Questions on the relation of the axis of a cone to its conic sections

Question

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below:

Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? Also for the parabola? If not, what is the significance of this point where the conic section intersects the axis of the cone? Does the diameter of the base of the cone pass through the focus of the hyperbola?

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(2) It is easy to see that the axis of the cone passes through the center of the circle, and that a plane passing through the vertex of a double cone would pass through the center point of a hyperbola. But how does one compute the center of the ellipse from this picture?

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(3) The center of a conic section is equidistant from its foci. One can think of a circle as having one focus at its center, where an ellipse's foci would coincide if its major and minor axes were equal in length. But a parabola also has one focus, so what is the "center" of a parabola?

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I am guessing all of these have something to do with Dandelin spheres as described here, but not sure that this answers my questions or not.

Answer 1

Converting comments to answer, as requested:

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The Dandelin spheres answer question (1): a focus of a conic section is the point of tangency of its plane with one of those spheres. Clearly, the point on tangency lies on the cone axis if and only if the plane is perpendicular to that axis; therefore, the axis contains a focus in, and only in, the case of a circle. The axis point has some relation to the conic section, but not one as interesting or useful as a focus. (As for "Does the diameter of the base of the cone pass through the focus of the hyperbola?": note that the cone has no base. The thing extends, and expands, infinitely-far.)

This notion of points seeming to move "out to infinity and come back on the other side" is not uncommon in analytical geometry. (It's kinda what happens along asymptotes, when you think about it.) You might be interested in studying Projective Geometry, which adds a "line at infinity" to the standard Euclidean plane; this broader context helps unify all the conic sections into a single kind of curve that has various different relations to that line. (The parabola, in particular, has its "second vertex" on that line.)

Answer 2

I have made some computations with the cone $C$ of equation $$z^2 = x^2 + y^2$$ and the plane $\Pi$ through the point $P = (0,0,1)$ generated by the vectors $v=(1,0,0)$ and $w=(0,\frac{4}{5},\frac{3}{5})$. Namely, $\Pi$ is given parametrically as $P + s v + t w$, where $s,t$ are the parameters.

So if your claim (1) is true then the point $P$ must be a foci of the intersection of $\Pi$ and $C$. I did all the computations and if I do not made mistakes this do not happen. Here are the computations:

Points of the plane are $(x,y,z) = (0,0,1) + s(1,0,0) + t(0,\frac{4}{5},\frac{3}{5}) = (s,t\frac{4}{5},1+t\frac{3}{5})$ for any $s,t \in \mathbb{R}$. So such a point is in the cone $C$ iff $$(1+t\frac{3}{5})^2 = s^2 + (t\frac{4}{5})^2 $$ Now by completing the square the above equation becomes $$1 = \frac{s^2}{\frac{16}{7}} + \frac{(t - \frac{15}{7})^2}{(\frac{20}{7})^2}$$ showing that the conic section is a ellipse centered at the point $Q$ of $\Pi$ of coordinates $(s,t) = (0,\frac{15}{7})$ i.e. $Q = (0, \frac{12}{7},\frac{16}{7})$. The foci are the points whose coordinates $(s,t)$ are $(0,c)$ where $$c = \frac{15}{7} \pm \sqrt{ (\frac{20}{7})^2 - \frac{16}{7}} = \frac{15 \pm 12 \sqrt{2}}{7}$$ So no foci has coordinates $(0,0)$ showing that indeed the point $P$ is not a foci of the ellipse.

Conceptually, it seems to me that your question (1) should be equivalent to the claim that foci are invariant by projective transformatons of the conic. But this is not true as you can check by rescaling the axis $(x,y)$ of a ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. The rescaling is the map $(x,y) \to (\alpha x, \beta y)$, where $\alpha,\beta > 0$ which is of course a projective map (but writen in affine coordinates $x,y$).