Joining curved seams

Sounds like you want to do it with a router. If so, an inlay bushing set will get it done with a perfect fit.
Here's an example of a project with a very curvy joint done that way

This tutorial covers the whole project. There are lots of photos but the router cutting stuff is about in the middle.

Much appreciated shipwright, I mentioned doing it with a router because I couldn't think of another way to do it. I do usually try to use my hand tools as much as possible but couldn't think of a reliable way to do it. What are some other methods?

In one of James Krenov's books he describes doing it with
hand planes. He found it tricky.

How would you do an concave curve with a hand plane?

You'd make special planes.

Ah, and thus the trickiness of it I assume.

The Krenov description mentioned by Loren is in The Impractical Cabinetmaker. You could do it with special hand-made planes or use an adjustable curved-bottom metal plane. The trickiness is less the fact that you need special planes and more in simply getting the two arcs to match up. Even though I dislike routers and look for alternative ways to do just about everything, I'd hesitate trying to do this by hand.

Krenov: "I planed them together with the help of one of my curved planes. And I tell you, I'm not at all sure I want to do that again! It took about a day and a half… I'd get it almost right, and then maybe take just one little stroke of the plane too much somewhere and spoil it, and then I'd have to start all over again." (p.75 in my copy)

It's using compass planes to get matched glue joint that
is likely to be tricky. I think he already had the planes.
They aren't that hard to make. You can buy an
adjustable compass plane - I've used a couple and
never got the hang of making a smooth cut with one.

You can also get wood compass planes from Lee Valley,
ebay, the Japan Woodworker.

Yea, I believe this will be a situation where I use my router in then name of making something and not using tools.

The other way is to stack cut on a scroll saw (or band saw for less dramatic curves) in double bevel style.
Again, the fit will be perfect. This blog describes the theory (although in veneer stock). The same idea works in thicker stock but the saw angle will be much closer to square.