It is well known that conic sections are the intersections of a plane with a cone. Let the cone be $z^2 = x^2 + y^2$ and the plane be $z = ex + b$. Project orthogonaly the resultant conic onto the $xy$ plane (projecting both up and down).
I conjecture that the projection of a non-degenerate conic (even a parabola or hyperbola) is an ellipse, but cannot prove this, or find a way to have Wolfram Alpha or similar display this visually.
- Is my conjecture correct?
- Is there a way to visualize this?
- What about degenerate conics? Clearly the point obtained for $b = 0$ projects to a point. What about other degenerate conics?
All you need to do is eliminate $z$ from the system cone-plane. Specifically, you get $$ (ex+b)^2=x^2+y^2 $$ or $$ (e^2-1)x^2+2bex-y^2+b^2=0 $$ Therefore, the type of the resulting conic-projection depends on $e$. If $|e|>1$ you get a hyperbola, if $|e|=1$, a parabola and, if $|e|<1$, an ellipse.
This is geometrically obvious. Just observe that when $|e|=1$, your plane is parallel to one of the generators of the cone.