What practical purpose -- or application -- do directrices serve?

Question

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.

Answer 1

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

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

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

In a reciprocal way, 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 connected to conic sections' drawing, more precisely, to their computer display (Have a look at the first graphic below).

The fact that a common definition for the 3 types of conic sections is possible allows for example to display 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. Here is a short Matlab program that has generated these curves by squaring equation (1) upwards under the ("put to the squares") form:

$$x^2+y^2=e^2 (x+1)^2 \tag{2}$$

The focus is taken at the origin ; the directrix is the vertical line at $x=-1$ ; It is fixed while $e$ is allowed to vary ; A different drawing was possible with a varying directrix and a fixed eccentricity $e$.

Letting eccentricity $e$ vary from $e=0.1$ till $e=1.2$, we cover the different cases. The first one is an ellipse with a very small eccentricity, therefore close to a circle, the last one is a hyperbola; the case $e=1$ gives 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
f=@(x,y,e)(x.^2+y.^2-e*(x+1).^2);
for e=0.1:0.1:1.2;            %eccentricity
   g=@(x,y)f(x,y,e); 
   ezplot(g(x,y));            %implicit equ. plot
end;

The second figure shows is a 3D representation explaining how the first figure can be obtained as the perspective (= projective) image of a ring of concentric circles.

Remark: If we set $e^2=1-\frac{1}{m}$, equation (2) can be written in this way :

$$m (x+1)^2+(1-m)(x^2+y^2)=0 \tag{3}$$

which is the general equation of a so-called "pencil of conics" generated by the by two "degenerated conic curves" with equations $(x+1)^2=0$ (the equation of the directrix) and $x^2+y^2=0$ (the "equation" of point $F$, that can be seen as a circle with radius $0$).