Hey Alan,
Having, not to long ago, made a
white oak door, I suggest that you plan to replace your door frame with white oak as well. I made the mistake of making the door and hanging it in the existing frame. The door was so heavy (I used ball bearing hinges) that the frame bowed under the weight, and I had to eventually rebuild it out of white oak as well. Unless you really need the security of a heavy door, I suggest a lighter wood, perhaps sapele?
To avoid warping, you want the straightest, quarter sawn pieces that you can buy; with the least runout possible. Again, in my region, a tropical hardwood like sapele is going to be most likely to fit that bill, but your choices might be different.
Framing up the round section, I would use floating tenons for the best strength, and I would use as many sections as I could and still have it look good, to avoid warping. Same for the frame as well.
Lastly, please be sure to use tempered glass in the door. It is expensive, but it is dangerous, foolish, and probably illegal to use anything else in a door. If you can get it, laminated safety glass would be even better. I am also quite sure that your home insurance company will not cover you if you don't, and someone is hurt by a shattered glass door. Your glass shop will be able to advise you on how thick the glass should be.
Don't forget to take pictures!
Cheers,
Tom
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"There is no path to peace, peace is the path."
Mohandas K. Ghandi