M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
Learning module LM 10.2: Calculus with Parametrized Curves:
Learning module LM 10.3: Polar Coordinates:
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Slicing a cone
Ellipses
Hyperbolas
Parabolas and directrices
Completing the square
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Ellipses
Ellipses
Ellipses can be elegantly
described in four ways.
 Via Cartesian (rectangular) coordinates.
 In terms of distances to two foci (plural of focus).
 In terms of distances to a focus and a directrix.
 In polar coordinates.

We will do the first two on this
page, and the third and fourth later on.
The simplest description of an
ellipse is as a squashed or stretched circle. Start with the unit
circle $x^2 + y^2 =1$, and stretch it by a factor of $a$ in the $x$
direction and $b$ in the $y$ direction to get:
The standard formula for an ellipse in rectangular coordinates is
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$

The points $(\pm a,0)$ (and
sometimes the points $(0,\pm b)$) are called vertices. If
$a > b > 0$, then the major axis is the line segment from
$(a,0)$ to $(a,0)$ and the semimajor axis is the line segment
from the origin to $(a,0)$. Likewise, the minor axis runs from
$(0,b)$ to $(0,b)$ and the semiminor axis runs from the
origin to $(0,b)$. If $b > a > 0$, then the major and semimajor
axes are vertical and the minor and semiminor axes are
horizontal. For now we'll stick with the case that $a > b$, so that the
ellipse is short and fat. The origin is the center of the
ellipse.
Now let $c = \sqrt{a^2b^2}$. The points $(\pm c, 0)$ are
called foci. These points are extremely important in astronomy,
since planets follow elliptical orbits with the sun at a
focus, not with the
sun at the center.
Let $F_1(c,0)$ and $F_2(c,0)$ be the two foci, let $P(x,y)$ be an
arbitrary point on the ellipse. Let $L_1$ be the distance from $F_1$
to $P$, and let $L_2$ be the distance from $F_2$ to $P$, as
in the figure on the right.


Amazing fact: The ellipse is the set of all points where
$L_1 + L_2 = 2a$.

This fact gives elliptical rooms
amazing acoustic properties. If you whisper at one focus of such a
room, the sound waves from your voice will bounce off the walls and
converge at the other focus  that's why it is called a focus. The
same goes for light reflecting off elliptical mirrors.
To understand the amazing fact,
let's convert the equation $L_1 + L_2 = 2a$ to rectangular
coordinates:
\begin{eqnarray*} L_1 + L_2 & = & 2a \cr\cr
L_1 & = & 2aL_2 \cr \cr
\sqrt{(x+c)^2+y^2} & = & 2a \sqrt{(xc)^2 + y^2} \cr\cr
(x+c)^2 + y^2 & = & 4a^2 + (xc)^2 + y^2  4a \sqrt{(xc)^2 + y^2}\cr\cr
4a\sqrt{(xc)^2 + y^2}&=& 4a^24cx \cr \cr
a \sqrt{(xc)^2 + y^2} &=& a^2cx \cr \cr
a^2(xc)^2+ a^2 y^2 &=& a^4+c^2x^2 2a^2cx \cr \cr
a^2x^2 + a^2c^2 2a^2cx + y^2 &=& a^4 + c^2x^2 2a^2cx \cr \cr
(a^2c^2)x^2 + a^2 y^2 &=& a^2(a^2c^2) \cr \cr
b^2 x^2 + a^2 y^2 &=& a^2b^2 \cr \cr
\frac{x^2}{a^2} + \frac{y^2}{b^2} &=& 1,\end{eqnarray*}
where we have used the fact that $b^2=a^2c^2$. That's a long and
messy calculation for a simple and elegant result.
You should be able to construct
the equation of an ellipse given any two of $a$, $b$ and $c$, since
you can get the third from $c^2=a^2b^2.$
Example 1: Find the location of the foci of the ellipse
$\displaystyle{\frac{x^2}{25} + \frac{y^2}{9}=1}$.

Solution:
We have $a=5$ and $b=3$, so $c = \sqrt{a^2b^2} = 4$. The foci are at
$(\pm 4,0)$.

Example 2: Find the equation of an ellipse
with foci at $(\pm 3,0)$ if $b=4$.

Solution:
Since $c=3$ and $b=4$, $a^2=3^2+4^2=25$, so $a=5$. This makes the equation
$$\frac{x^2}{25} + \frac{y^2}{16} = 1.$$

The ratio $c/a$ is called
the eccentricity of the ellipse, and is usually denoted
$e$. Note that $e < 1$. A circle can be viewed as an ellipse with
eccentricity zero, and with both foci at the origin.


It is easy to plot an ellipse as a parametrized curve. Just take
$$x = a \cos(t); \qquad y = b\sin(t),$$
with the parameter $t$ running from $0$ to $2\pi$.

