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According to many websites, including Wikipedia, the eccentricity of a conic section is defined as the ratio of (the distance from a fixed point called the focus) to (the distance from a fixed line called the directrix). How is this definition applicable to circles? What are the focus and directrix? Are they the center and a tangent line to the circle? In that case, the eccentricity should not be zero, it should be undefined, as far as my intuition goes.

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    $\begingroup$ The focus is the center but the directrix is a line at infinity en.wikipedia.org/wiki/… $\endgroup$ – mm-crj Dec 14 '18 at 22:51
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    $\begingroup$ See also this answer, which demonstrates that the eccentricity is also equal to $\sin\angle S/\sin\angle Q$, where $\angle S$ is the angle of inclination of the plane that cuts a cone to make a conic section, and $\angle Q$ gives the steepness of the cone itself. Since a circle is cut by a "horizontal" plane, $\sin\angle S = 0$. (Note, too, that the directrix is the line where the cutting plane meets the "horizontal" plane through the cone's vertex; a "horizontal" cutting plane meets that vertex plane "at infinity".) $\endgroup$ – Blue Dec 15 '18 at 1:16
  • $\begingroup$ Conic sections and eccentricity are defined in multiple articles in Wikipedia. Here is another: en.wikipedia.org/wiki/…, again with an explanation of the eccentricity of a circle. Where are all these web pages that fail to explain the eccentricity of a circle? Specific links might help. $\endgroup$ – David K Dec 15 '18 at 15:49
  • $\begingroup$ @Blue true, but I was hoping to apply the definition of the focus and the directrix to the circle. The definition that includes the angles is pretty clear. Thanks $\endgroup$ – user626542 Dec 15 '18 at 21:53
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a general ellipse with parameters $a,b, c=\sqrt{a^2-b^2}$, and eccentricity $e=\cfrac ca$,

a circle can be regarded as the case where $a=b=r\text{(radius)}, c=0$, hence $e=\cfrac ca=\cfrac 0r=0$

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    $\begingroup$ How does this answer the question? The asker specifically mentions the focus and the directrix, and you've just written the formula for eccentricity in terms of the parameters. $\endgroup$ – T. Bongers Dec 14 '18 at 23:02
  • $\begingroup$ @T.Bongers: To be fair, this does address what the title asks. The body of the Question can be construed as the OP's attempt to solve it. $\endgroup$ – hardmath Dec 15 '18 at 0:00
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The eccentricity of an ellipse measures how elongated it is compared to a circle. As defined, it lies in the open interval $]0,1[$, with increasing values indicating ever more elongated ellipses. As the eccentricity decreases, the ellipses become more and more circular, so a circle can be viewed as the limiting curve of this process. It then makes sense to define the eccentricity of a circle as the limit of the decreasing eccentricities, namely zero. Going the other way, as the eccentricity increases, the ellipses get more and more elongated, approaching the parabola obtained when the eccentricity is $1$.†

You can see this limiting process in action algebraically. Let $F=(-1,0)$ and $x=d$, $d\gt0$ be the focus and directrix of a conic that passes through the origin. Using the focus-directrix definition of a conic, an equation for the curve is $$(x+1)^2+y^2={(x-d)^2\over d^2}.$$ As $d\to1$, this approaches the parabola $y^2=-4x$, while as $d\to\infty$, the equation approaches $(x+1)^2+y^2=1$, which is clearly that of a circle, and $e=1/d\to0$. The center of this circle is at $F$—its foci coincide—and it has no finite directrix. It can be thought of as being “at infinity,” an idea that is made concrete in projective geometry.††


† The ellipses never actually become a parabola, at least not on the Euclidean plane, since the latter is not a closed curve; the eccentricity of an ellipse is strictly less than $1$. On the projective plane, however, parabolas and hyperbolas are also closed, and you can continue “stretching” the ellipse out to the line at infinity and beyond, “wrapping around” to become a hyperbola. On the other hand, eccentricity isn’t a meaningful concept in projective geometry: circles are not projectively distinguishable from ellipses—you have to impose a Euclidean geometry on the plane to do so. Then again, the different types of nondegenerate conics are indistinguishable in the first place until you’ve designated a particular line as the line at infinity.

†† The polar of a conic’s focus is the corresponding directrix; the polar of a circle’s center (the center of any conic, for that matter) is the line at infinity.

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