Alright I need to get this prepared for the MATH ARRAY of proofs, that the Ellipse is a Cylinder section, and that the Conic section is an oval, never an ellipse
PROOF that Cylinder Section is an Ellipse, never a Oval:: I would have proven it by Symmetry. Where I indulge the reader to place a circle inside the cylinder and have it mounted on a swivel, a tiny rod fastened to the circle so that you can pivot and rotate the circle. Then my proof argument would be to say--when the circle plate is parallel with base, it is a circle but rotate it slightly in the cylinder or cone and determine what figure is produced. When rotated at diameter, the extra area added to the upper portion equals the extra area added to bottom portion, hence a ellipse. QED
Now for proof that the Conic section cannot be an ellipse but an oval, I again would apply the same proof argument by symmetry.
Proof:: Take a cone in general, and build a circle that rotates on a axis. Rotate the circle just a tiny bit for it is bound to get stuck on the slanted walls of the cone upward. Rotate as far as you possible can. Now filling in the area upwards is far smaller than filling in the area downwards. Hence, only 1 axis of symmetry, not 2 axes of symmetry, thus a oval, never an ellipse. QED
The above two proofs are Synthetic Geometry proofs, which means they need no numbers, just some concepts and axioms to make the proof work. .
Now, further I should have a Analytic Geometry Proofs of these two, that uses numbers, or at least coordinate points. . .
Now I bet in Old Math, they could not allow such proofs as above, allow them to be accepted as proofs, partially because people in Old Math were too severely toilet trained when a toddler, and secondly, none in Old Math has an amount of logic that a apricot has logic.
In Old Math, the airheads always insist on you doing "numbers" petty numbers here and there, even when numbers are totally unnecessary. And the reason they want you to do numbers in every proof, is not for the sake of a valid proof, but only because in Old Math, losers want to inflict pain on those doing math, not understanding and not logic, but just pain because they are losers of math themselves and the only thing that math affords losers of math, is to inflict pain on those getting ahead in math.