Alright, if our knife in Conic and Cylinder sectioning was always the normal knife then the Conic section is always a Oval, never an Ellipse. The Cylinder section is always a ellipse, never an oval.
So, that is true and I proved it.
But, today, what we want to do, is prove that -- how in the world can we get a ellipse out of a Conic Sectioning. For I want to stop saying "never an ellipse".
Alright, we know the Conic Sectioning is two cones shaped like this
\ / \ / \/ /\ / \ / \
Now, we do a Descent of one cone into the other cone. It can be either the top cone downwards into the other cone or the bottom cone upwards. And the descent of one into the other follows the cone axis, so it is a symmetrical descent and keeps the angle constant. And now, the knife that cuts is shaped like a V or ^. What this generates is what can be called two parabolas with a dihedral angle (discussed last night in association with polyhedra-- but has given me this idea of ellipse generation from conics).
Now, we have a V shaped Conic Section, and now, imagine a iron board where we iron out that dihedral angle of the V conic with a parabola for the / and another identical parabola for the \.
We iron that out and produce a Ellipse from a conic. Now, of course, that hinges on the idea, if true, that you can take any Ellipse, fold it at its center line of minor-axis (not sure if the major axis centerline has two parabolas using the center as the focus-- not sure, but suspect so.
Now, to get a Oval out of the Cylinder section, we do the same process, without a descent, but rather with a nonsymmetrical V knife. What is that? It is a V knife where you take one side and the angle and cylinder axis and make different.
make one angle different from other
Now, in the proof, I have to show there is no "seam". That we can take a ellipse, cause a dihedral angle to appear as a folded ellipse, and then show that each half is a parabola. In the Oval from cylinder section here again I need to prove no seam is involved.