How I learned conic sections in high school was as intersections of planes with a double cone. This definition always seemed arbitrary and contrived to me since the double cone isn't really a "natural" object, and arguably the conic sections seem more natural than the double cone.
Recently, I have heard that all conic sections are the same in projective space (I am almost at the point of proving it in Algebraic Geometry: A Problem Solving Approach). I also know that projective space can be modeled as a quotient space for lines going through the origin.
This animation on YouTube strongly suggests that the notion of lines extending from the origin in 3-space and the double cone are intimately related notions, although I am not sure how.
Question: Is the double cone (from which conic sections derive the name) just how people describe projective space to high schoolers without scaring them?