i like this would like to it finish block to get a better idea

http://hyena-in-petticoats.blogspot....-tutorial.html
This tutorial uses the method you speak of (with a few variations). I have used it, quilted my windows and there is no way any fabric is going to escape. (Unless my window fabric should fray.)

dsb - This method has the 1/4 inch turn up and press at the very beginning. I'm wondering if I could get away without this step.

I just read the tutorial by hyena in petticoats. It was so amazing. Great pictures and instructions. A great aid for anyone wanting to try this pattern.

you would have to be so super-careful at the corners not to let any threads fray that imo it would be faster to press in that .25" fold-over. at least for me it would be. if you decide to test it, let us know. if it works, it would be a great time-saver.

Can someone tell me where the curved parts come from?

No, I do not think it will work without turning the corner tips in so they are under your 1/4" iron down on the sides.
When you roll the edges of the square's sides the tip will pop out at you if it isn't secured under the side iron downs. Remember we are rolling the sides to the center and holding the side over the top of your 'window' fabric.
I have one on the machine now but I am using the sewing method rather than the ironing method. Eddie's tutorial: http://www.quiltingboard.com/t-35742-1.htm
I really don't know if one method is faster or better than the other.
I am hoping to have enough fabric for a California King. I will need 780 squares. The first photo is the one on the machine now but I took the windows out. They were not speaking to me. I will know when I find just the right fabric.
The other is the test squares I made when I was learning the pattern. Both give you a good view of the tips and why they need to be secured under.

The curve is coming from the sides of the flaps that are shown in the last photo(s).

You're right. Since I hadn't actually stitched anything in those pictures I didn't see that. Which is why I posted on the site. So now I need to decide which method is quicker/better; Eddie's way as done on the tutorial or the pressing method I learned years ago.

BTW - your quilt looks good so far.

i think that turning up the corners is the important part. that's the part that's likely to make problems with loose threads. but if you have to turn up the corners anyway, what's the point of not turning in the .25" all the way around? also, i think that the double edge, after the pressing, gives it some substance that gets lost without it. i would still like to see one made this way, especially a close-up at the corners.