I wouldn't redraw each segment. The best way to do it is to set each of the circles down and then perform Pathfinder knockouts and merges. It's a good practice job to learn how those tools work.
Approach it as if you are a draftsperson using a compass with a pencil and and eraser.
rather obvious, I'd think... but just in case...
...rotate it first, then lock it, then set some horizontal and vertical guidelines... then you should be able to draw a pretty quick oval... then you should be able to use repeats of that in many (or all?) cases...
then do as Fred suggested!
send it to www.24hourartwork.com
Great work, $12.00-$15.00