Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

While I was reading about conic sections, I came across the following statement:

A parabola is an ellipse, but with one focal point at infinity.

But it is not clear to me. Can someone explain it clearly?

share|improve this question
See also projective plane/point at infinity. –  Ian Mallett May 2 at 19:08
The funny thing is I've always thought of conic sections this way (after I learned about them). A parabola is an ellipse with a focal point at infinity; it is also a hyperbola with a focal point at infinity. To get from an ellipse to a hyperbola, the point wraps around at infinity. This seemed even more logical when I learned about eccentricity. –  Quincunx May 3 at 3:14

4 Answers 4

up vote 63 down vote accepted

The equation for an ellipse with a focus at $(0,0)$ and the other at $(0,2ae)$ keeping $a(1-e)=f$ (where $f$ is distance from the vertex to the focus of the ellipse, which ends up being the focal length of the parabola) is $$ \frac{x^2}{a^2(1-e^2)}+\frac{(y-ae)^2}{a^2}=1 $$ which is equivalent to $$ \frac{x^2}{f(1+e)}+\frac{y^2-2aey}{a}=f(1+e) $$ If we let $a\to\infty$ (and therefore $e=1-\frac fa\to1$), we get $$ y=\frac{x^2}{4f}-f $$ which is a parabola.

$\hspace{3.4cm}$enter image description here

share|improve this answer
Following on this line-of-thought, an hyperbola is a ellipse which had its focal point "wrap-around" at infinity and then come back from the other side? –  NothingsImpossible May 3 at 0:40
@NothingsImpossible: indeed –  robjohn May 3 at 11:36
+1 just for the amazing graphic –  recursive recursion May 3 at 14:44

Imagine an ellipse made of reflective material. Light rays emanating from one focus and reflecting off the ellipse will all be reflected toward the other focus. (This, applied to sound waves rather than light rays, is the principle behind whispering galleries.) Now imagine instead a parabola made of reflective material. Light rays emanating from its focus and reflecting off the parabola will all be reflected in the direction parallel to the axis of the parabola. (An approximation to this seems to be involved in automobile headlights.) So, if you think, as in projective geometry, of parallel lines as "meeting at infinity", then the point at infinity on a parabola's axis plays the same role as the other focus of an ellipse.

share|improve this answer

Think of a pair of cones with vertical axes and identical cone-angles, one facing up and one facing down. Place them so they're touching at one point. I'll call this a "cone" for now.

If you slice this object with a plane perpendicular to the axis, you'll get a circle of some radius, or perhaps a single point, which you could call a circle of radius 0.

If you slice it with a slightly tilted plane, you'll get an ellipse (or a single point). Thus circules and ellipses are both "cross-sections" of a cone, or "conic sections".

If you tilt the slicing plane further, so it's nearly vertical, it'll intersect both cones, resulting in a hyperbola, or, if the plane passes through the cone-point, a pair of intersecting lines. So intersecting lines are kind of a "limit" of hyperbolas.

Go back to ellipses: place the plane to slice an ellipse from the upper half of the cone. Tilt your slicing plane more and more, making the ellipse more and more eccentric. There's a last possible amount of tilt before you start also slicing into the bottom part of the cone. At that tilt, the intersection is no longer an ellipse, but instead a parabola.

So it's reasonable to say that a parabola is a limit of ellipses.

Of course, if you tilt a tiny bit more, you start getting hyperbolas, so it's also reasonable to say a parabola is a limit of hyperbolas.

share|improve this answer

As other answers show it makes perfectly sense to consider a parabola as limiting member of a family of ellipses. One might also call it a conic section touching the line at infinity, and I'm sure that other visualizations are possible.

But I don't think that you can say that in the limit "one of the foci is at infinity". We have to face the fact that in the limit one of the foci has disappeared once and for all. Foci belong strictly to euclidean geometry and to "finite" ellipses and hyperbolas in the euclidean plane. Already affine mappings destroy their distinguished character, let alone projective transformations coming into play when we talk about points at infinity.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.