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Are there any Mathematicians or engineers out there?
I can picture the more flat domed Napoli style oven, with higher vertical sides and a flatter dome center as being a parabola, where the focal point is the top of the the course of full-height brick soldiers. But I don't think that is the norm. Aren't most peoples ovens more of a sphere, where the inward curve starts immediately after the first course of the shorter (half high) vertical bricks.
For fun, I got this from a math web site:
A parabola (plural "parabolas"; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by , where a is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.
I'm an engineer. Usually, as your diagram suggests, parabolic shapes are used to focus energy coming straight in upon a single point, as is done in a satellite dish to focus the energy from the sattelite spread over the diameter of the dish upon the antenna at the focus.
Placing the fire at the focus of a parabola would do the opposite: spread the energy evenly over the oven floor. In practice, I am guessing that in a closed chamber heat bounces around from surface to surface anyway. Any surface that heats up more than the others will radiate more, which will tend to balance things out. So maybe it really doesn't matter at all, or maybe it's best to stay constant radius so that, during cooking, heat off the floor doesn't get too focused back on one spot in the middle. I think I'll go constant radius.
Possibly, an added advantage of a dome is that there is no straight shot from the back wall out the door. In a barrel shaped oven, the portion of the back wall that's aligned with the open door will radiate heat straight out the door. It's hard to say whether this is a meaningful difference.
The oven is parabolic in shape just like the domes of the cathedrals in Europe. A hemisphere wouldn't work as well for the heating reasons noted above but also because they are not generally self-supporting. The parabolic dome focuses the force vectors from outwardly splaying down into the slight vertical bottom walls.
They also help direct the airflow better in a chimney-less oven as the air sweeps in low, up the back & sides and washes over (under?) the dome (assuming the proportions are correct) before exiting the upper 1/3rd of the doorway.