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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:

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

therefore,

$d(P,F)=0$

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?

Blue
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Aleksandar Stefanović
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    "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. – Blue Sep 16 '18 at 12:35
  • Thanks! I don't like using infinity in this context, but I guess that is the only explanation... – Aleksandar Stefanović Sep 16 '18 at 12:55

1 Answers1

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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.

Blue
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  • Excellent answer. In the illustration, the directrices appearing to come closer together as the conics become hyperbolas, and then appearing to space out when they became circles seemed to pose a problem; the directrices seemed to become more separate and not the "same"(and therefore useless in distinguishing different-sized circles) as you termed them. I suppose this is resolved by the fact that since infinity is not really a defined quantity, information on all three directrices is equally rendered nil there, and thus directrices are useless. Is that the case? – harry Aug 16 '21 at 10:26
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    @harry: Yes, directrices of circles are pretty useless for distinguishing different-sized instances. However, (in a manner of speaking) the directrices actually converge to "the same line" once they reach infinity. (Infinity is cool like that.) The best context for this kind of discussion is projective geometry, where "the line at infinity" is fully embraced as a peer of old-fashioned lines. – Blue Aug 16 '21 at 10:36