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A chain suspended on two points hanging under gravity forms a curve called a Catenary, which can look pretty similar to a parabola. I'm not sure if this generalizes to the two dimensional case where an inextensible, flexible cloth is hanged from and within a circular rim, but assuming it does:

If that material were made to be reflective could it be used (possibly in combination with a secondary mirror (of which shape?)) as a telescope?

After all, non-parabolic mirrors seem to be frequently used as primary mirrors (and corrected by some secondary mirror).

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    similar: https://www.researchgate.net/publication/279514418_Shape_control_of_slack_space_reflectors_using_modulated_solar_pressure – planetmaker Aug 22 '22 at 09:26
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    Interesting question! An initially flat fabric or film can't form a axially symmetric concave shape without elastic deformation, and the way woven fabrics from fibers, and continuous films will deform will e different. I'll bet there are engineering equations out there for the resulting shapes for "plates" deformed under gravity, but those will be small deviations, not "deep dish" shapes. Hmm... – uhoh Aug 22 '22 at 13:44
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    How would you deal with seams? You can’t smoothly wrap a flat sheet (think paper map) around a hemisphere without making wrinkles or needing to cut and stitch, like in cartography. – Ed V Aug 22 '22 at 16:57
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    @uhoh yes! bubble films FTW. Not that I see that as a simple or stable mechanism. Maybe some material which forms a bubble and then hardens to shape? The problem with common bubbles (soap bubbles eg) is that the surface density is nonuniform as fluid "drains" to the lowest point. We'll need a long-chain interlocked poly-something molecule! – Carl Witthoft Aug 22 '22 at 17:27
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    Related: https://mathoverflow.net/q/69817 – PM 2Ring Aug 23 '22 at 03:30
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    @PM2Ring excellent find! – uhoh Aug 23 '22 at 06:32
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    As Ed V & the mathoverflow post explain, your inextensible cloth isn't going to work. But FWIW, here's a rotated catenary. https://sagecell.sagemath.org/?z=eJxVkMFqwzAMhu-BvIPwLimY0qU7rAPDTrsNShnsGNxEbbwlkbGVrtnTz06yNfNBlu1f1vdLCAEHYs0IGsqwddoNoI_UMxj24Ie2RY5XV-PTBMJ6d4aDDo4D7F8hP5juDPkmz9ebx3W-TRMhRJqkybPpGJ0uOU0qPEGRefON6l6CbYgLS-HZq4eNhBZ9rd5cjxJKasgpcbetdtUOhQSyujQ8hLLY--R0i-pFNz5oK-0-5zKLzlss2Vz-XnXPVPS2Co6mq9XTRL8HBQ4v1PRsqCsizLbKSvK1hOwqIQBF0NV_zkUup39ghh3jDXTeZ1cxrOIwop5rbDF0F5FcgDmNFgADHIjGnGsWk9A6-ohuqIvqhbmxaHGea8lxTWenbW3K-Yv92tf0lU0DG6Oc-qsxykUPdUuDja4x3e_EfgB0QLAS&lang=sage – PM 2Ring Aug 23 '22 at 07:14
  • Does "inextensible" automatically exclude shear strains as well? My coffee hasn't kicked in yet, I haven't been able to wrap my rigid head conformally around this problem yet :-) – uhoh Aug 24 '22 at 01:44

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