# 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 $\displaystyle\left(\frac{x}{3}\right)^{2}+\left(\frac{y}{2}\right)^{2}=1$:

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 semi-axis length and $b$ the minor semi-axis 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.,

 $E=\left\{p\,|\,r=\lvert p-p_{1}\rvert+|p-p_{2}\rvert\right\}.$

In terms of the geometric look of $E$, there are three possible scenarios for $E$: $E=\varnothing$, $E=\overline{p_{1}p_{2}}$, the line segment with end-points $p_{1}$ 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 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).

 $\displaystyle x(t)$ $\displaystyle=$ $\displaystyle a\cos t,$ $\displaystyle y(t)$ $\displaystyle=$ $\displaystyle b\sin t,\quad t\in[0,\,2\pi).$

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. 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. 2.

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

3. 3.

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

4. 4.

## Eccentricity

By definition, the eccentricity $\epsilon$ ($0\leq\epsilon<1$) of the ellipse is given by

 $\epsilon=\frac{\sqrt{a^{2}-b^{2}}}{a}\cdot$

For $\epsilon=0$, the ellipse reduces to a circle. Further, $b=a\sqrt{1-\epsilon^{2}}$, and by assuming that foci are located on $x$-axis, $p_{1}$ on $x<0$ and $p_{2}$ on $x>0$, then $|O-p_{1}|=|O-p_{2}|=\epsilon a$, 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

 $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle x+\epsilon a,$ $\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle y,$

but from (1) we get

 $\left(\frac{x^{\prime}-\epsilon a}{a}\right)^{2}+\left(\frac{y^{\prime}}{b}% \right)^{2}=1.$ (2)

By using the transformation equations to polar coordinates

 $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle r\cos\theta,$ $\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle r\sin\theta,$

and through (2) we arrive at the polar equation

 $r(\theta)=\frac{(1-\epsilon^{2})a}{1-\epsilon\cos\theta}\cdot$ (3)

This equation allows us to determine some additional properties about the ellipse:

 $\displaystyle r_{max}:=r(0)=(1+\epsilon)a,\qquad\text{which is called the {\em aphelium% }};$ $\displaystyle r_{min}:=r(\pi)=(1-\epsilon)a,\qquad\text{which is called the {% \em perihelium}}.$

Hence, the general definition of the ellipse expressed above shows that $r_{min}+r_{max}=2a$ and also that the arithmetic mean  $\displaystyle\frac{r_{min}+r_{max}}{2}=a$ corresponds to the major semi-axis, while the geometric mean  $\sqrt{r_{min}r_{max}}=b$ corresponds to the minor semi-axis of the ellipse. Likewise, if $\theta_{\epsilon}$ 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 $\cos\theta_{\epsilon}=\epsilon$.

 Title ellipse Canonical name Ellipse Date of creation 2013-03-22 15:18:10 Last modified on 2013-03-22 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 51-00 Related topic SqueezingMathbbRn Related topic Ellipsoid   Defines major axis Defines minor axis Defines major semi-axis Defines minor semi-axis Defines focus Defines foci Defines aphelium Defines perihelium Defines eccentric anomaly Defines focal radius Defines focal radii