Hi there!
So this is basically what we call prototyping. This is huge job most of professional upholsterers struggle with at the beginning. I don't think there is a book for it but i'll have a look.
If someone bring me a modern rounded design chair like the one one your picture and i don't have the old fabric i would:
1- I would go for a stretchy fabric if the shape is serioulsy rounded
2- I would take the new fabric, cut all my pieces a little large, and just pin it on the seat allaround like i wanted to handsew it but wit the extra fabric up. I would put a pin every 5 cm on every sides on the exact line that need to be sewn.
3 - I would CUT the fabric 1,5 cm away all around my pins lines.
4- I would put everything off to SEW it with a 1,5 cm sewing allowance
Off course I would go for a "TEST FABRIC" if I wasn't 100% sure to nail it first go.
But don't use a white calico for it, you need something with the exact same stretch than your fabric if you want to get you pattern with a test fabric first.
ALSO: Ask yourself before cutting and sewing everything if you will be able attach it like a simple attached cover or if you have to handsew a part or maybe add a zip somewhere to put your cover on wihout tearing the fabric.
I hope it was usefull, I'll do a tutorial about this as soon as I have the opportunity 😉