One of the definitions of a conic section is that the conic section is a locus of points whose distance to the focus $F$ is a constant multiple of distance between them and the directrix $D$, i.e.

$e = \frac{d(P,F)}{d(P, D)}$

Where $e$ is eccentricity. It is said that the eccentricity of a circle is $0$, which means:

$0 = \frac{d(P,F)}{d(P, D)}$



What confuses me about this, is that, unlike with other conic sections, where there would be infinite number of points $P$ which satisfy the given equation, it seems that the only point that satisfies this equation is the point $P=F$. Yet, the circle is composed of infinite points, which is contradictory. Can someone explain what is the geometric interpretation of this, and where I'm wrong?

  • 1
    $\begingroup$ "Therefore, $d(P,F)=0$" ... or $d(P,D) = \infty$! The directrix of a (non-point) circle is "at infinity". In this case, the focus-directrix definition isn't helpful, since it can't distinguish between circles with the same center. $\endgroup$ – Blue Sep 16 '18 at 12:35
  • $\begingroup$ Thanks! I don't like using infinity in this context, but I guess that is the only explanation... $\endgroup$ – Aleksandar Stefanović Sep 16 '18 at 12:55

Consider the following illustration of a triple-family of conics, each with a common focus and matching eccentricities, but with distinct directrices. (The animation oscillates between eccentricities $0$ and $2$.)

enter image description here

As each conic gets more circular, its directrix moves further away. Consequently, it's not unreasonable to say that, as a limiting case, a (non-degenerate) circle, which has eccentricity $0$, has its directrix "at infinity", which is consistent with the relation

$$\text{eccentricity} = \frac{\text{distance from point to focus}}{\text{distance from point to directrix}} \tag{$\star$}$$

Of course, this makes a directrix completely useless for uniquely determining a circle, since the three concentric circles shown have the same fixed focus and "the same" infinitely-distant directrix. (Another wrinkle: A circle's directrix is really "at infinity" in any direction; so, it's really an indeterminate element.)

Otherwise, you're correct that, for a given (not-at-infinity) line and a given (not-at-infinity) focus, relation $(\star)$ with eccentricity $0$ defines a point-circle (ie, a circle of radius $0$) at the focus/center. (Related wrinkle: For such a circle, any line not passing through the focus/center serves as a directrix.)

This is one of the ways conics sections force people to confront ---and embrace--- infinity. As above, the key to understanding is to consider any particular conic, not as an isolated curve, but as part of an appropriate family of curves.


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