In Calculus II (and briefly in Trigonometry, if I remember correctly) the concept of a directrix began poking its head around conic sections. While covering parabolas, ellipses, and hyperbolas, the Eccentricity-Directrix Theorem waltzed into the party unannounced -- so it appeared to me anyhow.

Foci became a point of interest (no pun intended) around the same time. However, I understand the practical import of foci. As applied to a parabolic receiving reflector, for example, foci can be used to, well, focus incoming light. A property useful in telescope design.

But I don't understand what practical purpose the directrix serves. Does it have any real-world applications like foci? Also, I'm not certain I really understand what purpose it serves mathematically.

A well-defined mathematical purpose and useful real-world application of directrices would be much appreciated.

  • 2
    $\begingroup$ Two things 1) I have added to my answer a computer display application 2) Congratulations for the humor and the puns (that I have appreciated though I am not a native english speaker) $\endgroup$ – Jean Marie Apr 27 '16 at 23:53

Let me first recall at first the focus-directrix definition of a conic section:

Let $(D)$ be a line (the directrix) and $F$ a point (the focus, or one of the foci) and $e>0$ be a real number (the eccentricity).

The set of points $P$ such that

$$\dfrac{dist(P,F)}{dist(P,(D))}=e \ \ \ (1)$$

is a conic section, that is an ellipse, a parabola, or a hyperbola according to the 3 resp. cases $e<1, e=1, e>1$.

Reciprocally, any conic curve can be described in this way.

see for example http://www.maths.gla.ac.uk/wws/cabripages/conics/focus.html

Regarding the applications, let me give one that is connected to conic sections' drawing, more precisely, to their computer display (Have a look at the graphic below).

The fact that there is a unique definition for the 3 types of conic sections permits to simulate for example the image of a torch light on a wall in a continuous transition from a circle, to ellipses, to a parabola (very transitory!), then to hyperbolas. It took me a short time to build the program that has generated these curves by writing equation (1) upwards under the ("put to the squares") form:

$$x^2+y^2=e^2 (x+1)^2$$

(I have fixed the focus at the origin and the directrix at $x=-1$ ; only $e$ varies ; but I could have let the directrix vary for a fixed $e$, etc.).

I have taken a step 0.1 from $e=0.1$ till $e=1.2$ (the fist one is an ellipse with a very small eccentricity, theus close to a circle, the last one is a hyperbola; the case $e=1$ giving equation $y^2=2x+1$ which is clearly the equation of a parabola).

In the case you desire to program something similar, here is the Matlab script I have written:

clear all;close all;hold on;axis off
axis([-1.5,5.5,-4,4]);        %framebox
plot(0,0,'*k')                %focus
d=-1;plot([d,d],[-4,4],'k');  %directrix
for e=0.1:0.1:1.2;            %eccentricity
ezplot(['x.^2+y.^2-',num2str(e^2),'.*(x+1).^2']);  %implicit equ. plot

enter image description here


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