# Behaviour of foci from ellipse to parabola

This image shows a progression from a circle, with eccentricity $$e = 0$$, to a hyperbola, with eccentricity $$e > 1$$.

A circle can be considered as having two coincident foci; an ellipse has two distinct foci; a parabola has a single focus; a hyperbola has two distinct foci again.

As eccentricity increases from $$0$$ to $$e < 1$$, foci seem to move away from each other. But what does it happen in the limit case, when $$e$$ equals $$1$$? Where does the "second" focus move?

Unlike the other conic sections, the parabola has a single focus. It is obvious when observing the intersection of a cone with a plane in space. But when dealing with eccentricity, considering the parabola as a limit case of the ellipse, it is not so obvious.

Consider this definition of eccentricity:

$$e = \frac{c}{a}$$

where $$c$$ is the distance between the center of an ellipse and either of its two foci; $$a$$ is semimajor axis. The limit case $$e = 1$$ implies $$a = c$$: foci should somewhat be placed on the border of the ellipse. But this is not helpful.

• Related (duplicate?): "A parabola is an ellipse with one focus at infinity" – Blue Sep 18 '19 at 20:32
• Note that all the conic sections in the figure appear to be defined by the given point and line as focus and directrix, except the circle. The actual limiting case for $e=0$ is a "circle of radius $0$" around $F$; that is, the section defined by focus $F$ and the given directrix when $e=0$ consists of just one point, $F$ itself. – David K Sep 23 '19 at 11:58
• Also notice that in the figure you linked to, the second focus is never indicated. That's because these figures (except the circle) are constructed by the focus-directrix property, which relies on one focus and a straight line called the directrix. The other focus is completely irrelevant in this definition. – David K Sep 23 '19 at 12:02
• A unifying feature of all conic sections is that they can be formed by the intersections of planes with cones. (That's why ajotaxe's answer is such a good explanation.) All definitions involving words such as "focus" and "directrix" are derived from this, and none of them (as far as I know) applies in a uniform way over all conic sections, except possibly if you work in projective geometry (in which every conic section is a closed loop and a parabola has two foci). – David K Sep 23 '19 at 12:11

An ellipse is the intersection of a plane and a cone, provided that the plane intersects every generatrix, and this happens when the acute angle between the plane and the axis of the cone is greater that the angle between a generatrix and the axis.

A parabola is the intersection of a plane and a cone when those angles are equal.

The foci are the contact points between the plane and the spheres tangent to the cone and the plane.

From the ellipse configuration, if you rotate the plane to make the angle smaller, the focus corresponding to the sphere that is towards the "open" part of the cone goes farther and farther from the another, so...

We could say, informally, that this focus goes to infinity when the ellipse becomes a parabola.

Maybe a physicist could explain it with an orbit around a star.

A more formal approach:

For an ellipse, we know that $$a^2=b^2+c^2$$. Dividing this by $$a$$ we get $$1=\frac{b^2}{c^2}+e^2$$ So if $$e$$ is near to $$1$$, then the fraction is 'small'. If you fix $$b$$, that means that $$c$$ is 'large', and $$c$$ is precisely the distance between the foci. Well, half of it.