< The figure is said to be ideal - parabolic.
I was trying to derive it to see if it is a perfect parabola even for large r at fast rotational speeds/large y:
At any point r from the axis of rotation the acceleration of the liquid is equal to r times the square of the angular velocity. The pressure at any point r will be the integral of the fluid density times the acceleration with respect to r from r = 0 to r = r so y ~ r^2.
This should hold no matter the plate shape. To cut down on the mercury and weight, however, the plate itself should be an almost perfect parabola.
Maybe they could build one several hundred feet in diameter and eliminate the bearings by using the earth for rotation. The location would be limited -- only the N. or S. poles -- and the secondary mirrors would be in low earth orbit. They might be able to get away with a less reflective but cheaper more environmental friendly liquid than mercury with tens of thousands of square feet of surface area.
Right now they seem to prefer to use two or more primary mirrors to collect more light. I'm not even sure why they need to locate them on the same mountain top. It's not like a few hundred or few thousand miles will make any difference with astronomical distances.
Astronomers do some much computer enhancing it seems like they could just saw a solid mirror up into pieces and reassemble it after they haul the pieces up the mountain. Even if they did a sloppy job they could just point it at something they knew very well like the moon, then adjust it with software.
I don't see why all this money is being spent on telescopes in the cheap info age. They need a curved shiny surface but that's all they need.