If you wanted to build a face frame whose top member was arched, like this:

https://www.flickr.com/photos/gdguar...in/dateposted/

.... how would you form the tops of the stiles? My sense of geometry
tells me that over a 2.5" width, an arc with a radius of 36 feet would
deviate from a straight line by a very small amount. (the height of the
deviation in the middle of the arch is 1") So would you simply cut the
tops of the stiles at the proper angles, leaving the cut straight, or
would you somehow form a very slight curve to mate with the arched rail?

Now the Sketchup question:

I drew the bottom and top rails first, including the arch. Then I drew
the stiles in place, butting up to the bottom rail properly, but with
excess length at the top. I opened a stile for editing, selected the
front face and tried to "intersect faces with model". Sketchup told me
there was no intersection. I think I've had this happen before. If two
surfaces are in the same plane, even if they obviously overlap, Sketchup
does not consider that "intersection".

As a workaround, I moved the top rail forward by half the thickness of
the "wood" and then did the "Intersect Faces" procedure on the front
face of the stile. This time it drew the line I was looking for on the
front face of the stile. I used the "pull" tool to "trim" the stile to
mate with the arch. Then I moved the top rail back to its intended
position.

Is this simply how Sketchup works? Or is there some other way to make
parts whose faces are in the same plane "mate"?