Brian,
couple ways to go here. The "plastic" is really plexiglass mirrors or you can get Lexan in the mirror also.
I do not jig saw any plexi or Lexan. Instead I make a jig and use a router ending with a very smooth cut. Plexi will chip !
You can buy this material in 1/8 and up thicknesses. Been awhile since I have ordered mirrored plexi from my supplier but seems to me it was around 150.00 for a 1/4 by 4X8 sheet.
The properties :
Plexi is durable and can be washed/cleaned easily. If fastening by screw, make the hole larger than the actual screw threads. If not, it will spider crack. To apply without screws, a window shop can get you the heavy duty sticky film to cover the back side of the plexi, then use a mirror mastic to hold in place.
You can sand and torch the edges of plexi for a glossy finish or sand then buff the edges. You can route the edges with a round over bit and get a nice finish that was as well.
Lexan will not break. bend it, cut it, beat it with a hammer and it will not break. But Lexan has more water in it, and therefore is softer. It will and can scratch terrible, especially the mirror finish even with a wet and very soft paper towel.
It can be drilled and attached to the ceilings the same way as Plexi.
Again, like veneer, make pattern using rosin paper, and since there is no forgiveness on plexi or lexan, make the first cut with some door skin or Luan. Get it to fit right, then use that as your pattern to cut/route the actual piece.
If attaching with mastic, practice the test fit a couple times first. Then adhere it to the ceiling.
I use button caps when using screw to fasten to the ceiling. Those little plastic caps that cover the screw heads.