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I watched the video review and I thought it was very well done.
Although I'm fairly new strobes and modifiers, my background is in mathematics and I can explain a bit of the theory behind the parabolic umbrella. Hopefully this will also illustrate how certain designs and techniques may prove to be more effective than others.
The parabola has the property that there is a point (called the focus) where any line/light ray originating from the focus and reflecting off the parabola will result in a ray that travels parallel to the parabola's axis. This property does not depend on the direction the light emanates from the source.
In real-world applications such as this umbrella, the surface is three-dimensional and is more accurately described as a paraboloid. The axis is the shaft of the umbrella, the vertex is the umbrella's tip (where the shaft meets the fabric), and the focus is precisely where you should place the strobe. If you place the strobe too close or too far, or off axis, you will lose the sharp falloff property that is the purpose of the umbrella.
It is also important that the umbrella's shape is reasonably accurate. The Elinchrom octa doesn't seem particularly precise in this regard--I've seen the Broncolor ones and they look very precise in shape (but the price is obscene IMO). So my thinking is that more ribs = more precise light output. Also, the PLM does seem "dishlike" but this only affects one thing: the shallower the paraboloid, the farther away the focus and therefore the narrower the angle of captured light. So the Broncolor parabolics, for example, have the bulb deeply recessed within the umbrella, which should translate to much greater flash efficiency.
I hope this helps. As I mentioned, I don't know a whole lot about studio flash, but I find it interesting that photographers have found useful applications for mathematical concepts.
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