# What shapes would an elliptical conic section produce?

If the cone(s) we slice with a plane to produce the various conic section shapes (ellipses, circles, parabolas, hyperbolas, lines) were not cones but instead, cone like shapes that had an elliptical base instead of a circular one, what shapes would be produced?

(By an elliptical cone I mean one that would produce an ellipse on a plane positioned perpendicular to the midline of the conic instead of a circle.)

I'm an artist, and I've sketched some pictures to try to visualize the intersections. It looks like I can get the same set of conics, but I can't verify that my drawings are in proper perspective.

• What have you tried? Apr 30 at 5:54
• You still get "regular" conics. As with a circular cone, and elliptical cone corresponds to a degree-$2$ polynomial equation in $x$, $y$, $z$; a plane is defined by a degree-$1$ equation. Therefore, their intersection is then a plane curve of degree $2\cdot 1=2$ (relative to a $2$D coordinate system on the plane), and we *know* that all of those are our familiar conics. (BTW: The same argument works when the cone is replaced by *any* surface determined by a degree-$2$ polynomial. Such surfaces are called "quadrics".)
– Blue
Apr 30 at 6:07
• I am coming to the mathematics community as more of an artist than a mathematician. I am curious about this because I have wondered what shapes ellipses would make in a perspective drawing (imagine trying to draw a wall with ellipses inscribed into it). What I have tried is just drawing the shapes but don't actually know how to validate if my shapes are drawn in correct perspective, which is why I need the help of the mathematics community. @DavidG.Stork May 1 at 23:45
• Thank you @Blue, it sounds like this could potentially be an answer if you wanted to post it. I did not know about quadrics until now. May 1 at 23:46