As far as I can tell, there are about 3 basic techniques to making woodworking curves. The first is wood bending.... with wood bending you'd typically use a long pipe that you can run a steam hose too, put a piece of wood in the pipe to get steamed, and then after it's steamed for a while, you pull it out hot and force it into the shape you want with clamps and pre-made forms. The second way is to use laminate board, or at least really thin wood, and glue layer after layer together in the shape you want. The third way, and the way I'll describe here, is to use straight boards glued into the approximate shape you want, and then finish them off but cutting, typically with a router.
For all techniques, you really need to have a form you want to match, either the thing you clamp the wood to, or the form that guides your router. In my case, I built a custom jig for the router, and it was easy, as I was doing half circles about for the top grill work on the gazebo.
Each arch I built took 11 boards, 5 on the front glued to 6 on the back, and then I would use the router with a straight bit to cut them out. Here is a quick diagram, once again, as I am sans pictures for this part of the project.
The back is on the left, the front is in the middle, and the two of them stacked on one another is on the right. The only reason you need a double set like this is for strength of bond, as you glue and clamp the boards together before you use the router on them. I used pressure treated 1x6 stock for this project, with the goal of having a 1 1/2" wide arch when I was done... it was to match the width of a the 2x4 stock through the grillwork. I had to buy a $40- router bit for this project, I got a large chuck bit with a 2" depth, worked like a charm... but this project produces incredible amounts of sawdust... I think I filled my 50 gallon waste bucket 2 or 3 times.... if you have a shop vac you can attach to your router (many have vacuum attachments), get yourself a long hose and rig it up... it's worth it.
What I did to create a custom router jig was to take a piece of plywood the width of the router base, and the length I needed to cut the arch, and put a few holes in it to attach as a router base. Some routers will make this easy for you by having rigs so you can attach things easily to them, mine did not. You them put two holes at the end the diameter of the circle to be cut, one for the inside cut, amdn one for the outside cut. Don't forget to account for width of the bit, if it's a 1/2 inch wide, you'll need to shorten and extend those length by the 1/2 inch.
One mistake I made here was that I was too impatient.... when I get moving on a project I try not to let much slow me down, and the 1x6 PT stock I got from home depot was too "fresh", I should of let it dry a few week before attempting this project... the gluing and clamping wasn't that great, since the wood was too wet to let the glue absorb and dry properly, and it's also harder to cut. I tend to use what I buy immediately, especially when it's stock from home depot, as otherwise it tend to bow/warp/bend like crazy as it dries out. You won't usually see that if it's part of a deck say, as the fasteners will keep it in place and shape. Tho not always...
In case my words were unclear, he is a diagram with the router installed and doing a cut :)
When you cut with the router, typically you want to just shave off a bit, and make many passes, trying to cut the whole thing in one pass isn't likely to work unless it's for the thinest of boards. I learned the hard way the first time I used a router. Lucky I didn't get injured as well, it went pretty crazy. All that is left now is to talk about the additional wood work in the grill sections. Another mistake made there :)