1

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.

Jacob Manaker
  • 9,937
Audus
  • 383
  • 1
    What have you tried? – David G. Stork Apr 30 '22 at 05:54
  • 6
    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 '22 at 06:07
  • 1
    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 – Audus May 01 '22 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. – Audus May 01 '22 at 23:46

1 Answers1

3

The scale transformation of the (circular) cone becomes an elliptical cone.

enter image description here

The intersections are merely scaled versions of the quadratic intersections in the base case, i.e., circles, ellipses, hyperbolas, parabolas, ...

enter image description here

David G. Stork
  • 29,774