Step 1: What you need
Any saw that can make round cuts (bandsaw, jigsaw or even a scroll saw).
A lathe (or a sander, but this will take very long!).
compass (for drawing circles)
Paper and pencil:-)
Step 2: Design phase
After sketching it up rouhly i decided for a minimal wall thickness (~10mm in my case) and drew the lines accordingly. Now I basically got a cross-section of the vase.
From this cross-section you can easily meassure the diameters/radiuses of all the layers.
Draw the the circles (or half circles) onto the board using a large compass (or a string + pencil, if you don't have one).
As I had only long and narrow boards i cut out half-circles and glued those together. If you are using a jigsaw and you have got wider boards you can of course cut out the whole circles.
The circles don't have to be ultra precise, bettter make them a little bit too big than too small. You can remove material quite easily on the lathe