How can I cover my desk chair's seat? My desk chair has started peeling. I think the cat scratched it, and now more comes off after every seating. 

Here's what the seat shape looks like:



After looking into this a bit more, I realized that only the top &amp; front of the seat have this material (the four sections easily seen in the picture). The sides and back of the seat are vinyl. I don't know if this changes much.

I'd like to do something such as a fitted sheet, but I don't know how to go about measuring a pattern, especially if I plan on elastic edges. I'm also concerned about it sliding or becoming loose. The seat is concave. I don't want to sit on wrinkles. 

Is there a better solution than a fitted sheet? I don't want something I need to adjust all the time. 

I'm not asking for detailed instructions or total walk-throughs. I'm just not sure on what type of solution to use. Once I pick a solution, I'll ask specific questions if I get stuck.
You should be able to turn the chair over and unscrew the seat part. If the cover doesn't come off (probable), you'll need to trim the loose pieces and make patterns for each section. If you don't want wrinkles, you need to cut fabric out in the pattern of the original seat. You need to allow for seams and fold over the edge (measure the edge into the plate that screws the seat to the chair. An easier option without the pieces is to use a stretchy woven or a firm knit, but in time they will sag.
You can probably just use duct tape to attach the fabric under the seat plate.