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

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 semi-axis length and $b$ the minor semi-axis length; i.e., 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 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

$\displaystyle x(t)$ | $\displaystyle=$ | $\displaystyle a\cos t,$ | ||

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

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. 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 focal radii 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. 4. The length of the perimeter of an ellipse can be expressed using an elliptic integral.

# 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 $\cos\theta_{\epsilon}=\epsilon$.

## Mathematics Subject Classification

53A04*no label found*51N20

*no label found*51-00

*no label found*

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