Foci and Directrices

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dA = r dr (d theta)
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Bonus: Cylindrical and spherical coordinates

Ellipses and hyperbolas are usually defined using two foci, but they can also be defined using a focus and a directrix.

Definitions of conic sections: Let

$L$ be the distance to the focus and let

$D$ be the distance to the directrix. Pick a constant

$e>0$ . The set of all points with

$L=eD$ is

An ellipse of eccentricity $e$ if $e<1$ ,
A parabola if $e=1$ ,
A hyperbola of eccentricity $e$ if $e>1$ .

For instance, suppose that we have a conic section with focus at the origin, directrix at $y=-1$ , and eccentricity $e \ne 1$ . Then

$\begin{eqnarray*} \sqrt{x^2 + y^2} &=& e|y+1| \cr x^2 + y^2 & = & e^2 (y^2 + 2 y + 1) \cr x^2 + (1-e^2)y^2 -2e^2 y& = & e^2 . \end{eqnarray*}$ Notice that the coefficient of

$y^2$ in the last equation is positive if

$e<1$ , giving us an ellipse, and is negative when

$e>1$ , giving us a hyperbola. After completing the square and applying some more algebraic manipulations, we can put the equation in standard form:

$\left ( \frac{1-e^2}{e^2}\right) x^2 + \left( \frac{(1-e^2)^2}{e^2}\right )\left ( y - \frac{e^2}{1-e^2}\right )^2 = 1.$

If $e<1$ , this is an vertically aligned ("tall and skinny") ellipse with center at $(0,\frac{e^2}{1-e^2})$ , with $a = \frac{e}{1-e^2}$ , $b=\frac{e}{\sqrt{1-e^2}}$ and $c=\frac{e^2}{1-e^2}$ and eccentricity $e=c/a$ . As $e \to 1$ , the center and the size of the ellipse both go to infinity.

If $e>1$ the situation is analogous. Our curve is then a hyperbola of with center at $(0,-\frac{e^2}{e^2-1})$ , with $a = \frac{e}{e^2-1}$ , $b=\frac{e}{\sqrt{e^2-1}}$ and $c=\frac{e^2}{e^2-1}$ and eccentricity $e=c/a$ . As $e \to 1$ , the center and the size of the hyperbola both go to infinity.

The following video describes — without equations — what happens to a conic section as the eccentricity is increased from 0 to 1, and then beyond 1.