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[parent] ellipse (Definition)

An ellipse that is centered at the origin is the curve in the plane determined by

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


\begin{pspicture}(-3.2,-2.2)(3.5,2.5) \psaxes{<->}(0,0)(-3.2,-2.2)(3.5,2.5) \psellipse(0,0)(3,2) \rput[b](3.5,-0.5){$x$} \rput[r](0,2.5){$y$} \end{pspicture}

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 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.,

$\displaystyle E = \left\{ p\, \vert\, r=\lvert p-p_1\rvert + \vert p-p_2\rvert\right\}.$
In terms of the geometric look of $ E$, there are three possible scenarios for $ E$: $ E=\varnothing$, $ E=\overline{p_1p_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. The area of an ellipse is $ \pi a b$. (See this page.)
  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
$\displaystyle \epsilon=\frac{\sqrt{a^2-b^2}}{a}\cdot$    

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 $ \vert O-p_1\vert=\vert O-p_2\vert=\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
$\displaystyle x'$ $\displaystyle =$ $\displaystyle x+\epsilon a,$  
$\displaystyle y'$ $\displaystyle =$ $\displaystyle y,$  

but from (1) we get
$\displaystyle \left(\frac{x'-\epsilon a}{a}\right)^2 + \left(\frac{y'}{b}\right)^2 = 1.$ (2)

By using the transformation equations to polar coordinates
$\displaystyle x'$ $\displaystyle =$ $\displaystyle r\cos\theta,$  
$\displaystyle y'$ $\displaystyle =$ $\displaystyle r\sin\theta,$  

and through (2) we arrive at the polar equation
$\displaystyle r(\theta)=\frac{(1-\epsilon^2)a}{1-\epsilon\cos\theta}\cdot$ (3)

This equation allows us to determine some additional properties about the ellipse:
$\displaystyle r_{max}:=r(0)=(1+\epsilon)a,$   which is called the aphelium$\displaystyle ;$    
$\displaystyle r_{min}:=r(\pi)=(1-\epsilon)a,$   which is called the perihelium$\displaystyle .$    

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'$ and the radial distance $ \vert B-p_1\vert$, where $ B(0,b)$ is the point of the ellipse over the $ y$-axis, then we get the useful equation $ \cos\theta_\epsilon=\epsilon$.



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See Also: squeezing $\mathbb{R}^n$, ellipsoid

Also defines:  major axis, minor axis, major semi-axis, minor semi-axis, focus, foci, aphelium, perihelium, eccentric anomaly, focal radius, focal radii

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Attachments:
easy calculation of the area of an ellipse (Definition) by cvalente
perimeter of ellipse (Example) by pahio
conjugate diameters of ellipse (Topic) by pahio
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Cross-references: radial, polar axis, geometric mean, arithmetic mean, polar, equations, transformation, rectangular coordinate, eccentricity, elliptic integral, perimeter, area, angle, normal line, abscissa, circumscribed, polar angle, polar coordinates, algebraic, rotation, translation, linear transformations, distances, sum, property, real number, Euclidean, points, length, radius, circle, endpoints, line segment, graph, plane, curve, origin
There are 33 references to this entry.

This is version 29 of ellipse, born on 2005-05-22, modified 2007-05-29.
Object id is 7102, canonical name is Ellipse2.
Accessed 10434 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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