I'm intrigued by this, and how it would work:- 3 sub-questions if I may:
Construction: As I understand it's a flexible sphere constrained by a rigid edge.
a. Do we simply glue 2 flat circular pieces of flexible material together at the edges and inflate? (e.g. the inflatable dish on PE1RAH) Or,
b. do you have to cut out sections to make it parabolic, and the inflation simply gives it the rigidity? (e.g. page 4 of this document)
Spherical Antenna, as manufactured by e.g. GATR. Would the design for this inflated dome with internal parabola be in effect 2 inflatable structures - 1 sphere (to give rigid circumference, and an internal parabola as in question 1, or could you do something like this with only 1 inflatable structure?
Design. I'm assuming the parabola generated using question 1.a would be a factor of gas pressure + stretch of fabric given there is no slack in the material? Or are other factors involved?
Any suggestions for design or theory papers would be most appreciated. I'm going to have a try at building one of these for portable wifi :)
EDIT: I just threw together a prototype from 2 plastic bags/drinking straw/tape - it's a bit rough to see a definitive answer, but looks parabolic.
EDIT 2: on further digging, I confirmed that the inner parabola in a spherical antenna holds its shape by continually topping up the pressure in the 'top' of the sphere to keep the parabolic shape bowed the right way.