Greg Guarino wrote:
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"?

You might rethink how your rails and stiles intersect. Look at this.
Typically rails, outer rails fit between stiles. This does not address the
inner stiles but I think you cab rethink a bit and come up with a simpler
solution.

https://flic.kr/p/e5u1yW

Anyway if you stay with this design exactly, cut the top rail first. Cut
the stiles next and long. Clamp stiles precisely where you want them,
under the top rail. With a top bearing flush cut bit in your router use
the clamped top rail as the guide to cut a groove on the face of the
stile.
Once you have all stiles grooved, remove and cut off the excess, close to
the line you want to keep. Now use that edge again to guide the bit to
remove the remainder. Make several passes.