Producing Concave Surfaces With A Vertical Mill

I can't believe I'm wading into this, but here goes.

There is no Parabola. There is no part of a parabola. There is nothing parabolic about any of it.

With two rotating axes, there can only be circular (spherical) and ellipse (ellipsoidal) surfaces produced. There cannot even be radii produced larger than the sweep of the cutter. The cannot be any hyberbolics.

What is produced is an ellipsoidal (swept 3D ellipse) that can be used to *approximate* a spherical surface. Close enough for even aircraft work (that is typically .03 at best. Careful probing on your local CMM will reveal this. Yes, I've seen the charts and "formulas", but the geometry is not up for opinion.

This is because all of these fall into a class of curve known as 'conics'. Someone posted a cone that had been cut up. That's why the cone matters.

Analytically, these curves all follow what is known as the "2nd degree general equation, or <deep breath> Ax^2+By^2 +Cxy+Dx+Ey+F=0
So when you have, say, A,B,and F coefficients, you have a circle. (where F = radius^2).
Mathematically, without a distinct Cxy component (and a particular proportion at that) no parabola can exist. If you're working with circles, like rotating things are, no Cxy component enters into the equation. Even if they're eccentrically rotated, the resultant intersection is still just elliptical.
So, returning to the cone, we can see that when the cone is sliced exactly perpendicular to the axis, a circle results. Only then. When sliced exactly perpendicular to the base, a parabola results. Only then. All the infinite other slicings that are not normal to base or axis are ellipse and hyperbola.
I got into this both as an Engineer and as a Lofter before that. I lofted portions of some small planes you've heard of. Everything subsonic, including airfoils and propellers, are lofted to 2nd degree equations or combinations thereof. As an Engineer, i investigated 'best-fit conics' for surveillance algorithms. (There are no true circles in nature, so finding one with your UAV means you found something manmade... like a truck.)
In defense of anyone i might have offended, the word "parabolic" is like "aircraft aluminum" or "surgical steel" and is thrown around with reckless abandon.. often by folks that really ought to know better but choose to go with the flow instead of correcting a roomful of people (BT;DT). So it's no one's shortcoming if they haven't ventured deep into GeekLand to sort all this out.
Also, very often the differences between these geometric distinctions are microscopic. Indeed, unless your cutter is *perfect*, which none can be, you're cutting an ellipse, anyway.
All i'm offering is analytical theory. Not how to cut stuff.

Wrat
Thank you so much for finally putting forth a proper explanation of this procedure.
 
I know this is an old post but I just joined and thought I would share a mathematical proof for anyone interested, please see attachment.
 

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So does mine.
I was really hoping a simple solution would be forthcoming to get the real parabolic surface.
I will stick to sweeping my pendulum grinder across the rotating glass to get my spherical concave surface.
 
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