Projection of cone plane intersection onto the plane perpendicular to cone axis.

Question

Throughout this post I'm assuming that the cone is pointing "up" in the $Z$ axis.

It's known that the intersection of a plane and cone is an ellipse. But what about the projection of this ellipse onto the $X,Y$ plane - is it always a circle? Or equivalently, does every intersection of a plane and cone lie in some cylinder pointing in the $Z$ direction? It is true for a plane and paraloboid intersection, but it doesn't seem like it for a cone plane intersection:

For a cone $z^2 = x^2 + y^2$ and a plane $z=\frac{x+1}{2}$ we have

$4 x^2 + 4 y^2 = 4z^2 = x^2 + 2x + 1$

$3x^2 -2x =-4y^2 + 1$

$x^2 - \frac{2}{3}x = \frac{-4}{3}y^2 + \frac{1}{3}$

$(x- \frac{1}{3})^2 - \frac{1}{9} = \frac{-4}{3}y^2 + \frac{1}{3}$

$(x- \frac{1}{3})^2 + \frac{4}{3}y^2 = \frac{4}{9}$

Which is not the equation for a circle.

Is there perhaps a cleaner way of showing that the projection of a plane and cone is not always a circle?

Answer 1

If an intersection curve’s projection is a circle, then, as you pointed out, this curve must lie on a vertical cylinder that passes through the circle. So, the intersection curve must be the intersection curve $C$ of the cone and this cylinder.

If $C$ is a planar section of the cone, then it must be planar, obviously. It’s highly unlikely that intersecting a vertical cone and a vertical cylinder will give a planar curve. You get a planar curve if the cone and cylinder have the same centerline, but rarely (if ever) otherwise.

In fact, the intersection of two quadric surfaces is planar if and only if they are both tangent to some third quadric. See this question.

Also, see this question.

Answer 2

Suppose we have an elliptical intersection curve $E_1$ whose projection is a circle. Now we change the slicing plane a little bit — we rotate it around the minor axis of $E_1$, making its angle steeper. This will give us a new elliptical intersection $E_2$ that’s more elongated than $E_1$. $E_2$ will have the same minor radius as $E_1$, but a longer major radius.

When we project $E_1$ and $E_2$ onto a horizontal plane, we again get two ellipses that have the same minor radius but different major radii. So, they can’t both be circles.

Edit:
The statement in italics above is incorrect. Rotating the slicing plane around the minor axis changes the minor radius of the ellipse, as the following picture clearly shows.