ellipse
An ellipse^{} that is centered at the origin is the curve in the plane determined by
$${\left(\frac{x}{a}\right)}^{2}+{\left(\frac{y}{b}\right)}^{2}=1,$$  (1) 
where $a,b>0$.
Below is a graph of the ellipse ${\left({\displaystyle \frac{x}{3}}\right)}^{2}+{\left({\displaystyle \frac{y}{2}}\right)}^{2}=1$:
The major axis of an ellipse is the longest line segment^{} whose endpoints are on the ellipse. The minor axis of an ellipse is the shortest line segment through the midpoint^{} of the ellipse whose endpoints are on the ellipse.
In the first equation given above, if $a=b$, the ellipse reduces to a circle of radius $a$, whereas if $a>b$ (as in the graph above), $a$ is said to be the major semiaxis length and $b$ the minor semiaxis length; i.e. (http://planetmath.org/Ie), the lengths of the major axis and minor axis are $2a$ and $2b$, respectively.
More generally, given any two points ${p}_{1}$ and ${p}_{2}$ in the (Euclidean) plane and any real number $r$, let $E$ be the set of points $p$ having the property that the sum of the distances^{} from $p$ to ${p}_{1}$ and ${p}_{2}$ is $r$; i.e.,
$$E=\{pr=p{p}_{1}+p{p}_{2}\}.$$ 
In terms of the geometric look of $E$, there are three possible scenarios for $E$: $E=\mathrm{\varnothing}$, $E=\overline{{p}_{1}{p}_{2}}$, the line segment with endpoints ${p}_{1}$ and ${p}_{2}$, or $E$ is an ellipse. Points ${p}_{1}$ and ${p}_{2}$ are called foci of the ellipse; the line segments connecting a point of the ellipse to the foci are the focal radii^{} belonging to that point. When ${p}_{1}={p}_{2}$ and $r>0$, $E$ is a circle. Under appropriate linear transformations (a translation^{} followed by a rotation^{}), $E$ has an algebraic appearance expressed in (1).
In polar coordinates^{}, the ellipse is parametrized as
$x(t)$  $=$  $a\mathrm{cos}t,$  
$y(t)$  $=$  $b\mathrm{sin}t,t\in [0,\mathrm{\hspace{0.17em}2}\pi ).$ 
If $a>b$, then $t$ is the eccentric anomaly; i.e., the polar angle^{} of the point on the circumscribed circle having the same abscissa^{} as the point of the ellipse.
Properties

1.
If $a>b$, the foci of the ellipse (1) are on the $x$axis with distances $\sqrt{{a}^{2}{b}^{2}}$ from the origin. The constant sum of the of a point $p$ is equal to $2a$.

2.
The normal line^{} of the ellipse at its point $p$ halves the angle between the focal radii drawn from $p$.

3.
The area of an ellipse is $\pi ab$. (See this page (http://planetmath.org/AreaOfPlaneRegion).)

4.
The length of the perimeter^{} of an ellipse can be expressed using an elliptic integral^{}.
Eccentricity
By definition, the eccentricity $\u03f5$ ($$) of the ellipse is given by
$$\u03f5=\frac{\sqrt{{a}^{2}{b}^{2}}}{a}\cdot $$ 
For $\u03f5=0$, the ellipse reduces to a circle. Further, $b=a\sqrt{1{\u03f5}^{2}}$, and by assuming that foci are located on $x$axis, ${p}_{1}$ on $$ and ${p}_{2}$ on $x>0$, then $O{p}_{1}=O{p}_{2}=\u03f5a$, where $O(0,0)$ is the origin of the rectangular coordinate system.
Polar equation of the ellipse
By translating the $y$axis towards the focus ${p}_{1}$, we have
${x}^{\prime}$  $=$  $x+\u03f5a,$  
${y}^{\prime}$  $=$  $y,$ 
but from (1) we get
$${\left(\frac{{x}^{\prime}\u03f5a}{a}\right)}^{2}+{\left(\frac{{y}^{\prime}}{b}\right)}^{2}=1.$$  (2) 
By using the transformation equations to polar coordinates
${x}^{\prime}$  $=$  $r\mathrm{cos}\theta ,$  
${y}^{\prime}$  $=$  $r\mathrm{sin}\theta ,$ 
and through (2) we arrive at the polar equation
$$r(\theta )=\frac{(1{\u03f5}^{2})a}{1\u03f5\mathrm{cos}\theta}\cdot $$  (3) 
This equation allows us to determine some additional properties about the ellipse:
${r}_{max}:=r(0)=(1+\u03f5)a,\text{which is called the}\text{\mathit{a}\mathit{p}\u210e\mathit{e}\mathit{l}\mathit{i}\mathit{u}\mathit{m}};$  
${r}_{min}:=r(\pi )=(1\u03f5)a,\text{which is called the}\text{\mathit{p}\mathit{e}\mathit{r}\mathit{i}\u210e\mathit{e}\mathit{l}\mathit{i}\mathit{u}\mathit{m}}.$ 
Hence, the general definition of the ellipse expressed above shows that ${r}_{min}+{r}_{max}=2a$ and also that the arithmetic mean^{} $\frac{{r}_{min}+{r}_{max}}{2}}=a$ corresponds to the major semiaxis, while the geometric mean^{} $\sqrt{{r}_{min}{r}_{max}}=b$ corresponds to the minor semiaxis of the ellipse. Likewise, if ${\theta}_{\u03f5}$ is the angle between the polar axis ${x}^{\prime}$ and the radial distance $B{p}_{1}$, where $B(0,b)$ is the point of the ellipse over the $y$axis, then we get the useful equation $\mathrm{cos}{\theta}_{\u03f5}=\u03f5$.
Title  ellipse 
Canonical name  Ellipse 
Date of creation  20130322 15:18:10 
Last modified on  20130322 15:18:10 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  33 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 53A04 
Classification  msc 51N20 
Classification  msc 5100 
Related topic  SqueezingMathbbRn 
Related topic  Ellipsoid^{} 
Defines  major axis 
Defines  minor axis 
Defines  major semiaxis 
Defines  minor semiaxis 
Defines  focus 
Defines  foci 
Defines  aphelium 
Defines  perihelium 
Defines  eccentric anomaly 
Defines  focal radius 
Defines  focal radii 