$, too. ${\displaystyle d_1}$ is the altitude of point ${\displaystyle P_1}$ and ${\displaystyle \alpha }$ the angle between the half diameters. Hence the area of the ellipse (see section metric properties) can be written as ${\displaystyle A_{el}=\pi ab=\pi c_2{d}_1=\pi c_1{c}_2\sin \alpha }$.
3. The parallelogram of tangents adjacent to the given conjugate diameters has the ${\displaystyle {\text{Area}}_{12}=4ab.}$

Proof

Let the ellipse be in the canonical form with parametric equation $${\displaystyle \overrightarrow{p}(t)=(a\cos t,b\sin t).}$$

The two points ${\overrightarrow{c}}_1=\overrightarrow{p}(t),{\overrightarrow{c}}_2=\overrightarrow{p}\left(t+\frac{\pi }{2}\right)$ are on conjugate diameters (see previous section). From trigonometric formulae one obtains ${\displaystyle {\overrightarrow{c}}_2=(-a\sin t,b\cos t{)}^{\mathsf{T}}}$ and $${\displaystyle {\left|{\overrightarrow{c}}_1\right|}^2+{\left|{\overrightarrow{c}}_2\right|}^2=\cdots =a^2+b^2.}$$

The area of the triangle generated by ${\displaystyle {\overrightarrow{c}}_1,{\overrightarrow{c}}_2}$ is $${\displaystyle A_{\mathrm{\Delta }}=\frac{1}{2}det\left({\overrightarrow{c}}_1,{\overrightarrow{c}}_2\right)=\cdots =\frac{1}{2}ab}$$

and from the diagram it can be seen that the area of the parallelogram is 8 times that of ${\displaystyle A_{\mathrm{\Delta }}}$. Hence $${\displaystyle {\text{Area}}_{12}=4ab.}$$

Orthogonal tangents

For the ellipse ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ the intersection points of orthogonal tangents lie on the circle ${\displaystyle x^2+y^2=a^2+b^2}$.

This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).

Drawing ellipses

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 (ellipsographs) 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.

If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.

For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.

de La Hire's point construction

The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation ${\displaystyle (a\cos t,b\sin t)}$ of an ellipse:

1. Draw the two circles centered at the center of the ellipse with radii ${\displaystyle a,b}$ and the axes of the ellipse.
2. Draw a line through the center, which intersects the two circles at point ${\displaystyle A}$ and ${\displaystyle B}$, respectively.
3. Draw a line through ${\displaystyle A}$ that is parallel to the minor axis and a line through ${\displaystyle B}$ that is parallel to the major axis. These lines meet at an ellipse point ${\displaystyle P}$ (see diagram).
4. Repeat steps (2) and (3) with different lines through the center.

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 pins, 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 ${\displaystyle 2a}$. 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, and it was elaborated in a now-lost 9th-century treatise by Al-Ḥasan ibn Mūsā.

A similar method for drawing confocal ellipses with a closed string is due to the Irish bishop Charles Graves.

Paper strip methods

The two following methods rely on the parametric representation (see § Standard parametric representation, above): $${\displaystyle (a\cos t,b\sin t)}$$

This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes ${\displaystyle a,b}$ have to be known.

Method 1

The first method starts with

a strip of paper of length ${\displaystyle a+b}$.

The point, where the semi axes meet is marked by ${\displaystyle P}$. If the strip slides with both ends on the axes of the desired ellipse, then point ${\displaystyle P}$ traces the ellipse. For the proof one shows that point ${\displaystyle P}$ has the parametric representation ${\displaystyle (a\cos t,b\sin t)}$, where parameter ${\displaystyle t}$ 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 ${\displaystyle a+b}$, 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.

A variation of the paper strip method 1 uses the observation that the midpoint ${\displaystyle N}$ of the paper strip is moving on the circle with center ${\displaystyle M}$ (of the ellipse) and radius ${\displaystyle \frac{a+b}{2}}$. Hence, the paperstrip can be cut at point ${\displaystyle N}$ into halves, connected again by a joint at ${\displaystyle N}$ and the sliding end ${\displaystyle K}$ fixed at the center ${\displaystyle M}$ (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe.

Method 2

The second method starts with

a strip of paper of length ${\displaystyle a}$.

One marks the point, which divides the strip into two substrips of length ${\displaystyle b}$ and ${\displaystyle a-b}$. 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 ${\displaystyle (a\cos t,b\sin t)}$, where parameter ${\displaystyle t}$ 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.

Most ellipsograph drafting instruments are based on the second paperstrip method.

Approximation by osculating circles

From Metric properties below, one obtains:

• The radius of curvature at the vertices ${\displaystyle V_1,V_2}$ is: ${\displaystyle \frac{b^2}{a}}$
• The radius of curvature at the co-vertices ${\displaystyle V_3,V_4}$ is: ${\displaystyle \frac{a^2}{b}.}$

The diagram shows an easy way to find the centers of curvature ${\displaystyle C_1=\left(a-\frac{b^2}{a},0\right),C_3=\left(0,b-\frac{a^2}{b}\right)}$ at vertex ${\displaystyle V_1}$ and co-vertex ${\displaystyle V_3}$, respectively:

1. mark the auxiliary point ${\displaystyle H=(a,b)}$ and draw the line segment ${\displaystyle V_1{V}_3,}$
2. draw the line through ${\displaystyle H}$, which is perpendicular to the line ${\displaystyle V_1{V}_3,}$
3. 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 circles.

Steiner generation

The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:

Given two pencils ${\displaystyle B(U),B(V)}$ of lines at two points ${\displaystyle U,V}$ (all lines containing ${\displaystyle U}$ and ${\displaystyle V}$, respectively) and a projective but not perspective mapping ${\displaystyle \pi }$ of ${\displaystyle B(U)}$ onto ${\displaystyle B(V)}$, then the intersection points of corresponding lines form a non-degenerate projective conic section.

For the generation of points of the ellipse ${\displaystyle \frac{\phantom{y^2}{x}^2}{a^2}+\frac{y^2}{b^2}=1}$ one uses the pencils at the vertices ${\displaystyle V_1,V_2}$. Let ${\displaystyle P=(0,b)}$ be an upper co-vertex of the ellipse and ${\displaystyle A=(-a,2b),B=(a,2b)}$.

${\displaystyle P}$ is the center of the rectangle ${\displaystyle V_1,V_2,B,A}$. The side ${\displaystyle \overline{AB}}$ of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal ${\displaystyle A{V}_2}$ as direction onto the line segment ${\displaystyle \overline{V_{1}B}}$ 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 ${\displaystyle V_1}$ and ${\displaystyle V_2}$ needed. The intersection points of any two related lines ${\displaystyle V_1{B}_i}$ and ${\displaystyle V_2{A}_i}$ are points of the uniquely defined ellipse. With help of the points ${\displaystyle C_1,\dots }$ 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

The ellipse is a special case of the hypotrochoid when ${\displaystyle R=2r}$, as shown in the adjacent image. The special case of a moving circle with radius ${\displaystyle r}$ inside a circle with radius ${\displaystyle R=2r}$ is called a Tusi couple.

Inscribed angles and three-point form

Circles

A circle with equation ${\displaystyle {\left(x-x_{\circ }\right)}^2+{\left(y-y_{\circ }\right)}^2=r^2}$ is uniquely determined by three points ${\displaystyle \left(x_1,y_1\right),\left(x_2,y_2\right),\left(x_3,y_3\right)}$ not on a line. A simple way to determine the parameters ${\displaystyle x_{\circ },y_{\circ },r}$ uses the inscribed angle theorem for circles:

For four points ${\displaystyle P_i=\left(x_i,y_i\right),i=1,2,3,4,}$ (see diagram) the following statement is true:

The four points are on a circle if and only if the angles at ${\displaystyle P_3}$ and ${\displaystyle P_4}$ 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 ${\displaystyle y=m_{1}x+d_1,y=m_{2}x+d_2,m_1\ne m_2,}$ one uses the quotient: $${\displaystyle \frac{1+m_1{m}_2}{m_2-m_1}=\cot \theta .}$$

Inscribed angle theorem for circles

For four points ${\displaystyle P_i=\left(x_i,y_i\right),i=1,2,3,4,}$ 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 ${\displaystyle P_3}$ and ${\displaystyle P_4}$ are equal. In terms of the angle measurement above, this means: $${\displaystyle \frac{(x_4-x_1)(x_4-x_2)+(y_4-y_1)(y_4-y_2)}{(y_4-y_1)(x_4-x_2)-(y_4-y_2)(x_4-x_1)}=\frac{(x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)}{(y_3-y_1)(x_3-x_2)-(y_3-y_2)(x_3-x_1)}.}$$

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 ${\displaystyle P_i=\left(x_i,y_i\right)}$: $${\displaystyle \frac{(x-x_1)(x-x_2)+(y-y_1)(y-y_2)}{(y-y_1)(x-x_2)-(y-y_2)(x-x_1)}=\frac{(x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)}{(y_3-y_1)(x_3-x_2)-(y_3-y_2)(x_3-x_1)}.}$$