Apparently this is what's termed the Hencky-Campbell problem, and has been thought about for nearly a century. This paper from the USAF Research Laboratory at Kirtland AFB goes over the mathematics in eye-watering detail. The problem was also considered by NASA at around the same time. The answer seems to be that the curve is not a simple function but rather a power series that depends on factors including the elasticity of the membrane.
The practical application is using Mylar or similar reflective foil to make a cheap near-parabolic mirror.
UK amateur astronomer Maurice Gavin - who I knew back in the early 1980s as founder of an APA for swapping home computer programs for astronomy - did some experiments in this line. The results were optically awful but adequate for simple photometry.
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timeasmymeasure
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Date: 2011-12-21 12:09 pm (UTC)The practical application is using Mylar or similar reflective foil to make a cheap near-parabolic mirror.
UK amateur astronomer Maurice Gavin - who I knew back in the early 1980s as founder of an APA for swapping home computer programs for astronomy - did some experiments in this line. The results were optically awful but adequate for simple photometry.