This question might fall in a gray area between here and the DIY Stack Exchange site but I'm attaching it to cabinets so I'll post it here.
How do you cope out the profile for a crown molding joint when the crown molding sticks out considerably like the one below? (To add to the complexity, this kind of molding has a hollow where the back would be.)
I haven't installed crown molding myself but it sounds like coping the joints is an alternative to cutting bevels on both pieces of molding on a miter saw. Coping can give you a better fit than a miter joint because walls are often textured and aren't always perfectly square to one another, making for a sloppy fit.
Family Handyman outlines a process whereby you butt the first piece all the way into the corner. After that you cut an initial bevel on the second piece using your miter saw to reveal the profile, then use a coping saw to clear out the profile and fine-tune the fit with a file and/or sandpaper.
There are a great many websites and YouTube videos that have tutorials on how to install crown molding. I've looked at a lot of them and tried to implement the skills in my own house. Here is my very favorite article along with a series of videos [youtube]
It's tricky because even the slightest irregularities in your joints are glaringly obvious. As a result, I used a lot of caulk. My brother-in-law is a former home builder and he assured me that caulk is our friend in this regard. Still, I observed a few things that are not obvious from tutorials.
It is pretty important to measure the angle of your walls (or cabinets in this case) since if they are not square, you can at least use one of the online charts and set your miter saw to the appropriate angles to match that of your walls.
If you do not have a powered miter saw (some call it a chop saw), then you will need to use a coping saw to cut the inside miters. This again is a bit tricky because the slightest mistake is visible. I found that my coping saw was of a pretty low quality and so bought one made by Olson. It worked considerably better. Realistically, it was the blades that worked better; they seemed sharper.
Any little irregularities in your wall will cause the molding to sit proud of the surface and cause gaps (edit: I should point out that my walls are textured - smooth walls would not have this problem as much). The best way I found of dealing with this was to go over the wall where the molding will sit with a putty knife to make sure it was smooth. On cabinets, this will be less of an issue. Then I placed the molding up and nailed it in the very center. As I worked from the center to the miters in the corners, I could "roll" or "twist" the molding so that the miters would match up in the corners. Basically, I was correcting any inaccuracies as I worked out from the center of the molding.
As I read back over this, it seems a poor explanation of what I was doing but I can't think of how to improve it. It is a very tactile thing and would be easy to demonstrate but is hard to describe.
Precision and accurate marking and measuring are key.
The most expensive option is commissioning a router bit to make that cut in one pass.
However You can do this with multiple passes using a tilted router and a variety of bits. For some straight corners a few passes over a lowered table saw will do.
